differential equation. The number of equations equals the number of states minus 1. The probability of being in the last state equals 1 minus the sum of the probabilities of being in the other states. Examples of systems analyzed with Markov analysis are load sharing systems, standby systems, degraded systems, and multistate systems (Ebeling 1997:108ff). For example, if one component fails in a load sharing system, the failure rates increase for the remaining components. A standby component may experience a low or zero failure rate in its standby state, and a higher failure rate when operational (which may or may not equal the failure rate of the originally operating component).

2 Another definition is that of memorylessness where, conditional on the present state of the system, its future and past are independent. See e.g. Taylor and Karlin (1998) for further definitions.

a stochastic process with the Markov property is that the conditional probability distribution of future states of the process depends only upon the present state.2 Markov analysis has proven highly successful applied to reliability analysis. This paper is concerned with two limitations of Markov analysis. First, enabling each of two players to choose strategies in each of two periods violates the Markov property since players are free to choose future strategies that are not conditioned on their present strategies. Generally, any theory involving intentional action (e.g. game theory) violates the Markov property. Second, this paper relaxes the constraint in Markov modeling where the transition rates between

This paper enables players to exert efforts to impact the system reliability as time progresses. That is, we analyze how players choose strategies through time to impact the reliability of dependent systems. We consider a dependent system consisting of a main system and a standby system. Both the main system and the standby system can be in two states which are to operate or fail. The dependent system is analyzed for general parameter values with backward induction as a two period game. We determine how two players make strategic decisions through time to impact the system reliability. Players allocate resources in the sense of substituting efforts across components and across time. Determining the nature of such substitutions is of substantial interest, see e.g. Enders and

The paper answers questions such as whether players exert high efforts in the first period to position themselves for the second period, whether they are so weakened that they withdraw from the game, or whether they prefer the game to last one period or two periods. One player, the defender, maximizes the system reliability. The other player, the attacker, minimizes the system reliability. Both the main system's and the standby system's reliabilities depend on the relative levels of defense and attack and on the contest intensities. Each player's utility depends additively on the system reliability in the two periods, with a discount parameter varying between 0 and 1 for the second period. The unit costs of defense and attack, and the contest intensities, are different for the main system and the standby system, analogously to failure rates being different in Markov analysis dependent on the

Hausken (2010) analyzes complex systems applying game theory. Hausken (2011) provides a game theoretic analysis of a two period dependent system of two components which can

For multi-state system reliability, see Lisnianski and Levitin (2003). For degraded systems see Ebeling's (1997:117ff) Markov analysis of a system which can be fully operational,

differential equation. The number of equations equals the number of states minus 1. The probability of being in the last state equals 1 minus the sum of the probabilities of being in the other states. Examples of systems analyzed with Markov analysis are load sharing systems, standby systems, degraded systems, and multistate systems (Ebeling 1997:108ff). For example, if one component fails in a load sharing system, the failure rates increase for the remaining components. A standby component may experience a low or zero failure rate in its standby state, and a higher failure rate when operational

2 Another definition is that of memorylessness where, conditional on the present state of the system, its

Sandler (2003), Hausken (2006), and Keohane and Zeckhauser (2003).

be fully operational, in two states of intermediate degradation, or fail.

(which may or may not equal the failure rate of the originally operating component).

future and past are independent. See e.g. Taylor and Karlin (1998) for further definitions.

different states are kept constant through time.

system state.

degraded, ot failed. See Zio and Podofillini (2003) for Monte Carlo simulation of the effects of different system performance levels on the importance of multi-state components. Ramirez-Marquez and Coit (2005) use Monte-Carlo simulation to approximate multi-state two-terminal reliability. A next step is to incorporate strategic defenders and attackers into the analysis of multi-state and degraded systems.

In earlier research Levitin (2007) considers the optimal element separation and protection in a complex multi-state series-parallel system, and suggests an algorithm for determining the expected damage caused by a strategic attacker. Hausken and Levitin (2009) present a minmax optimization algorithm. The defender minimizes the maximum damage the attacker can inflict thereafter. The defender has multiple defense strategies which involve separation and protection of system elements. The attacker also has multiple attack strategies against different groups of system elements. A universal generating function technique is applied for evaluating the losses caused by system performance reduction. Levitin and Hausken (2009) introduce three defensive measures, i.e. providing redundancy, protecting genuine elements and deploying false elements and analyze the optimal resource distribution among these measures in parallel and k-out of-N systems. Levitin (2009) considers optimizing defense strategies for complex multi-state systems.

Azaiez and Bier (2007) consider the optimal resource allocation for security in reliability systems. Bier et al. (2005) analyze the protection of series and parallel systems with components of different values. They specify optimal defenses against intentional threats to system reliability, focusing on the tradeoff between investment cost and security. Bier et al. (2006) assume that a defender allocates defense to a collection of locations while an attacker chooses a location to attack. Hausken (2008) considers defense and attack for series and parallel reliability systems. Dighe et al. (2009) consider secrecy in defensive allocations as a strategy for achieving more cost-effective attacker deterrence.

Section 2 presents the model. Section 3 solves the model. Section 4 analyzes three special cases. Section 5 simulates the solution. Section 6 considers examples. Section 7 concludes.

#### **2. The model**

Consider a main system and a standby system. A defender and an attacker play a two period game. In period *j*, *j*=1,2, the defender exerts effort *t*Mj at unit cost *c*M to defend the main system, where *t*Mj is the defender's free choice variable. Analogously, the attacker exerts effort *T*Mj at unit cost *C*M to attack the main system, where *T*Mj is the attacker's free choice variable. If the main system is successfully defended in period 1 it continues to operate in period 2, while the standby system continues to be on standby. If the main system is successfully attacked in period 1 it does not operate in period 2, and instead the standby system operates in period 2. This means that the defender exerts effort *t*S2 at unit cost cS to defend the standby system, and the attacker exerts effort *T*S2 at unit cost *C*S to attack the standby system. Defense and attack are interpreted broadly. Defense means protecting against the attack, and maintaining and adjusting the system to prevent that it breaks down. Attacking means attacking the system, which may get aided by natural factors (technology, weather, temperature, humidity, etc.) to ensure that the system breaks down. We assume that the standby system does not operate in period 1 but is located in a secure place (e.g. underground bunker) from which it can be accessed if it is needed in period 2. We do not

Game Theoretic Analysis of Standby Systems 81

time for the onset of period 2. This assumption is justified since repairing or replacing failed components can be complicated for economical and logistical reasons, and may require competence and time, which we assume is impossible both during the periods and in the transition from period 1 to period 2.3 Hence the strategies the players choose for period 1 have to account for the combinations of possibilities in which the main system may operate or fail in the two periods, and the standby system may operate or fail in period 2. If the main system fails in period 1, since it cannot be repaired or replaced before the commencement of period 2, the players need to assess their defense and attack in period 2 to account for which of the two states (operation or failure) follows after period 1. In period 2 the players also make their strategic choices simultaneously and independently, knowing the outcome and

If the main system operates after period 1 (state 1), then the unit costs of defense and attack remain unchanged and the players make strategic choices *t*M2 and *T*M2. If the main system fails in period 1 (state 2), we assume that the unit costs of defense and attack for the standby system are *c*S and *C*S, and the contest intensity is *m*S. The defender and attacker values of the standby system are *v*S and *V*S, where *v*S≤*v*M and *V*S≤*V*M. The

> and

(*V*M(1-*p*M2)-*C*M*T*M2)

(*V*S(1-*p*S2)-*C*S*T*S2)

are time

defender and attacker values after period 2 are shown in Table 2, where

State Main system Reliability Defender value Attacker value after period 1 after period 1 in period 2 in period 2

1 1 1 2 2 122

survive period 1. Fig. 1 shows the two-period game as an extensive form game tree.

*M M MM M M M MM M S S SS*

*u v p c t p v p c t p vp ct*

*v*M*p*M2*-c*M*t*M2)

1 11 2 2 1 2 2

( ) (1 )( ), (1 ) ( (1 ) ) (1 )( (1 ) )

(3)

*M M MM M M M MM M S S SS*

The third term on the right hand side in both utilities contains *p*M1 which is the probability that the main system survives period 1. The fourth and rightmost term on the right hand side in both utilities contains 1-*p*M1 which is the probability that the main system does not

The two players have two strategic choice variables *t*M1 and *T*M1 in period 1, and four strategic choice variables *t*M2,*T*M2, *t*S2, and *T*S2 in period 2. We analyze pure strategy Nash equilibria. We solve the game with backward induction starting with period 2. Differentiating gives *u t* / *M*2 = *u t* / *S*2 =0 and *U T* / *M*2 = *U T* / *S*2 =0. Solving the four

*U V p C T p V p C T p V p CT*

*v*S*p*S2*-c*S*t*S2)

choices in period 1.

discount parameters.

1 operates *p*M1

**3. Solving the model**

equations gives

2 fails 1- *p*M1

Table 2. Defender and attacker values after period 2.

3 Future research may model the repair of the main system.

We thus express the players' utilities over the two periods as

model efforts the defender and attacker may exert with respect to the standby system in period 1. In both periods both players make their strategic choices simultaneously and independently. Before the second period both players know the strategies chosen and the outcome of the first period. We formulate the reliability *p*Mj of the main system in period *j*, and the reliability *p*S2 of the standby system in period 2, as contests between the defender and attacker. The most common functional form is the ratio form (Tullock 1980)

$$p\_{M\circ} = \frac{t\_{M\circ}^{m\_M}}{t\_{M\circ}^{m\_M} + T\_{M\circ}^{m\_M}}, \mathbf{j} = \mathbf{1}, \mathbf{2}, \quad p\_{S2} = \frac{t\_{S2}^{m\_S}}{t\_{S2}^{m\_S} + T\_{S2}^{m\_S}}.\tag{1}$$

where *p t Mj Mj* / 0 , / 0 *Mj Mj p T* , 2 2 / 0 *S S p t* , 2 2 / 0 *S S p T* , and *m*I 0, *I=M,S* is a parameter for the contest intensities of the main system (*M*) and standby system (*S*). The reliabilities *p*Mj and *p*S2 can also be interpreted as probabilities of system survival. Equation (1) is common in the rent seeking literature where the rent is an asset which corresponds to reliability in this paper. Conflict exists over reliability between the defender and the attacker, just as conflict exists over a rent between contending players. See Tullock (1980) for the use of *m*I, Skaperdas (1996) for an axiomatization, Nitzan (1994) for a review, Hirshleifer (1995) for illustration, usefulness, and application, and Hausken (2005) for recent literature. At the limit, with infinitely much defensive effort, and finite offensive effort, system *I* is 100% reliable. The same result follows with finite defensive effort and zero offensive effort. At the other limit, with infinitely much offensive effort, and finite defensive effort, component i is 0% reliable. The same result follows with finite offensive effort and zero defensive effort. The sensitivity of *p*ij to *t*ij increases as *m*I increases. When *m*I =0, the efforts *t*ij and *T*ij have equal impact on the reliability regardless of their size which gives 50% reliability, *p*ij=1/2. 0< *m*I <1 gives a disproportional advantage of exerting less effort than one's opponent. When *m*I =1, the efforts have proportional impact on the reliability. *m*I >1 gives a disproportional advantage of exerting more effort than one's opponent. This is often realistic in praxis, as evidenced by benefits from economies of scale. Finally, *m*I = gives a step function where "winner-takes-all".

The main system can in period 1 be in the two states shown in Table 1, where *v*M and *V*M are the defender's and attacker's values of an operational main system given presence of a standby system.

Table 1. Main system in two states in period 1.

We express the players' period 1 utilities as

$$\begin{aligned} \mu\_1 &= v\_M p\_{M1} - c\_M t\_{M1} \\ \mu\_1 &= V\_M (1 - p\_{M1}) - C\_M T\_{M1} \end{aligned} \tag{2}$$

The players' period 1 strategic choices determine both their first period utilities and the system state as the start of period 2. Each time period can be short or long, e.g. one minute, one month, one shift, one season. If the main system fails in period 1, it cannot be repaired in

model efforts the defender and attacker may exert with respect to the standby system in period 1. In both periods both players make their strategic choices simultaneously and independently. Before the second period both players know the strategies chosen and the outcome of the first period. We formulate the reliability *p*Mj of the main system in period *j*, and the reliability *p*S2 of the standby system in period 2, as contests between the defender

2

(1)

2 2

2

*M S M M S S m m Mj S Mj m m S m m Mj Mj S S t t p jp t T t T*

where *p t Mj Mj* / 0 , / 0 *Mj Mj p T* , 2 2 / 0 *S S p t* , 2 2 / 0 *S S p T* , and *m*I 0, *I=M,S* is a parameter for the contest intensities of the main system (*M*) and standby system (*S*). The reliabilities *p*Mj and *p*S2 can also be interpreted as probabilities of system survival. Equation (1) is common in the rent seeking literature where the rent is an asset which corresponds to reliability in this paper. Conflict exists over reliability between the defender and the attacker, just as conflict exists over a rent between contending players. See Tullock (1980) for the use of *m*I, Skaperdas (1996) for an axiomatization, Nitzan (1994) for a review, Hirshleifer (1995) for illustration, usefulness, and application, and Hausken (2005) for recent literature. At the limit, with infinitely much defensive effort, and finite offensive effort, system *I* is 100% reliable. The same result follows with finite defensive effort and zero offensive effort. At the other limit, with infinitely much offensive effort, and finite defensive effort, component i is 0% reliable. The same result follows with finite offensive effort and zero defensive effort. The sensitivity of *p*ij to *t*ij increases as *m*I increases. When *m*I =0, the efforts *t*ij and *T*ij have equal impact on the reliability regardless of their size which gives 50% reliability, *p*ij=1/2. 0< *m*I <1 gives a disproportional advantage of exerting less effort than one's opponent. When *m*I =1, the efforts have proportional impact on the reliability. *m*I >1 gives a disproportional advantage of exerting more effort than one's opponent. This is often realistic in praxis, as evidenced by benefits from economies of scale. Finally, *m*I = gives a

The main system can in period 1 be in the two states shown in Table 1, where *v*M and *V*M are the defender's and attacker's values of an operational main system given presence of a

State Main system Reliability Defender value Attacker value

1 11

*u vp ct U V p CT*

1 11

The players' period 1 strategic choices determine both their first period utilities and the system state as the start of period 2. Each time period can be short or long, e.g. one minute, one month, one shift, one season. If the main system fails in period 1, it cannot be repaired in

*M M MM*

(1 ) *M M MM* ,

(2)

1 operates *p*M1 *v*M-*c*M*t*M1 -*C*M*T*M1 2 fails 1- *p*M1 -*c*M*t*M1 *V*M-*C*M*T*M1

, 1,2, ,

and attacker. The most common functional form is the ratio form (Tullock 1980)

step function where "winner-takes-all".

Table 1. Main system in two states in period 1.

We express the players' period 1 utilities as

standby system.

time for the onset of period 2. This assumption is justified since repairing or replacing failed components can be complicated for economical and logistical reasons, and may require competence and time, which we assume is impossible both during the periods and in the transition from period 1 to period 2.3 Hence the strategies the players choose for period 1 have to account for the combinations of possibilities in which the main system may operate or fail in the two periods, and the standby system may operate or fail in period 2. If the main system fails in period 1, since it cannot be repaired or replaced before the commencement of period 2, the players need to assess their defense and attack in period 2 to account for which of the two states (operation or failure) follows after period 1. In period 2 the players also make their strategic choices simultaneously and independently, knowing the outcome and choices in period 1.

If the main system operates after period 1 (state 1), then the unit costs of defense and attack remain unchanged and the players make strategic choices *t*M2 and *T*M2. If the main system fails in period 1 (state 2), we assume that the unit costs of defense and attack for the standby system are *c*S and *C*S, and the contest intensity is *m*S. The defender and attacker values of the standby system are *v*S and *V*S, where *v*S≤*v*M and *V*S≤*V*M. The defender and attacker values after period 2 are shown in Table 2, where and are time discount parameters.

Table 2. Defender and attacker values after period 2.

We thus express the players' utilities over the two periods as

$$\begin{aligned} \mu &= \upsilon\_M p\_{M1} - \upsilon\_M t\_{M1} + \delta p\_{M1} (\upsilon\_M p\_{M2} - \upsilon\_M t\_{M2}) + \delta (1 - p\_{M1}) (\upsilon\_S p\_{S2} - \upsilon\_S t\_{S2}) \\ \mathcal{U} &= V\_M (1 - p\_{M1}) - \mathsf{C}\_M T\_{M1} + \Delta p\_{M1} (V\_M (1 - p\_{M2}) - \mathsf{C}\_M T\_{M2}) + \Delta (1 - p\_{M1}) (V\_S (1 - p\_{S2}) - \mathsf{C}\_S T\_{S2}) \end{aligned} \tag{3}$$

The third term on the right hand side in both utilities contains *p*M1 which is the probability that the main system survives period 1. The fourth and rightmost term on the right hand side in both utilities contains 1-*p*M1 which is the probability that the main system does not survive period 1. Fig. 1 shows the two-period game as an extensive form game tree.

#### **3. Solving the model**

The two players have two strategic choice variables *t*M1 and *T*M1 in period 1, and four strategic choice variables *t*M2,*T*M2, *t*S2, and *T*S2 in period 2. We analyze pure strategy Nash equilibria. We solve the game with backward induction starting with period 2. Differentiating gives *u t* / *M*2 = *u t* / *S*2 =0 and *U T* / *M*2 = *U T* / *S*2 =0. Solving the four equations gives

3 Future research may model the repair of the main system.

Game Theoretic Analysis of Standby Systems 83

Equations (3) and (4) show that the second period strategic choice variables do not depend on the first period strategic choice variables, only on the parameters. This means that the two-period game gives the same result as a corresponding one-period game where the players choose their six strategies simultaneously and independently. This result follows since the players' strategic choices in period 2 are independent for the two states that are possible after period 1. For state 1 the strategic choice variables are *t*M2 and *T*M2. For state 2 the strategic choice variables are *t*S2 and *T*S2. Consequently the players do not need to know the outcome of period 1 in order to play period 2. But, the probabilities of the two states depend on how period 1 is played, so the players account for the outcome of period 2 for each of the two states when determining their strategies in period 1. Thus the expressions for *t*M2 and *T*M2 are valid for a one period main system as described in Table 1, and the expressions for *t*S2 and *T*S2 are valid for a one period system

of a standby system.

the two equations gives

where

*M*

To solve period 1 we rewrite (3) as

11 1 1

*a v v p c t vp ct b vp ct*

1 1 2 1 1

*M M*

*c a C A T t Tp*

1 1 2

*u U*

/

*C A m A*

*M*

*M*

*m*

*M MM M M M*

*u ap b c t U A p B C T*

2 2 22 22

*A V V p C T V p CT B V p C T*

, (1 ) , ( ), ( ),

*M M M MM S S SS S S SS*

2 2 22 2 2

(6)

( (1 ) (1 ) ), ( (1 ) )

*M M M MM S S SS M M MM*

where a,b,A,B are parameters determined by inserting (4) into (6). We interpret *a* and *A* as the defender's and attacker's values of an operational two period system consisting of a main system and a standby system. The parameters *b* and *B* provide direct values to the defender and attacker, respectively. Differentiating gives *u t* / *M*1 = *U T* / *M*1 =0. Solving

/ / 1 ,, , / / / 1 1 / /

*c a C A C A*

*M M*

*M M*

*c a c a C A u ab U A B c a C A C A*

*m m*

*M M m M M M m m M*

*M M*

*C A C A*

*c a c a*

*M M m m*

2 2

*M M*

*M M*

2

/ , /

*M*

*M M*

*C A c a*

*M*

(7)

*m*

*m m*

/ / ( ) / / ,

*M M MM m m M M M M M M*

*C c a c a*

/ /

/ / / , / / / 1 1

*M M*

/ / 1 1

/ /

*c a c a*

*M M M M*

*M M M M*

*M M*

*C A C A*

*C A C A c a c a*

*MM M M MM*

*v v am V Am V*

/ / 1 (1 ) 1

*m m*

$$\begin{split} T\_{12} &= \frac{m\_l \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l} V\_l}{\mathbf{C}\_I \left(1 + \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l}\right)^2}, \quad I\_{12} = \frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l} T\_{12}, \quad p\_{12} = \frac{1}{1 + \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l}}, \quad I = M\_r S\_l\\ \mathbf{u}\_{I2} &= v\_l p\_{I2} - c\_1 t\_{12} = \frac{1 + (1 - m\_l) \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l}}{\left(1 + \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^2\right)^2} v\_{I'}\\ \mathbf{U}\_{I2} &= V\_I (1 - p\_{I2}) - \mathbf{C}\_I T\_{12} = \frac{1 - m\_l + \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l}}{\left(1 + \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^2\right)^2} \left(\frac{\mathbf{C}\_I / V\_I}{c\_1 / v\_l}\right)^{m\_l} V\_{I} \end{split} \tag{4}$$

where *u*I2 and *U*I2 are the period 2 utilities for system *I, I=M,S*. The second order conditions are satisfied when

$$\left(\frac{m\_l - 1}{m\_l + 1}\right)^{1/m\_l} < \frac{C\_I}{c\_I} \Big/ \frac{V\_I}{v\_I} < \left(\frac{m\_l + 1}{m\_l - 1}\right)^{1/m\_l}, I = M, S \tag{5}$$

Equation (5) is satisfied with an infinitely large range for the commonly used contest intensity *m*j=1, and generally stretches from below to above / / *I I I I C V c v* =1.

Fig. 1. The two-period extensive form game tree.

*C V m V c v C V T t T p I M S c v C V C V*

/ / 1 , , , , / / / 1 1 / /

*I*

*m*

*I I*

/ /

*C V m c v C V V c v C V*

*I I I I I m I I I I*

*I*

/ *I I I I C V c v*

Defender loses: -

Attacker wins:

Defender wins:

Attacker loses: -

Defender wins:

Defender loses: -

Attacker loses: -

*c*M*t*M2

Attacker wins: (VM-CMTM2)

*C*M*T*M2

*v*M*-c*M*t*M2)

=1.

*c*S*t*S2

(*V*S-*C*S*T*S2)

*v*S*-c*S*t*S2)

*C*S*T*S2 (4)

(5)

*I*

*m I I I m*

/ / 1 /

where *u*I2 and *U*I2 are the period 2 utilities for system *I, I=M,S*. The second order conditions

1/ 1/ 1/ 1 , , 1/ 1 *m m I I*

(1*-p*M1)(1- *p*S2)

(1*-p*M1)*p*S2

Period 2

Period 2

*p*M1(1- *p*M2)

*p*M1*p*M2

*m CV m I MS*

Equation (5) is satisfied with an infinitely large range for the commonly used contest

*I I*

*c v*

*I*

2 2 2 2 2

/ 1 /

/ 1 (1 ) / ,

*I I I I*

/ 1

*I II I I II I*

*m cv m*

*C V c v*

*I I I I I*

*I I I I I I II m m I I I I I I I I I I I*

*C c v c v*

*I*

2 22 2

*I I I II I m*

2

intensity *m*j=1, and generally stretches from below to above /

Fig. 1. The two-period extensive form game tree.

Period 1

*p*M1

1- *p*M1

*C V m c v u vp ct v*

*I*

*m I I I I*

/

2 22

(1 )

*U V p CT*

are satisfied when

Defender loses: -*c*M*t*M1 Attacker wins: *V*M-*C*M*T*M1

Defender wins: *v*M-*c*M*t*M1 Attacker loses: -*C*M*T*M1

*I I I II*

Equations (3) and (4) show that the second period strategic choice variables do not depend on the first period strategic choice variables, only on the parameters. This means that the two-period game gives the same result as a corresponding one-period game where the players choose their six strategies simultaneously and independently. This result follows since the players' strategic choices in period 2 are independent for the two states that are possible after period 1. For state 1 the strategic choice variables are *t*M2 and *T*M2. For state 2 the strategic choice variables are *t*S2 and *T*S2. Consequently the players do not need to know the outcome of period 1 in order to play period 2. But, the probabilities of the two states depend on how period 1 is played, so the players account for the outcome of period 2 for each of the two states when determining their strategies in period 1. Thus the expressions for *t*M2 and *T*M2 are valid for a one period main system as described in Table 1, and the expressions for *t*S2 and *T*S2 are valid for a one period system of a standby system.

To solve period 1 we rewrite (3) as

$$\begin{aligned} u &= ap\_{M1} + b - c\_M t\_{M1}, \quad \mathcal{U} = A(1 - p\_{M1}) + B - \mathcal{C}\_M T\_{M1}, \\ a &= v\_M + \mathcal{S}(v\_M p\_{M2} - c\_M t\_{M2} - v\_S p\_{S2} + c\_S t\_{S2}), \quad b = \mathcal{S}(v\_S p\_{S2} - c\_S t\_{S2}), \\ A &= V\_M + \Delta(-V\_M(1 - p\_{M2}) + \mathcal{C}\_M T\_{M2} + V\_S(1 - p\_{S2}) - \mathcal{C}\_S T\_{S2}), \quad B = \Delta(V\_M(1 - p\_{M2}) - \mathcal{C}\_M T\_{M2}) \end{aligned} \tag{6}$$

where a,b,A,B are parameters determined by inserting (4) into (6). We interpret *a* and *A* as the defender's and attacker's values of an operational two period system consisting of a main system and a standby system. The parameters *b* and *B* provide direct values to the defender and attacker, respectively. Differentiating gives *u t* / *M*1 = *U T* / *M*1 =0. Solving the two equations gives

1 1 2 1 1 1 1 2 / / / 1 ,, , / / / 1 1 / / / / ( ) / / , / / 1 1 / / *M M M M M M M m M M M M M M MM m m M M M M M M m m M M MM M M MM M M m m M M M M C A m A c a C A T t Tp c a C A C A C c a c a C A C A v v am V Am V c a c a u U C A C A c a c a* 2 2 2 / , / / / 1 (1 ) 1 / / / , / / / 1 1 / / *M M M M M M m M M m m M M M M m M M M m m M M M M M C A c a C A C A m m c a c a C A u ab U A B c a C A C A c a c a* (7)

where

Game Theoretic Analysis of Standby Systems 85

utility. In the absence of a pure-strategy Nash equilibrium, the assumption of withdrawal is plausible. Analogously, for the case *U*<0, we assume that the attacker withdraws exerting zero effort and earning zero utility, while the defender exerts negligible effort earning utility

Let us consider three special benchmark cases with straightforward interpretations. The first is the egalitarian case *m*M=*m*S=0 causing zero efforts and thus 50% probability of failure for the main system in period 1, and for the main system or standby system in period 2. This case illustrates how the players' utilities depend on the main system and standby system

*V*I=*v*i, and equal contest intensities for the main system and the standby system, *m*M=*m*S=*m*. For case 2 we show how this impacts the values *a,A,b,B* of the two period system dependent

, ,, , ( ), ( ) 2 2 2 2 24 2 4

2 22 2 ( ), ( ), , 4 44 4 *M M SM S MSM m mm m av v v AV V V b v B V*

giving rise to three observations. 1. The defender gets increased value of the two period system and the attacker gets decreased value of the two period system with low contest intensity *m*<2, since *v*S≤*v*M and *V*S≤*V*M. High contest intensity *m*>2 causes the reverse result and is costly for the defender. 2. Contest intensity *m*=2 gives *a*=*v*M, *A*=*V*M, *b*=*B*=0, *T*M1=*T*M2, and *t*M1=*t*M2. 3. Equal values *v*S=*v*M and *V*S=*V*M give *a*=*v*M, *A*=*V*M, *T*M1=*T*M2, and *t*M1=*t*M2.

(11)

*I I M S S M*

=1 into (4),(7),(8) gives

1 0, , ,, 2

*M M I I M M*

=1 and *m*M=*m*S=*m* into (8) gives

reliability at the end of period 2. The utilities are positive in both periods.

=1 and *m*M=*m*S=*m*=

*v V v V v V u U u U u v v U VV*

The utilities are not affected by efforts in egalitarian contests, so the players choose zero efforts which cause 50% reliability for the main system in period 1. The main system has 50% probability of surviving into period 2, and thus 25% reliability at the end of period 2. The standby system has 50% probability of being implemented in period 2, and thus 25%

=1, which occurs e.g. when the players have equal unit costs *C*I=*c*i and evaluations

 and

. For case 3 we furthermore show the impact of no

. Cases 2 and 3 assume

(10)

(1+

> / / *I I I I C V c v*

discounting

)*v*M since all reliabilities are one.

**4. Analyzing three special cases**

on the discount parameters

==1.

Second, inserting /

Third, inserting /

and the weights assigned to period 2 expressed with

 and

*T t T t p p I MS*

First, inserting *m*M=*m*S=0 into (4),(7),(8) gives

1 1 22 1 2

*MM II M I*

11 22

/ *I I I I C V c v*

/ *I I I I C V c v*

2 2 2 / / 1 (1 ) 1 (1 ) / / , / / 1 1 / / / 1 / / 1 / *M S M S S S m m M M S S M S M M S S M MS m m M M S S M M S S m S S S S S S M m S S S S C V C V m m c v c v a v v v C V C V c v c v C V m c v C A V C V c v* 2 2 / 1 / / / , / / / 1 / / / 1 (1 ) 1 / / , / 1 1 / *M S M M S M S m M M m m M S M M M M S M m S S M M M M M M m m S S M M S M S S M M S m S S S S C V m V c v C V V V c v c v C V c v C V C V m m c v c v b v B C V C c v* 2 / / / / *M M m M M M m M M M M M M C V V c v V c v* (8)

Inserting =0 into (8) gives *a*=*v*M, *A=V*M, *b=B*=0, *T*M1=*T*M2, and *t*M1=*t*M2, confirming that period 1 operates as the last period 2 with maximum discounting. Equation (8) implies lim 0 *mM a* (since a cannot be negative) and lim *mM A* and thus high contest intensity for the main system causes the defender and attacker to perceive the two period system as valueless and very valuable, respectively. Conversely, (8) also implies lim*mS a* (since *a* cannot be negative) and lim 0 *mS A* (since A cannot be negative) and thus high contest intensity for the standby system causes the defender and attacker to perceive the two period system as very valuable and valueless, respectively.

The second order conditions are satisfied when

$$\left(\frac{m\_M - 1}{m\_M + 1}\right)^{1/m\_M} < \frac{C\_M \,/\, A}{c\_M \,/\, a} < \left(\frac{m\_M + 1}{m\_M - 1}\right)^{1/m\_M} \tag{9}$$

The boundary solutions are as follows. The interior solution above is valid when *u*≥0 and *U*≥0 in (7). When *u*<0 or *U*<0, no pure-strategy Nash equilibrium exists. Analyzing mixed strategy equilibria is beyond the scope of this paper. Equation (7) and the next sections show that *u*<0 and *U*<0 are possible when the contest intensity of the main system *m*M is large which induces large costly efforts. The case *u*<0 is calamitous for the defender since it cannot earn positive utility. We assume that the defender withdraws in this case, exerting zero effort and earning zero utility, while the attacker exerts negligible effort earning utility *V*M+*V*S since all reliabilities are zero. One may reason that if the defender knows that the attacker exerts negligible effort, the defender can exert positive effort and earn positive utility. However, if the attacker knows that, it can exert positive effort and earn positive

2 2

*M S*

*M M S S*

*M M S S*

*S M*

*m m*

*M S*

/

*A*

*c v*

period 1 operates as the last period 2 with maximum discounting. Equation (8) implies

the main system causes the defender and attacker to perceive the two period system as

intensity for the standby system causes the defender and attacker to perceive the two period

The boundary solutions are as follows. The interior solution above is valid when *u*≥0 and *U*≥0 in (7). When *u*<0 or *U*<0, no pure-strategy Nash equilibrium exists. Analyzing mixed strategy equilibria is beyond the scope of this paper. Equation (7) and the next sections show that *u*<0 and *U*<0 are possible when the contest intensity of the main system *m*M is large which induces large costly efforts. The case *u*<0 is calamitous for the defender since it cannot earn positive utility. We assume that the defender withdraws in this case, exerting zero effort and earning zero utility, while the attacker exerts negligible effort earning utility

*M MM M MM m C Am m ca m*

1/ 1/ 1 /1 1 /1

*V*S since all reliabilities are zero. One may reason that if the defender knows that the attacker exerts negligible effort, the defender can exert positive effort and earn positive utility. However, if the attacker knows that, it can exert positive effort and earn positive

*m m M M*

*M M M M*

=0 into (8) gives *a*=*v*M, *A=V*M, *b=B*=0, *T*M1=*T*M2, and *t*M1=*t*M2, confirming that

/ 1

2

*M*

and thus high contest intensity for

*a*

(since *a*

(8)

*m*

*M*

/

*M M M m M M*

*M*

*m*

/ / / , / / / 1

*C V V c v V*

(since A cannot be negative) and thus high contest

(9)

*S M*

*M M m m M S M M M M S M m S S M M M M M M*

/

*c v*

2

/ /

*M*

*C V m V c v C V V V c v c v C V*

*m m*

/ / 1 (1 ) 1 (1 ) / / ,

/ / 1 1 / /

*C V C V m m c v c v a v v v*

*M S*

*M MS m m M M S S M M S S*

*C V C V c v c v*

/ / ,

*S S M M*

valueless and very valuable, respectively. Conversely, (8) also implies lim*mS*

*S S M M*

2

*S S S*

/ / 1 (1 ) 1

*S M*

*C V C*

*C V C V m m c v c v b v B*

*S*

*S*

*m*

/

2

*S m*

*S*

/ 1 1

(since a cannot be negative) and lim *mM*

system as very valuable and valueless, respectively.

The second order conditions are satisfied when

*A*

> *S S S S*

*C V c v*

*S S*

/ 1

*S*

Inserting

lim 0

*mM a*

*V*M+ *C V m c v C A V*

*M m*

/

cannot be negative) and lim 0 *mS*

*c v*

*S S S S*

/ 1 /

utility. In the absence of a pure-strategy Nash equilibrium, the assumption of withdrawal is plausible. Analogously, for the case *U*<0, we assume that the attacker withdraws exerting zero effort and earning zero utility, while the defender exerts negligible effort earning utility (1+)*v*M since all reliabilities are one.

#### **4. Analyzing three special cases**

Let us consider three special benchmark cases with straightforward interpretations. The first is the egalitarian case *m*M=*m*S=0 causing zero efforts and thus 50% probability of failure for the main system in period 1, and for the main system or standby system in period 2. This case illustrates how the players' utilities depend on the main system and standby system and the weights assigned to period 2 expressed with and . Cases 2 and 3 assume / / *I I I I C V c v* =1, which occurs e.g. when the players have equal unit costs *C*I=*c*i and evaluations

*V*I=*v*i, and equal contest intensities for the main system and the standby system, *m*M=*m*S=*m*. For case 2 we show how this impacts the values *a,A,b,B* of the two period system dependent on the discount parameters and . For case 3 we furthermore show the impact of no discounting ==1.

First, inserting *m*M=*m*S=0 into (4),(7),(8) gives

$$\begin{aligned} T\_{M1} &= t\_{M1} = T\_{12} = t\_{12} = 0, \quad p\_{M1} = p\_{12} = \frac{1}{2}, \quad I = M, S\_r\\ \mathbf{u}\_1 &= \frac{\mathbf{v}\_M}{2}, \mathbf{U}\_1 = \frac{V\_M}{2}, \mathbf{u}\_{12} = \frac{\mathbf{v}\_l}{2}, \mathbf{U}\_{12} = \frac{V\_l}{2}, \quad \mathbf{u} = \frac{\mathbf{v}\_M}{2} + \frac{\delta}{4}(\mathbf{v}\_M + \mathbf{v}\_S), \quad \mathbf{U} = \frac{V\_M}{2} + \frac{\Lambda}{4}(V\_S + V\_M) \end{aligned} \tag{10}$$

The utilities are not affected by efforts in egalitarian contests, so the players choose zero efforts which cause 50% reliability for the main system in period 1. The main system has 50% probability of surviving into period 2, and thus 25% reliability at the end of period 2. The standby system has 50% probability of being implemented in period 2, and thus 25% reliability at the end of period 2. The utilities are positive in both periods.

$$\text{Second, inserting } \frac{C\_I \;/\; V\_I}{c\_l \;/\; v\_l} = 1 \text{ and } m\_\text{M} = m\_\text{S} = m \text{ into (8) gives}$$

$$a = v\_M + \delta \frac{2 - m}{4} (v\_M - v\_S)\_\prime \quad A = V\_M + \Delta \frac{2 - m}{4} (V\_S - V\_M)\_\prime \quad b = \delta \frac{2 - m}{4} v\_{S\prime} \quad B = \Delta \frac{2 - m}{4} V\_M \tag{11}$$

giving rise to three observations. 1. The defender gets increased value of the two period system and the attacker gets decreased value of the two period system with low contest intensity *m*<2, since *v*S≤*v*M and *V*S≤*V*M. High contest intensity *m*>2 causes the reverse result and is costly for the defender. 2. Contest intensity *m*=2 gives *a*=*v*M, *A*=*V*M, *b*=*B*=0, *T*M1=*T*M2, and *t*M1=*t*M2. 3. Equal values *v*S=*v*M and *V*S=*V*M give *a*=*v*M, *A*=*V*M, *T*M1=*T*M2, and *t*M1=*t*M2.

$$\text{Third, inserting } \frac{\mathbb{C}\_{\text{I}} / V\_{\text{I}}}{c\_{\text{I}} / v\_{\text{I}}} = 1 \text{ and } m\_{\text{M}} = m\_{\text{S}} = m = \text{\S} = 1 \text{ into (4),(7),(8) gives } v\_{\text{I}} = \frac{1}{c\_{\text{I}} - v\_{\text{I}}} = 1 \text{ and } v\_{\text{I}} = \frac{1}{c\_{\text{I}} - v\_{\text{I}}} = 1 \text{ for } \text{V} = \text{I}$$

Game Theoretic Analysis of Standby Systems 87

Fig. 2. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *v*S=*V*S and *v*M=*V*M.

Fig. 3. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *c*M=*c*S and *c*M.

Fig. 4 panel 1 plots as functions of equivalent contest intensities *m*M=*m*S for both systems. High contest intensities induce higher efforts which increase linearly in *m*M=*m*S. The higher efforts are costly causing the utilities to decrease linearly reaching zero when *m*M=*m*S=2. Fig. 4 panel 2 plots as functions of the contest intensity *m*M for the main system, keeping *m*S=1. Thus *t*S2=*T*S2 remain constant at 0.25. When *m*M<1, the results are similar but the comparatively higher *m*S=1 causes lower utilities in panel 2. For the main system in period 2 the efforts increase linearly in *m*M as seen from (4) inserting *c*M=*C*M=*v*M=*V*M=1. However, for the main system in period 1 the efforts do not increase linearly since *A* and *a*, instead of *v*M and *V*M, operate in the efforts *t*M1 and *T*M1 in (7). As *m*M increases above 1, *t*M1 and *T*M1 increase in a decreasing manner, reaching maxima and thereafter decreasing towards zero.

larger in period 1 than in period 2, and conversely when *c*M>1.

Fig. 3 panel 1 plots as functions of the defender's unit costs *c*M=*c*S changing equally for both systems. The three defender efforts *t*M1=*t*M2=*t*S2 decrease convexly and equivalently as the defense becomes more costly. The three attacker efforts *T*M1=*T*M2=*T*S2 are inverse *U* shaped. When *c*M=*c*S is low, the inferior attacker provides modest efforts against the defender cheaply producing a substantial defense. When *c*M=*c*S is high, the defender efforts are low and the attacker does not need to attack substantially. Hence the attacker efforts are largest for intermediate *c*M=*c*S. The defender utility decreases, and the attacker utility increases, in *c*M=*c*S. Fig. 3 panel 2 plots as functions of the defender's unit cost *c*M for the main system, keeping *c*S=1 for the standby system. The results are similar but *t*S2=*T*S2 remain constant at 0.25. When *c*M<1, both the defender's and the attacker's efforts for the main system are

1 1 2 1 1 22 2 1 1 2 2 2 2 / / / 1 ,,, / / / 1 1 / / 1 , , ,, 4 2 / / ( ) / / / , , / / / 1 1 / / , / 1 / *M M M M M MM M M M M M M I II I I M M M M M M M M M M M M M M I I I I I I C A A c a C A T t T p c a C A C A C c a c a V T t p I MS C C A C A v va V AV c a c aC A u U c a C A C A c a c a v u C V c v* 2 2 2 2 2 2 / / , / 1 / / / , , / / 1 1 / / 5 3 , ,, 4 4 44 *I I I I I I I I I I M M M M M M MS MS S M C V c v U V C V c v C A A a c a u b U B C A C A c a c a v v VV v V a A bB* (12)

#### **5. Simulating the solution**

Figs. 2-5 plot the six efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and two utilities *u* and *U* as functions of one parameter relative to the baseline *c*M=*C*M=*c*S=*C*S=*v*M=*V*M=*vS*=*V*S=*m*M=*m*S==1. The titles on the vertical axis are as specified in the legend box. Fig. 1 panel 1 plots as functions of the standby system values *vS*=*VS* varying between 0 and 1. When the standby system has its maximum value *v*S=*V*S=*v*M=*V*M=1, the six efforts equal 0.25 and the utilities are 0.5, as also seen from (12) where *a=A*=1 and *b=B*=0.25. As *v*S=*V*S decrease below 1, the efforts *t*S2=*T*S2 decrease linearly toward zero, while *t*M2=*T*M2 remain constant at 0.25. However, the defender compensates for the decreased value of the standby system by defending the main system more thoroughly in period 1, and thus *t*M1 increases when *v*S=*V*S decrease. The attacker responds to this by decreasing *T*M1 when *v*S=*V*S decrease, and thus the attacker's utility *U* also decreases when *v*S=*V*S decrease. The defender's utility *u* is almost constant (slightly *U* shaped) since the defender compensates for the decreasing value of the standby system by defending the main system more thoroughly in period 1. Fig. 1 panel 2 plots as functions of the main system values *v*M=*V*M increasing upwards from *v*S=*V*S=1. Now the standby system efforts *t*S2=*T*S2 remain constant at 0.25, and all the other variables increase in *v*M=*V*M. Increasing *v*M=*V*M relative to the fixed *v*S=*V*S=1 induces the defender to defend the more valuable main system more thoroughly in period 1, and thus *t*M1 increases more than *T*M1, and the defender's utility u increases more than the attacker's utility *U*, though *t*M2=*T*M2 for the main system in period 2 increase equivalently.

*M M M*

*c a C A C A C c a*

/ / 1 ,,, / / / 1 1 / /

*M M M*

/ / ( ) / / / , , / / / 1 1

*c a C A C A*

*C A C A*

*M M M M M M*

2

/ ,

2

(12)

=1. The

*M M*

/ /

2 2

/ 1 /

/ / , ,

*C A A*

*M M*

Figs. 2-5 plot the six efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and two utilities *u* and *U* as functions of

titles on the vertical axis are as specified in the legend box. Fig. 1 panel 1 plots as functions of the standby system values *vS*=*VS* varying between 0 and 1. When the standby system has its maximum value *v*S=*V*S=*v*M=*V*M=1, the six efforts equal 0.25 and the utilities are 0.5, as also seen from (12) where *a=A*=1 and *b=B*=0.25. As *v*S=*V*S decrease below 1, the efforts *t*S2=*T*S2 decrease linearly toward zero, while *t*M2=*T*M2 remain constant at 0.25. However, the defender compensates for the decreased value of the standby system by defending the main system more thoroughly in period 1, and thus *t*M1 increases when *v*S=*V*S decrease. The attacker responds to this by decreasing *T*M1 when *v*S=*V*S decrease, and thus the attacker's utility *U* also decreases when *v*S=*V*S decrease. The defender's utility *u* is almost constant (slightly *U* shaped) since the defender compensates for the decreasing value of the standby system by defending the main system more thoroughly in period 1. Fig. 1 panel 2 plots as functions of the main system values *v*M=*V*M increasing upwards from *v*S=*V*S=1. Now the standby system efforts *t*S2=*T*S2 remain constant at 0.25, and all the other variables increase in *v*M=*V*M. Increasing *v*M=*V*M relative to the fixed *v*S=*V*S=1 induces the defender to defend the more valuable main system more thoroughly in period 1, and thus *t*M1 increases more than *T*M1, and the defender's utility u increases more than the attacker's utility *U*, though *t*M2=*T*M2

/

*C V c v U V C V c v*

*I I I I I I I I I I*

*c a c a*

*c a c aC A u U*

*M M*

2 2

1 1 2 1 1

*M M M M MM*

*c a C A T t T p*

1 1 2 2

*M M M M*

*v va V AV*

1 , , ,, 4 2

,

/ / 1 1 / /

*M M M M*

*a c a u b U B C A C A c a c a v v VV v V a A bB*

5 3 , ,, 4 4 44

*MS MS S M*

one parameter relative to the baseline *c*M=*C*M=*c*S=*C*S=*v*M=*V*M=*vS*=*V*S=*m*M=*m*S=

22 2

*I II I I*

/

*C A A*

*c a*

*V T t p I MS C*

*M*

2 2

*I*

**5. Simulating the solution**

*u*

*I*

*C V c v*

for the main system in period 2 increase equivalently.

*v*

*I I I I*

/ 1 /

Fig. 2. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *v*S=*V*S and *v*M=*V*M.

Fig. 3 panel 1 plots as functions of the defender's unit costs *c*M=*c*S changing equally for both systems. The three defender efforts *t*M1=*t*M2=*t*S2 decrease convexly and equivalently as the defense becomes more costly. The three attacker efforts *T*M1=*T*M2=*T*S2 are inverse *U* shaped. When *c*M=*c*S is low, the inferior attacker provides modest efforts against the defender cheaply producing a substantial defense. When *c*M=*c*S is high, the defender efforts are low and the attacker does not need to attack substantially. Hence the attacker efforts are largest for intermediate *c*M=*c*S. The defender utility decreases, and the attacker utility increases, in *c*M=*c*S. Fig. 3 panel 2 plots as functions of the defender's unit cost *c*M for the main system, keeping *c*S=1 for the standby system. The results are similar but *t*S2=*T*S2 remain constant at 0.25. When *c*M<1, both the defender's and the attacker's efforts for the main system are larger in period 1 than in period 2, and conversely when *c*M>1.

Fig. 3. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *c*M=*c*S and *c*M.

Fig. 4 panel 1 plots as functions of equivalent contest intensities *m*M=*m*S for both systems. High contest intensities induce higher efforts which increase linearly in *m*M=*m*S. The higher efforts are costly causing the utilities to decrease linearly reaching zero when *m*M=*m*S=2. Fig. 4 panel 2 plots as functions of the contest intensity *m*M for the main system, keeping *m*S=1. Thus *t*S2=*T*S2 remain constant at 0.25. When *m*M<1, the results are similar but the comparatively higher *m*S=1 causes lower utilities in panel 2. For the main system in period 2 the efforts increase linearly in *m*M as seen from (4) inserting *c*M=*C*M=*v*M=*V*M=1. However, for the main system in period 1 the efforts do not increase linearly since *A* and *a*, instead of *v*M and *V*M, operate in the efforts *t*M1 and *T*M1 in (7). As *m*M increases above 1, *t*M1 and *T*M1 increase in a decreasing manner, reaching maxima and thereafter decreasing towards zero.

Game Theoretic Analysis of Standby Systems 89

There are no limits to the kinds of standby systems that can be envisioned. In fact, any system produced to deliver some function, can be supplemented with a standby system to deliver the same function in the event that the main system breaks down. Examples of systems are power supply, telecommunications systems, water supply, roads, bridges, tunnels, political and economic institutions, businesses, schools, hospitals, recreational facilities, and various assets. One example of a standby system is a standby generator which is a back-up electrical system that operates automatically (Hickey 2002). Within seconds of a utility outage, an automatic transfer switch senses the power loss, directs the standby generator to start, and transfers the electrical load to the standby generator. The standby generator thereafter supplies power to the circuits. To ensure a proper response to a power outage, a standby generator runs weekly self-tests. Most units run on diesel, natural gas or liquid propane gas. Automatic standby generators may be required by building codes for critical safety systems. Examples are building elevators, fire protection systems, standby lighting, or medical and life support equipment. Residential standby generators are common, providing backup electrical power to security systems, household appliances such as refrigerators, stoves, and hot water heaters, and HVAC systems. To determine the quality, design, and maintenance regime for the main system and standby system in each

We consider two players choosing strategies through time to impact the reliability of a dependent system which consists of a main system and a standby system. Each system can be in two states, i.e. it can operate or fail. If the main system operates successfully through period 1, it continues to operate into period 2 and the standby system remains in standby. If

Each system is protected by a defender which maximizes its reliability subtracting the defense costs, and attacked by an attacker which maximizes its unreliability subtracting the attack costs. Each system's reliability depends on the relative levels of defense and attack and on the contest intensity. Each player's utility depends additively on the system reliability in two time periods, with a time discount parameter for the second period. The unit costs of effort and the contest intensities are different for the two players and the two

In period 1 the defender chooses a defense effort and the attacker chooses an attack effort for the main system. In period 2 the defender chooses one defense effort for the main system and one defense effort for the standby system, not knowing before the game starts whether the main system or the standby system is the system to be defended in period 2. Analogously in period 2, the attacker chooses one attack effort for the main system and one attack system for the standby system. Hence six strategic decisions are made by the two

The players assign different values to the main system and the standby system. We present analytical solutions and simulations to illustrate the players' efforts in the two periods and

the main system fails in period 1, the standby system is implemented in period 2.

systems. The two period game is analyzed with backward induction.

the utilities dependent on parametric changes.

particular example, the analysis in this paper can be used.

**6. Examples**

**7. Conclusion**

players.

The defender accepts its maximum for *t*M1 being lower than the attacker's maximum for *T*M1. This gives low reliability *p*M1 for the main system in period 1 as *m*M increases above 1. The high contest intensity for the main system makes it too costly to defend and attack compared with defending and attacking the standby system in period 2. The defender's utility decreases below 0.25 as *m*M increases above 1.61, reaches a minimum 0.14 for *m*M=2.22 as the defender realizes that the main system is too costly to defend, and increases asymptotically towards 0.25. Recall from (3) that if the defender does not defend the main system in period 1, it is guaranteed to fail, *p*M1=0, causing utility 0.25 generated by the standby system since *p*S2=0.5 with the given parameter values. Hence the attacker's utility eventually increases towards 1.25.

Fig. 4. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *m*M=*m*S and *m*M.

Fig. 5 panel 1 plots as functions of the discount parameters . All efforts are constant at 0.25. The utilities increase from 0.25 when period 2 is discounted (=0) and reliability is *p*M1=0.5 for period 1, to 0.5 when both periods have equal weight (=1) and all reliabilities are *p*M1= *p*M2=*p*S2=0.5. Fig. 5 panel 2 also plots as functions of , but decreases the values of the standby system to *v*S=*V*S=0.5. This decreases the efforts for the standby system to *t*S2=*T*S2=0.125. The utilities remain at 0.25 when =0 and period 2 is irrelevant. As increase, the defender compensates for the less valuable standby system by increasing *t*M1 from 0.25 to 0.28 when =1, which increases the reliability of the main system in period 1. The attacker decreases *T*M1 from 0.25 to 0.22 when =1, while *t*M2=*T*M2 remain constant at 0.25. The defender's utility thus increases more than the attacker's utility, but both utilities increase less than in panel 1 where the standby system is more valuable at *v*S=*V*S=1.

Fig. 5. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of .

#### **6. Examples**

88 Advanced Topics in Applied Operations Management

The defender accepts its maximum for *t*M1 being lower than the attacker's maximum for *T*M1. This gives low reliability *p*M1 for the main system in period 1 as *m*M increases above 1. The high contest intensity for the main system makes it too costly to defend and attack compared with defending and attacking the standby system in period 2. The defender's utility decreases below 0.25 as *m*M increases above 1.61, reaches a minimum 0.14 for *m*M=2.22 as the defender realizes that the main system is too costly to defend, and increases asymptotically towards 0.25. Recall from (3) that if the defender does not defend the main system in period 1, it is guaranteed to fail, *p*M1=0, causing utility 0.25 generated by the standby system since *p*S2=0.5 with the given parameter values. Hence the attacker's utility

Fig. 4. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of *m*M=*m*S and *m*M.

standby system to *v*S=*V*S=0.5. This decreases the efforts for the standby system to *t*S2=*T*S2=0.125.

defender compensates for the less valuable standby system by increasing *t*M1 from 0.25 to 0.28

defender's utility thus increases more than the attacker's utility, but both utilities increase less

=1, which increases the reliability of the main system in period 1. The attacker

=0 and period 2 is irrelevant. As

. All efforts are constant at 0.25.

=1) and all reliabilities are *p*M1=

, but decreases the values of the

=1, while *t*M2=*T*M2 remain constant at 0.25. The

. increase, the

=0) and reliability is *p*M1=0.5

Fig. 5 panel 1 plots as functions of the discount parameters

for period 1, to 0.5 when both periods have equal weight (

*p*M2=*p*S2=0.5. Fig. 5 panel 2 also plots as functions of

The utilities remain at 0.25 when

decreases *T*M1 from 0.25 to 0.22 when

when

The utilities increase from 0.25 when period 2 is discounted (

than in panel 1 where the standby system is more valuable at *v*S=*V*S=1.

Fig. 5. Efforts *t*M1,*T*M1,*t*M2,*T*M2,*t*S2,*T*S2 and utilities *u* and *U* as functions of

eventually increases towards 1.25.

There are no limits to the kinds of standby systems that can be envisioned. In fact, any system produced to deliver some function, can be supplemented with a standby system to deliver the same function in the event that the main system breaks down. Examples of systems are power supply, telecommunications systems, water supply, roads, bridges, tunnels, political and economic institutions, businesses, schools, hospitals, recreational facilities, and various assets. One example of a standby system is a standby generator which is a back-up electrical system that operates automatically (Hickey 2002). Within seconds of a utility outage, an automatic transfer switch senses the power loss, directs the standby generator to start, and transfers the electrical load to the standby generator. The standby generator thereafter supplies power to the circuits. To ensure a proper response to a power outage, a standby generator runs weekly self-tests. Most units run on diesel, natural gas or liquid propane gas. Automatic standby generators may be required by building codes for critical safety systems. Examples are building elevators, fire protection systems, standby lighting, or medical and life support equipment. Residential standby generators are common, providing backup electrical power to security systems, household appliances such as refrigerators, stoves, and hot water heaters, and HVAC systems. To determine the quality, design, and maintenance regime for the main system and standby system in each particular example, the analysis in this paper can be used.

#### **7. Conclusion**

We consider two players choosing strategies through time to impact the reliability of a dependent system which consists of a main system and a standby system. Each system can be in two states, i.e. it can operate or fail. If the main system operates successfully through period 1, it continues to operate into period 2 and the standby system remains in standby. If the main system fails in period 1, the standby system is implemented in period 2.

Each system is protected by a defender which maximizes its reliability subtracting the defense costs, and attacked by an attacker which maximizes its unreliability subtracting the attack costs. Each system's reliability depends on the relative levels of defense and attack and on the contest intensity. Each player's utility depends additively on the system reliability in two time periods, with a time discount parameter for the second period. The unit costs of effort and the contest intensities are different for the two players and the two systems. The two period game is analyzed with backward induction.

In period 1 the defender chooses a defense effort and the attacker chooses an attack effort for the main system. In period 2 the defender chooses one defense effort for the main system and one defense effort for the standby system, not knowing before the game starts whether the main system or the standby system is the system to be defended in period 2. Analogously in period 2, the attacker chooses one attack effort for the main system and one attack system for the standby system. Hence six strategic decisions are made by the two players.

The players assign different values to the main system and the standby system. We present analytical solutions and simulations to illustrate the players' efforts in the two periods and the utilities dependent on parametric changes.

Game Theoretic Analysis of Standby Systems 91

Azaiez, N., Bier, V.M. (2007). Optimal Resource Allocation for Security in Reliability

Bier, V.M., Nagaraj, A., Abhichandani, V. (2005). Protection of Simple Series and Parallel

Bier, V.M., Oliveros, S., Samuelson, L. (2006). Choosing What to Protect: Strategic Defense

Dighe, N., Zhuang J., and Bier V.M. (2009). Secrecy in defensive allocations as a strategy for

Ebeling, C. (1997). An introduction to Reliability and Maintainability Engineering, McGraw-

Enders, W., Sandler, T. (2003). What do we know about the substitution effect in

Hausken, K., (2005) Production and conflict models versus rent seeking models. Public

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Hausken, K. (2008), Strategic Defense and Attack for Series and Parallel Reliability Systems,

Hausken, K. (2010), Defense and Attack of Complex and Dependent System, Reliability

Hausken, K. (2011), Game Theoretic Analysis of Two Period Dependent Degraded Multistate Reliability Systems, International Game Theory Review Forthcoming. Hausken, K. (2011), Protecting Complex Infrastructures Against Multiple Strategic

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*v*S defender's value of standby system *V*S attacker's value of standby system

*u* defender's utility *U* attacker's utility

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rcf.usc.edu/~tsandler/substitution2ms.pdf

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**9. References**

*m*M attacker-defender contest intensity for main system *m*S attacker-defender contest intensity for standby system

 defender's time discount parameter for period 2 attacker's time discount parameter for period 2 *a* defender's value of an operational two period system *A* attacker's value of an operational two period system *b* additional value to defender of two period system *B* additional value to attacker of two period system

Each time period can be short or long, e.g. minutes, days, months, as determined by the nature of the system and how failures occur dependent on the players' efforts. Since the main system fails in period 1 with positive probability, period 2 starts with the standby system with this positive probability.

Before period 1 each player assesses its effort for period 1 knowing that this effort impacts the probability that the main system survives into period 2. Exerting low defense effort in period 1 increases the probability that the standby system is implemented in period 2. We show that as the value of the standby system decreases below that of the main system, the defender increases its defense of the main system in period 1. Increasing the value of the main system increases the defender's utility more than the attacker's utility since the defender defends the main system more thoroughly in period 1. High defense effort in period 1 is an investment into the future for the defender.

As the defender's unit defense costs increases, its efforts and utilities decrease while the attacker's efforts are inverse U shaped and its utility increases. This follows since low unit defense costs make it not worthwile for the inferior attacker to attack, while high unit defense costs make it unnecessary for the superior attacker to attack substantially.

Increasing contest intensities for both systems causes all efforts to increase driving utilities downwards eventually reaching zero. Increasing the contest intensity only for the main system causes both efforts for the main system in period 1 to be inverse U shaped but taller for the attacker. This increases the probability that the main system fails in period 1. This benefits the attacker and does not benefit the defender which resorts to defending the standby system in period 2. Increasing discount parameters, making period 2 more valuable, benefit both players.

Two limitations of Markov analysis have been illustrated in this paper. First, we have enabled players to choose efforts strategically, which violates the Markov property. Second, we have relaxed the constraint in Markov modeling where the transition rates between different states are kept constant through time. The parameter values for the standby system in period 2 may differ from the parameter values for the main system in period 1. Future research may model in a multi-period game multiple states of operation for the main system and the standby system, and repair of the main system.

#### **8. Notation**

#### **9. References**

90 Advanced Topics in Applied Operations Management

Each time period can be short or long, e.g. minutes, days, months, as determined by the nature of the system and how failures occur dependent on the players' efforts. Since the main system fails in period 1 with positive probability, period 2 starts with the standby

Before period 1 each player assesses its effort for period 1 knowing that this effort impacts the probability that the main system survives into period 2. Exerting low defense effort in period 1 increases the probability that the standby system is implemented in period 2. We show that as the value of the standby system decreases below that of the main system, the defender increases its defense of the main system in period 1. Increasing the value of the main system increases the defender's utility more than the attacker's utility since the defender defends the main system more thoroughly in period 1. High defense effort in

As the defender's unit defense costs increases, its efforts and utilities decrease while the attacker's efforts are inverse U shaped and its utility increases. This follows since low unit defense costs make it not worthwile for the inferior attacker to attack, while high unit

Increasing contest intensities for both systems causes all efforts to increase driving utilities downwards eventually reaching zero. Increasing the contest intensity only for the main system causes both efforts for the main system in period 1 to be inverse U shaped but taller for the attacker. This increases the probability that the main system fails in period 1. This benefits the attacker and does not benefit the defender which resorts to defending the standby system in period 2. Increasing discount parameters, making period 2 more

Two limitations of Markov analysis have been illustrated in this paper. First, we have enabled players to choose efforts strategically, which violates the Markov property. Second, we have relaxed the constraint in Markov modeling where the transition rates between different states are kept constant through time. The parameter values for the standby system in period 2 may differ from the parameter values for the main system in period 1. Future research may model in a multi-period game multiple states of operation for the main system

*v*M defender's value of operational main system given presence of a standby system *V*M attacker's value of operational main system given presence of a standby system

defense costs make it unnecessary for the superior attacker to attack substantially.

system with this positive probability.

valuable, benefit both players.

**8. Notation**

period 1 is an investment into the future for the defender.

and the standby system, and repair of the main system.

*p*Mj reliability of main system in period j *p*S2 reliability of standby system in period 2 *c*M defender's unit cost of effort for main system *C*M attacker's unit cost of effort for main system *c*S defender's unit cost of effort for standby system *C*S attacker's unit cost of effort for standby system

*t*Mj defender's effort to protect main system in period j, j=1,2 *T*Mj attacker's effort to attack main system in period j, j=1,2 *t*S2 defender's effort to protect standby system in period 2 *T*S2 attacker's effort to attack standby system in period 2

**6**

*USA*

**Harnessing Efficiency and Building**

**Effectiveness in the Tax Department**

*Director WTP Advisors, Business Advisory Services Practice Leader*

Organizations employ a number of formulas to improve their business processes. These actions typically involve searching for internal cost-savings opportunities, developing departmental strategic relevance and efficiencies, and demonstrating an enhanced focus on increasing profitability. The financial crisis that began in 2007 continues to put stress on companies in the global marketplace. As a result, these companies face extreme pressure to tighten budgets and improve their practices. Nevertheless, the tax function is rarely taken into consideration as a viable arena through which to operate more profitably when organizations look to improve the efficiency and effectiveness of their internal operations. This omission can largely be attributed to the perception of the tax department held by the majority of business executives. That is, the tax department is typically viewed strictly as a cost center, a necessary component for the continued functioning of the organization but unable in itself to generate any revenue. This cost center mentality does not need to be the case. The tax department has the potential to produce data analysis regarding future periods rather than simply reporting on the historical data of the company. This perspective enables the tax department to take on an active role in contributing to the organization's strategic, forward-looking directives. The selective implementation and integration of operations strategy tools and methods, with a focus on maximizing process improvement and efficiency enhancement, can lead to increased value within the tax department. Further, these management and strategy practices will expand the role the tax function plays in facilitating business process improvements, thus leading to increased value within the

A tax department holds four primary responsibilities: compliance activity, tax accounting, risk mitigation, and strategic tax planning. The tax department's domain historically has been a highly specialized and discrete world in which professionals focus primarily on fulfilling a complex web of tax requirements from federal, state, and foreign tax authorities. As a result, complying with legislation has dominated the tax department's agenda in recent years. Surveys indicate that tax professionals spend approximately 84 percent of their time on compliance activity. This "low value added" data management and compliance work inundates many tax personnel preventing them from contributing to strategic organizational

**1. Introduction**

organization as a whole.

**2. The current state of the tax department**

Yair Holtzman and Laura Wells

Hirshleifer, J. (1995). Anarchy and Its Breakdown. Journal of Political Economy 103, 1, 26-52.

## **Harnessing Efficiency and Building Effectiveness in the Tax Department**

Yair Holtzman and Laura Wells

*Director WTP Advisors, Business Advisory Services Practice Leader USA*

#### **1. Introduction**

92 Advanced Topics in Applied Operations Management

Hausken, K. and Levitin, G. (2009), Minmax defense strategy for complex multi-state

Hickey, R.B. (2002). Electrical Construction Databook, McGraw Hill, New York, Chapter 14. Hirshleifer, J. (1995). Anarchy and Its Breakdown. Journal of Political Economy 103, 1, 26-52. Keohane N., Zeckhauser R.J. (2003). The ecology of terror defense. Journal of Risk and

Levitin, G., (2007). Optimal Defense Strategy Against Intentional Attacks. IEEE Transactions

Levitin, G. (2009). Optimizing Defense Strategies for Complex Multi-State Systems. In: Bier

Levitin, G. and Hausken, K. (2009), Redundancy vs. Protection vs. False Targets for Systems

Lisnianski, A. and Levitin, G. (2003), Multi-state system reliability. Assessment,

Nitzan, S. (1994), Modelling Rent-Seeking Contests, European Journal of Political Economy

Ramirez-Marquez, J.E. and Coit, D.W., (2005). A Monte-Carlo simulation approach for

Taylor H.M., Karlin S., (1998), An Introduction To Stochastic Modeling, Third Edition,

Tullock, G. (1980). Efficient Rent-Seeking. In Buchanan, J.M., Tollison, R.D., and Tullock, G.,

Zio, E. and Podofillini, L. (2003), Monte Carlo simulation analysis of the effects of different

approximating multi-state two-terminal reliability, Reliability Engineering &

Toward a Theory of the Rent-Seeking Society, Texas A. & M. University Press,

system performance levels on the importance of multi-state components, Reliability

under Attack, IEEE Transactions on Reliability 58, 1, 58-68.

optimization and applications, World Scientific, New Jersey.

Skaperdas, S., (1996). Contest success functions. Economic Theory 7, 2, 283-290.

VM and Azaiez MN (eds.), Game Theoretic Risk Analysis of Security Threats,

systems, Reliability Engineering & System Safety 94, 2, 577-587.

Uncertainty 26, 201-229.

on Reliability 56, 1, 148-156.

Springer, New York, 33-64.

System Safety 87, 2, 253-264.

Academic Press, New York.

Engineering & System Safety 82, 1, 63-73.

College Station, 97-112.

10, 1, 41-60.

Organizations employ a number of formulas to improve their business processes. These actions typically involve searching for internal cost-savings opportunities, developing departmental strategic relevance and efficiencies, and demonstrating an enhanced focus on increasing profitability. The financial crisis that began in 2007 continues to put stress on companies in the global marketplace. As a result, these companies face extreme pressure to tighten budgets and improve their practices. Nevertheless, the tax function is rarely taken into consideration as a viable arena through which to operate more profitably when organizations look to improve the efficiency and effectiveness of their internal operations. This omission can largely be attributed to the perception of the tax department held by the majority of business executives. That is, the tax department is typically viewed strictly as a cost center, a necessary component for the continued functioning of the organization but unable in itself to generate any revenue. This cost center mentality does not need to be the case. The tax department has the potential to produce data analysis regarding future periods rather than simply reporting on the historical data of the company. This perspective enables the tax department to take on an active role in contributing to the organization's strategic, forward-looking directives. The selective implementation and integration of operations strategy tools and methods, with a focus on maximizing process improvement and efficiency enhancement, can lead to increased value within the tax department. Further, these management and strategy practices will expand the role the tax function plays in facilitating business process improvements, thus leading to increased value within the organization as a whole.

#### **2. The current state of the tax department**

A tax department holds four primary responsibilities: compliance activity, tax accounting, risk mitigation, and strategic tax planning. The tax department's domain historically has been a highly specialized and discrete world in which professionals focus primarily on fulfilling a complex web of tax requirements from federal, state, and foreign tax authorities. As a result, complying with legislation has dominated the tax department's agenda in recent years. Surveys indicate that tax professionals spend approximately 84 percent of their time on compliance activity. This "low value added" data management and compliance work inundates many tax personnel preventing them from contributing to strategic organizational

Harnessing Efficiency and Building Effectiveness in the Tax Department 95

tax accounting. These weak organizational practices play into the inefficiencies of the tax department and highlight an area of immense opportunity for improvement through internal transparency and integration of higher quality control. The poor controls currently in place and the tightened regulations being enacted demonstrate a significant need within the tax department for the production of consistent information and data that is reliable,

Growth and restructuring strategies often result in disorganized and inconsistent production of information, particularly in the area of mergers and acquisitions. Various tax personnel, software systems, and compliance processes are also merged and result in this disorganization and inconsistency. Mergers and acquisitions can lead to each location or business entity reporting using different accounting systems, formatting, or even interpretation of accounting rules. Newly acquired information management systems are often incongruous with a company's established systems. The use of varied systems results in inconsistent data, and often considerable manual work is required to standardize, consolidate, and integrate the data. This incongruity typically results in a conglomeration of manual methods and complex spreadsheets. As a result, accounting and tax personnel are spending an inordinate amount of time and effort collecting, validating, and manipulating data, and they are unable to contribute to the profitability of the company through higher

Furthermore, the prior system often continues to be run concurrently with the new system when new software is implemented. This practice is beneficial in the short term, but the duration of this concurrent operation takes place much longer than is necessary for implementation purposes. Delaying the completion of the transition adds additional steps rather than simplifying the process. Compiling data for reporting cycles in multiple systems

Many companies report for management and tax purposes on different bases, whether that is by business unit or legal entity. This split in reporting further complicates the procurement of quality tax information. The two sets of information do not always reconcile easily. Handling data procurement issues and irreconcilable reports leaves little opportunity for the tax department to become deeply involved in higher level analysis and high value added activities such as tax planning and strategizing. Accordingly, tax is disconnected from contributing to fundamental corporate strategies or forward-looking directives. For obvious reasons, the absence of tax professionals in the strategic decision making process is

The integration of tax accounting systems and financial reporting systems enables tax to take on a more active role in organizational planning and strategy. An enterprise needs systems in place that produce the data and information required for tax planning, tax compliance, and developing and evaluating tax strategy. In most companies, numerous systems feed into a core general ledger, and no single integrated system can easily produce data for all required finance, accounting, and tax purposes. The resulting process inefficiency leads to data complexity, manual procedures, and a lack of data control. Companies can create consistency between the data sets of various departments by linking their general ledger to a single accounting system. Data requirements differ among departments, yet all parties benefit from pulling information within a consolidated, complete set of data. Leveraging its use of technology significantly impacts a tax

detrimental not only to the department but also to the company as a whole.

organized, and coherent.

value added activities.

becomes an extremely complex, laborious process.

goals. Contributing to strategic organizational objectives requires that the appropriate tax employees have the capacity to perform forward looking and "high value added" analysis including strategic planning activity as part of the leadership team. Assume this figure of 84 percent is correct and that these individuals work a 40 hour work week. Cutting the time spent on compliance activity by a conservative 25 percent would provide each individual with the opportunity to expend between 400 and 500 additional hours per year on higher value added activity.

The increased examination of the tax function by shareholders, the audit committee, and the executive suite, in conjunction with increased regulatory requirements, is trapping the tax department in a low value loop, thus preventing the department from achieving its potential to contribute strategic value to the organization. This low value loop results as companies attempt to collect and report information in accordance with tightened governmental control.

The prevalence of increased governmental involvement is demonstrated through regulations such as the Sarbanes-Oxley Act of 2002. The Sarbanes-Oxley Act (HR 4173) set new standards for the corporate management and accounting processes of publicly traded companies, and it requires strict documentation of business process controls affecting financial reporting. Section 404(b) is considered one of the most pertinent to compliance because it requires companies to report on the managerial assessment of their internal controls. Sarbanes-Oxley-related compliance initiatives play a major role in driving change within documentation, compliance, and reporting measures of the tax function in recent years.

Additionally, the Federal Accounting Standards Board, through measures such as FAS 109 and FIN 48, has involved itself in working to create more transparency. The Board determined that "financial statements should reflect the current and deferred tax consequences of all events that have been recognized in the financial statements or tax returns" (63). FAS 109 requires companies to establish deferred taxes for temporary, or timing, differences in their assets and liabilities. That is, as defined in the FAS 109 objectives, companies must account for differences "between the tax basis of an asset or a liability and its reported amount in the statement of financial position [that] will result in taxable or deductible amounts in some future year(s) when the reported amounts of assets are recovered and the reported amounts of liabilities are settled." This measure affects the amount of disclosure required of companies.

Legislative changes and updated regulations affecting tax department responsibilities continue to transpire. More recently, in 2010, the IRS released Schedule UTP (Uncertain Tax Positions), requiring designated businesses to report uncertain tax positions on their tax returns. Schedule UTP is being phased-in over a five year period. This has allowed many tax departments to delay addressing the issue.

Amidst their efforts to comply with these legislative measures, many companies have found their tax processes to be poorly controlled. Several factors play into these poor controls in addition to legislative measures. The tax function adheres to calendars and timetables unique from other departments, and it uses different processes and systems, accounting methods, and reporting standards than the rest of the finance function or the business at large. Studies have identified that one third of material weaknesses in controls were tied to

goals. Contributing to strategic organizational objectives requires that the appropriate tax employees have the capacity to perform forward looking and "high value added" analysis including strategic planning activity as part of the leadership team. Assume this figure of 84 percent is correct and that these individuals work a 40 hour work week. Cutting the time spent on compliance activity by a conservative 25 percent would provide each individual with the opportunity to expend between 400 and 500 additional hours per year on higher

The increased examination of the tax function by shareholders, the audit committee, and the executive suite, in conjunction with increased regulatory requirements, is trapping the tax department in a low value loop, thus preventing the department from achieving its potential to contribute strategic value to the organization. This low value loop results as companies attempt to collect and report information in accordance with tightened governmental

The prevalence of increased governmental involvement is demonstrated through regulations such as the Sarbanes-Oxley Act of 2002. The Sarbanes-Oxley Act (HR 4173) set new standards for the corporate management and accounting processes of publicly traded companies, and it requires strict documentation of business process controls affecting financial reporting. Section 404(b) is considered one of the most pertinent to compliance because it requires companies to report on the managerial assessment of their internal controls. Sarbanes-Oxley-related compliance initiatives play a major role in driving change within documentation, compliance, and reporting measures of the tax function in recent

Additionally, the Federal Accounting Standards Board, through measures such as FAS 109 and FIN 48, has involved itself in working to create more transparency. The Board determined that "financial statements should reflect the current and deferred tax consequences of all events that have been recognized in the financial statements or tax returns" (63). FAS 109 requires companies to establish deferred taxes for temporary, or timing, differences in their assets and liabilities. That is, as defined in the FAS 109 objectives, companies must account for differences "between the tax basis of an asset or a liability and its reported amount in the statement of financial position [that] will result in taxable or deductible amounts in some future year(s) when the reported amounts of assets are recovered and the reported amounts of liabilities are settled." This measure affects the

Legislative changes and updated regulations affecting tax department responsibilities continue to transpire. More recently, in 2010, the IRS released Schedule UTP (Uncertain Tax Positions), requiring designated businesses to report uncertain tax positions on their tax returns. Schedule UTP is being phased-in over a five year period. This has allowed many tax

Amidst their efforts to comply with these legislative measures, many companies have found their tax processes to be poorly controlled. Several factors play into these poor controls in addition to legislative measures. The tax function adheres to calendars and timetables unique from other departments, and it uses different processes and systems, accounting methods, and reporting standards than the rest of the finance function or the business at large. Studies have identified that one third of material weaknesses in controls were tied to

value added activity.

control.

years.

amount of disclosure required of companies.

departments to delay addressing the issue.

tax accounting. These weak organizational practices play into the inefficiencies of the tax department and highlight an area of immense opportunity for improvement through internal transparency and integration of higher quality control. The poor controls currently in place and the tightened regulations being enacted demonstrate a significant need within the tax department for the production of consistent information and data that is reliable, organized, and coherent.

Growth and restructuring strategies often result in disorganized and inconsistent production of information, particularly in the area of mergers and acquisitions. Various tax personnel, software systems, and compliance processes are also merged and result in this disorganization and inconsistency. Mergers and acquisitions can lead to each location or business entity reporting using different accounting systems, formatting, or even interpretation of accounting rules. Newly acquired information management systems are often incongruous with a company's established systems. The use of varied systems results in inconsistent data, and often considerable manual work is required to standardize, consolidate, and integrate the data. This incongruity typically results in a conglomeration of manual methods and complex spreadsheets. As a result, accounting and tax personnel are spending an inordinate amount of time and effort collecting, validating, and manipulating data, and they are unable to contribute to the profitability of the company through higher value added activities.

Furthermore, the prior system often continues to be run concurrently with the new system when new software is implemented. This practice is beneficial in the short term, but the duration of this concurrent operation takes place much longer than is necessary for implementation purposes. Delaying the completion of the transition adds additional steps rather than simplifying the process. Compiling data for reporting cycles in multiple systems becomes an extremely complex, laborious process.

Many companies report for management and tax purposes on different bases, whether that is by business unit or legal entity. This split in reporting further complicates the procurement of quality tax information. The two sets of information do not always reconcile easily. Handling data procurement issues and irreconcilable reports leaves little opportunity for the tax department to become deeply involved in higher level analysis and high value added activities such as tax planning and strategizing. Accordingly, tax is disconnected from contributing to fundamental corporate strategies or forward-looking directives. For obvious reasons, the absence of tax professionals in the strategic decision making process is detrimental not only to the department but also to the company as a whole.

The integration of tax accounting systems and financial reporting systems enables tax to take on a more active role in organizational planning and strategy. An enterprise needs systems in place that produce the data and information required for tax planning, tax compliance, and developing and evaluating tax strategy. In most companies, numerous systems feed into a core general ledger, and no single integrated system can easily produce data for all required finance, accounting, and tax purposes. The resulting process inefficiency leads to data complexity, manual procedures, and a lack of data control. Companies can create consistency between the data sets of various departments by linking their general ledger to a single accounting system. Data requirements differ among departments, yet all parties benefit from pulling information within a consolidated, complete set of data. Leveraging its use of technology significantly impacts a tax

Harnessing Efficiency and Building Effectiveness in the Tax Department 97

contributing to internal controls and risk management initiatives, and collaborating with

The tax department depends on management investment to avoid tax-related errors in financial statements and to contribute to tax planning activities. Additionally, investment contributes to building managerial and technical skills among senior tax executives. These skills are critical to the future success of the team. Consequently, management must be willing to expand investment if it would like to see the tax department move forward.

The department can immerse itself in high value-adding strategy and planning activity as it moves forward. Such immersion must be layered on top of the tax function's core compliance and tax planning activities. Lowering the effective tax rate through strategic tax planning could serve as a channel for tax to play a comprehensive strategic role. Other potential contributions include facilitating greater efficiency and standardization of processes, recording and analyzing data, and reporting. As a result, tax involvement early on in strategic business operations processes is worthwhile because the initial framing of business decisions will not only affect the overall success of the business but also affect their future core tax activities. For example, downstream activities, such as tax compliance, accounting, and record keeping, would be more efficient, less risky, and less time

Funding, automation, and technology require additional input and tools to achieve optimal results in regard to tax efficiency. Organizational responsibilities and related tax resources must be aligned with the vision and objectives of the business to create exceptional results. A framework can provide several dimensions of analysis, including value drivers, processes, and enablers that help optimize tax decision making. In global organizations, management must examine decisions in light of the tax value drivers that align to broader

In this section we introduce some of the key concepts and mathematical tools that can be effectively utilized in facilitating the implementation of the improvements discussed

1. Scheduling –Typically, each resource unit is scheduled for operation only a designated portion of total time (e.g., eight hours per day, five days per week). The amount of time a resource is scheduled for operation is called the **scheduled availability of the resource**. Scheduled availability of various resources in a process may differ. For example, in a manufacturing setting, some areas within a plant operate only one shift per day (8 hours) while others operate two (16 hours). The same time availability analysis could apply to the tax department due to the use of co-sourcing and outside consultants. Moreover, the choice of one day as the time period of measurement assumes that availability patterns repeat on a daily basis. More complicated patterns are possible as well. Some resource pools, for example, may be available only two days a week, with the pattern repeating every week. In that case, measure scheduled

others on finance and accounting matters that are not directly linked to tax.

consuming for the tax department and for the company at large.

corporate goals and objectives.

above.

Quantitative Tools:

**4. Key tax processes and tax process tools**

availability as the number of hours per week.

department's ability to achieve a higher performance level and move beyond its role as a cost center. In their present form, tax controls and competencies rely heavily on individuals' skill sets. As a result, problems arise if staff turnover is high or available resources are mixed inappropriately. The automation of processes and consolidation of systems not only saves time and enables employees to contribute to more value-add activities, but it also leads to increased quality control.

A tax department loses a great deal of time as regulatory requirements tighten audit processes and increase the need for careful documentation. Detailed documentation ensures the presence of proper financial controls. Problematic audits arise when a company's various branches and entities lack integration or use several different accounting and tax systems. Misaligned data leads to wasted steps, thus wasted time and money, and generates risk of reporting errors, penalties, interest, and additional fees. Automation puts controls in place to monitor and better control data and information flow. This control improves audit trails and reduces risk levels.

Departments can effectively manage data and mitigate tax risk through carefully crafted tax processes involving technology and automation. The organization potentially faces high implementation costs; nevertheless, the resulting benefit of risk reduction offsets those costs. The department can detect errors more rapidly, avoid penalties and interest, and improve return on technological investment. Though technology in itself does not guarantee improvement, it assists in the betterment of the organization when applied appropriately. When mapped out, return completion and compliance are simply a function of processes. Consequently, the tax department can direct technological improvements appropriately by gaining a complete understanding of compliance processes and determining specific points of inefficiency. The department then can reduce and eliminate these inefficiencies.

A focused and concerted effort to improve processes around the areas of tax planning, tax risk management, and audit defense results in reducing both cost and time required to perform basic activity. At the same time, process improvement advances quality and reduces risk. It enables employees to contribute to additional high value-adding activities. Furthermore, as the tax function expands capability and increases productivity through internal process enhancements, improved technological resources, the department will develop a stronger and more collaborative relationship with other sectors of the corporation. As this indicates, the tax department contains potential to expand beyond its role as a cost center and instead become a profit center. Operations management techniques and strategies hold potential to transform the tax department into an efficient and effective revenue-generating contributor to the organization.

#### **3. The future state of the tax department**

Legislative developments, such as Sarbanes-Oxley, force the tax function to expand capacity beyond traditional core areas of compliance. In addition, since 2003, external forces such as new accounting standards and increased regulatory scrutiny require new capabilities among tax teams. In response to these dynamics, the tax department is undergoing a metamorphosis. Tax departments are recasting business processes, building stronger more heterogeneous leadership teams, and investing in new technology. The tax function is moving toward a more active role in supporting major transactions and operating decisions, contributing to internal controls and risk management initiatives, and collaborating with others on finance and accounting matters that are not directly linked to tax.

The tax department depends on management investment to avoid tax-related errors in financial statements and to contribute to tax planning activities. Additionally, investment contributes to building managerial and technical skills among senior tax executives. These skills are critical to the future success of the team. Consequently, management must be willing to expand investment if it would like to see the tax department move forward.

The department can immerse itself in high value-adding strategy and planning activity as it moves forward. Such immersion must be layered on top of the tax function's core compliance and tax planning activities. Lowering the effective tax rate through strategic tax planning could serve as a channel for tax to play a comprehensive strategic role. Other potential contributions include facilitating greater efficiency and standardization of processes, recording and analyzing data, and reporting. As a result, tax involvement early on in strategic business operations processes is worthwhile because the initial framing of business decisions will not only affect the overall success of the business but also affect their future core tax activities. For example, downstream activities, such as tax compliance, accounting, and record keeping, would be more efficient, less risky, and less time consuming for the tax department and for the company at large.

Funding, automation, and technology require additional input and tools to achieve optimal results in regard to tax efficiency. Organizational responsibilities and related tax resources must be aligned with the vision and objectives of the business to create exceptional results. A framework can provide several dimensions of analysis, including value drivers, processes, and enablers that help optimize tax decision making. In global organizations, management must examine decisions in light of the tax value drivers that align to broader corporate goals and objectives.

#### **4. Key tax processes and tax process tools**

In this section we introduce some of the key concepts and mathematical tools that can be effectively utilized in facilitating the implementation of the improvements discussed above.

Quantitative Tools:

96 Advanced Topics in Applied Operations Management

department's ability to achieve a higher performance level and move beyond its role as a cost center. In their present form, tax controls and competencies rely heavily on individuals' skill sets. As a result, problems arise if staff turnover is high or available resources are mixed inappropriately. The automation of processes and consolidation of systems not only saves time and enables employees to contribute to more value-add activities, but it also leads to

A tax department loses a great deal of time as regulatory requirements tighten audit processes and increase the need for careful documentation. Detailed documentation ensures the presence of proper financial controls. Problematic audits arise when a company's various branches and entities lack integration or use several different accounting and tax systems. Misaligned data leads to wasted steps, thus wasted time and money, and generates risk of reporting errors, penalties, interest, and additional fees. Automation puts controls in place to monitor and better control data and information flow. This control improves audit

Departments can effectively manage data and mitigate tax risk through carefully crafted tax processes involving technology and automation. The organization potentially faces high implementation costs; nevertheless, the resulting benefit of risk reduction offsets those costs. The department can detect errors more rapidly, avoid penalties and interest, and improve return on technological investment. Though technology in itself does not guarantee improvement, it assists in the betterment of the organization when applied appropriately. When mapped out, return completion and compliance are simply a function of processes. Consequently, the tax department can direct technological improvements appropriately by gaining a complete understanding of compliance processes and determining specific points

A focused and concerted effort to improve processes around the areas of tax planning, tax risk management, and audit defense results in reducing both cost and time required to perform basic activity. At the same time, process improvement advances quality and reduces risk. It enables employees to contribute to additional high value-adding activities. Furthermore, as the tax function expands capability and increases productivity through internal process enhancements, improved technological resources, the department will develop a stronger and more collaborative relationship with other sectors of the corporation. As this indicates, the tax department contains potential to expand beyond its role as a cost center and instead become a profit center. Operations management techniques and strategies hold potential to transform the tax department into an efficient and effective

Legislative developments, such as Sarbanes-Oxley, force the tax function to expand capacity beyond traditional core areas of compliance. In addition, since 2003, external forces such as new accounting standards and increased regulatory scrutiny require new capabilities among tax teams. In response to these dynamics, the tax department is undergoing a metamorphosis. Tax departments are recasting business processes, building stronger more heterogeneous leadership teams, and investing in new technology. The tax function is moving toward a more active role in supporting major transactions and operating decisions,

of inefficiency. The department then can reduce and eliminate these inefficiencies.

revenue-generating contributor to the organization.

**3. The future state of the tax department**

increased quality control.

trails and reduces risk levels.

1. Scheduling –Typically, each resource unit is scheduled for operation only a designated portion of total time (e.g., eight hours per day, five days per week). The amount of time a resource is scheduled for operation is called the **scheduled availability of the resource**. Scheduled availability of various resources in a process may differ. For example, in a manufacturing setting, some areas within a plant operate only one shift per day (8 hours) while others operate two (16 hours). The same time availability analysis could apply to the tax department due to the use of co-sourcing and outside consultants. Moreover, the choice of one day as the time period of measurement assumes that availability patterns repeat on a daily basis. More complicated patterns are possible as well. Some resource pools, for example, may be available only two days a week, with the pattern repeating every week. In that case, measure scheduled availability as the number of hours per week.

Harnessing Efficiency and Building Effectiveness in the Tax Department 99

1. Early Start Time (ES): The earliest time at which an activity can start, considering the

2. Early Finish Time (EF): The sum of the early start time (ES) and the time required to

3. Late Start Time (LS): The latest time at which an activity can start, considering all of the precedence relationships, without delaying the completion time for the project. 4. Late Finish Time (LF): The sum of the late start time and the time required to complete

CPM requires calculations of the four parameters (ES, EF, LS, and LF) for each project activity. The implementation of the CPM algorithm begins by identifying ES and EF for all activities. The procedure starts with the first activity of a project, which has no predecessor. Once an analyst identifies ES and EF for the first activityhe repeats the same procedure for subsequent activity. . Relationships with previously completed activities must be taken into consideration beginning with the second activity. As a result, complete the calculations of

The essential technique for using CPM is to construct a model of the project that includes: 1. A list of all activities required to complete the project, typically categorized within a

Using these values, CPM calculates the longest path of planned activities to the end of the project, and the earliest and latest that each activity can start and finish without making the project longer. This process determines which activities are "critical" (i.e. on the longest path) and which have "total float" (i.e. can be delayed without making the project longer). A critical path in project management is the sequence of project network activities that add up to the longest overall duration. This sequence determines the shortest time possible to complete the project. Any delay of an activity on the critical path directly impacts the planned project completion date (i.e. there is no float on the critical path). A project also can have several parallel, near critical paths. An additional parallel path through the network of

total duration shorter than the critical path is called a sub-critical or non-critical path.

These results allow managers to prioritize activities for the effective management of project completion and to shorten the planned critical path of a project. Managers shorten the planned critical path by pruning critical path activities, "fast tracking" (i.e. performing more activities in parallel), and "crashing the critical path" (i.e. shortening the durations of critical path activities by adding resources). Managers can apply these concepts with success

Just as in manufacturing, principles exist to create flow in the tax function. The concept of a workflow cycle sets in place physical pathways and information flows and provides the timing of information flow down these pathways. As a result, tax personnel know where and when information should flow, and they can determine whether this flow is on time. Connecting processes with fixed pathways and binary signals creates this value stream flow. Essentially, individuals design a system for flow of data and information by flow paths with binary switches, similar to those in an electric circuit. Information and data are moved based on signals (completed/not-completed) along fixed pathways. Implementation of this

beginning and ending times for each of the preceding activities.

ES and EF carefully so that none of the precedence relationships are ignored.

2. The time (duration) that each activity will take to completion.

complete the activity.

work breakdown structure.

3. The requirements of each particular activity.

toward the optimization of the tax department.

the activity.

Taking into account the load batching and the scheduled availability of a given resource provides a general expression for the theoretical capacity of a resource unit:

 **Theoretical capacity of a resource unit** = (1/Tp) X Load Batch X Scheduled Availability (1)

The Theoretical Capacity of resource pool p (Rp) is given by the number of resources in resource pool p (Cp) times the theoretical capacity of a resource unit in a pool. This expression translates to:

$$\mathbf{Rp} = \left(\mathbf{Cp}/\mathbf{Tp}\right) \mathbf{X} \text{ Load batch } \mathbf{X} \text{Scheduled Available} \tag{2}$$

For example, consider a resource pool containing two tax managers, where each manager can handle a maximum of 10 tasks simultaneously. On average, the time to research and resolve each task is 15 minutes. Finally, assume that the two managers are scheduled to work 7.5 hours per day (450 minutes per day). Identify the following parameters to calculate the theoretical capacity of the pool:

Cp = Number of resources = 2 tax managers (3)

$$\text{Tp} = \text{Unit Load} = 15 \text{ minutes} \tag{4}$$

$$\text{Load batch} = 10 \text{ tasks to be completed} \tag{5}$$

$$\text{Scheduled availability} = \text{450 minutes per day} \tag{6}$$

Calculate the theoretical capacity of the two tax managers' pool using the Formula (2) above:

Rp = (2/15) X 10 X 450 = 600 tasks can be addressed per day by the group of two. (7)

2. Critical Path Analysis— The critical path method (CPM) is an algorithm for scheduling activities within a project. CPM determines the fastest and most efficient execution of that project. The algorithm, originally developed by DuPont and Remington Rand Corporation in the 1950s, is an essential project management technique. The CPM algorithm estimates the time necessary to complete each part of the project utilizing the information presented in the work breakdown structure and the precedence relationship and time estimates. All paths within a project need to be finished before the project can be considered complete. The **critical path**, also known as the bottleneck path or the binding constraint, is the path that takes the longest time to complete. This critical path dictates the project's duration. The activities making up a critical path are known as **critical activities.** Any delay in the execution of a critical activity results in delaying the entire project. The analyst also needs to understand how much slack, or flexibility, exists in scheduling noncritical activities. Slack is the estimate of the maximum amount of time that a noncritical activity can be delayed without affecting the entire project schedule. Therefore, analysts use a systematic algorithm to calculate the critical path and identify slack for each activity. A computerized systematic algorithm implements the CPM approach for large projects because they contain a sizeable number of paths and activities.

The algorithm for identifying the critical path and slack involves calculating the following four parameters for each activity:

Taking into account the load batching and the scheduled availability of a given resource

 **Theoretical capacity of a resource unit** = (1/Tp) X Load Batch X Scheduled Availability (1) The Theoretical Capacity of resource pool p (Rp) is given by the number of resources in resource pool p (Cp) times the theoretical capacity of a resource unit in a pool. This

For example, consider a resource pool containing two tax managers, where each manager can handle a maximum of 10 tasks simultaneously. On average, the time to research and resolve each task is 15 minutes. Finally, assume that the two managers are scheduled to work 7.5 hours per day (450 minutes per day). Identify the following parameters to calculate

Calculate the theoretical capacity of the two tax managers' pool using the Formula (2) above:

 Rp = (2/15) X 10 X 450 = 600 tasks can be addressed per day by the group of two. (7) 2. Critical Path Analysis— The critical path method (CPM) is an algorithm for scheduling activities within a project. CPM determines the fastest and most efficient execution of that project. The algorithm, originally developed by DuPont and Remington Rand Corporation in the 1950s, is an essential project management technique. The CPM algorithm estimates the time necessary to complete each part of the project utilizing the information presented in the work breakdown structure and the precedence relationship and time estimates. All paths within a project need to be finished before the project can be considered complete. The **critical path**, also known as the bottleneck path or the binding constraint, is the path that takes the longest time to complete. This critical path dictates the project's duration. The activities making up a critical path are known as **critical activities.** Any delay in the execution of a critical activity results in delaying the entire project. The analyst also needs to understand how much slack, or flexibility, exists in scheduling noncritical activities. Slack is the estimate of the maximum amount of time that a noncritical activity can be delayed without affecting the entire project schedule. Therefore, analysts use a systematic algorithm to calculate the critical path and identify slack for each activity. A computerized systematic algorithm implements the CPM approach for large projects because they contain a

The algorithm for identifying the critical path and slack involves calculating the following

Rp = (Cp/Tp) X Load Batch X Scheduled Availability (2)

Cp = Number of resources = 2 tax managers (3)

Tp= Unit Load = 15 minutes (4)

Load batch = 10 tasks to be completed (5)

Scheduled availability = 450 minutes per day (6)

provides a general expression for the theoretical capacity of a resource unit:

expression translates to:

the theoretical capacity of the pool:

sizeable number of paths and activities.

four parameters for each activity:

CPM requires calculations of the four parameters (ES, EF, LS, and LF) for each project activity. The implementation of the CPM algorithm begins by identifying ES and EF for all activities. The procedure starts with the first activity of a project, which has no predecessor. Once an analyst identifies ES and EF for the first activityhe repeats the same procedure for subsequent activity. . Relationships with previously completed activities must be taken into consideration beginning with the second activity. As a result, complete the calculations of ES and EF carefully so that none of the precedence relationships are ignored.

The essential technique for using CPM is to construct a model of the project that includes:

Using these values, CPM calculates the longest path of planned activities to the end of the project, and the earliest and latest that each activity can start and finish without making the project longer. This process determines which activities are "critical" (i.e. on the longest path) and which have "total float" (i.e. can be delayed without making the project longer). A critical path in project management is the sequence of project network activities that add up to the longest overall duration. This sequence determines the shortest time possible to complete the project. Any delay of an activity on the critical path directly impacts the planned project completion date (i.e. there is no float on the critical path). A project also can have several parallel, near critical paths. An additional parallel path through the network of total duration shorter than the critical path is called a sub-critical or non-critical path.

These results allow managers to prioritize activities for the effective management of project completion and to shorten the planned critical path of a project. Managers shorten the planned critical path by pruning critical path activities, "fast tracking" (i.e. performing more activities in parallel), and "crashing the critical path" (i.e. shortening the durations of critical path activities by adding resources). Managers can apply these concepts with success toward the optimization of the tax department.

Just as in manufacturing, principles exist to create flow in the tax function. The concept of a workflow cycle sets in place physical pathways and information flows and provides the timing of information flow down these pathways. As a result, tax personnel know where and when information should flow, and they can determine whether this flow is on time. Connecting processes with fixed pathways and binary signals creates this value stream flow. Essentially, individuals design a system for flow of data and information by flow paths with binary switches, similar to those in an electric circuit. Information and data are moved based on signals (completed/not-completed) along fixed pathways. Implementation of this

Harnessing Efficiency and Building Effectiveness in the Tax Department 101

Tax departments want to maximize (8) subject to (9) and (10). A company with scarce tax resources needs to address and optimize this standard maximum problem in its desire to advance its delivery of error free on time tax return deliverables and development of tax planning and tax strategy capabilities. A company also needs to address and optimize the

The Optimal Assignment Problem.  The value of person i

An assignment is a choice of numbers, xij , for i = 1, . . . , I, and j = 1, . . . , J, where xij

Equation (11) reflects the fact that a person cannot spend more than 100% of his time working, (12) means that only one person is allowed on a job at a time, and (13) says that no one can work a negative amount of time on any job. Maximize the total value subject to (11),

This optimal assignment calculation represents a standard maximum problem with m = I + J

A vector, x for the standard maximum problem or y for the standard minimum problem, is said to be feasible if it satisfies the corresponding constraints.The set of feasible vectors is

A linear programming problem is feasible if the constraint set is not empty. The problem is

A feasible maximum (resp. minimum) problem is unbounded if the objective function can assume arbitrarily large positive (resp. negative) values at feasible vectors; otherwise, the feasible maximum problem is considered bounded. Thus, three possibilities are available for

The function to be maximized or minimized is called the objective function.

(14)

j = 1, . . . , J . Choose an assignment of persons to jobs to maximize the total value.

represents the proportion of person i 's time that is to be spent on job j. Thus,

assignment of these scarce resources.

(12) and (13),

and n = IJ .

**5. Terminology**

called the constraint set.

infeasible if the constraint set is empty.

working 1 day at job j is aij , for i = 1, . . . , I, and

technique proves successful, for instance, in regard to the timely and accurate review of tax provision. Certainly it takes time to design and implement a robust value stream flow, but once it is in place, the benefits dramatically outweigh this initial effort.

3. Constrained resource allocation--- Many project management techniques assume that a project team has the necessary resources to complete each activity within the specified function and budget. In reality, this might not be the case. Consequently, a number of supplementary techniques allocate resources to different project activities effectively. A resource breakdown structure and resource leveling are two commonly used resource management techniques. A **resource breakdown structure (RBS)** is a standardized list of personnel required to complete various activities in a project. This technique is often used in combination with the work breakdown structure. **Resource leveling** is an approach used to reduce the number of fluctuations in day-to-day resource requirements within an organization. This approach is especially useful when employees in an organization work on multiple projects simultaneously. Each project might have separate deadlines and a different set of resource requirements during different time periods. The number of employees typically remains constant. Therefore, the resource leveling approach is used to adjust the project schedule so that almost the same amount of personnel time is required every day to work on different projects.

Two problems arise in the deployment of scarce resources: the activity analysis problem and the optimal assignment problem

The Activity Analysis Problem. A company can employ n activities, A1, A2. .. . ,An , using the available supply of m resources, R1, R2. . . . , Rm (labor hours, computing capability, tax software availability, etc.). Let bi be the available supply of resource Ri. Let aij be the amount of resource Ri used in operating activity Aj at unit intensity. Let cj be the net value to the company of operating activity Aj at unit intensity. Choose the intensities with which the various activities are to be operated to maximize the value of the output to the company subject to the given resources.

Let xj be the intensity at which Aj is to be operated. The value of such an activity allocation is

$$\sum\_{j=1}^{n\_c} c\_j x\_j. \tag{8}$$

The amount of resource Ri used in this activity allocation must be no greater than the supply, bi; that is,

$$\sum\_{j=1} a\_{ij} x\_j \le b\_i \qquad \text{for } i = 1, \dots, m. \tag{9}$$

It is assumed that an activity at negative intensity cannot be operated; that is,

$$x\_1 \ge 0, x\_2 \ge 0, \dots, x\_n \ge 0. \tag{10}$$

Tax departments want to maximize (8) subject to (9) and (10). A company with scarce tax resources needs to address and optimize this standard maximum problem in its desire to advance its delivery of error free on time tax return deliverables and development of tax planning and tax strategy capabilities. A company also needs to address and optimize the assignment of these scarce resources.

The Optimal Assignment Problem.  The value of person i working 1 day at job j is aij , for i = 1, . . . , I, and

j = 1, . . . , J . Choose an assignment of persons to jobs to maximize the total value.

An assignment is a choice of numbers, xij , for i = 1, . . . , I, and j = 1, . . . , J, where xij represents the proportion of person i 's time that is to be spent on job j. Thus,

$$\sum\_{j=1}^{J} x\_{ij} \le 1 \qquad \text{for } i = 1, \ldots, I \tag{11}$$

$$\sum\_{i=1}^{I} x\_{ij} \le 1 \qquad \text{for } j = 1, \dots, J \tag{12}$$

100 Advanced Topics in Applied Operations Management

technique proves successful, for instance, in regard to the timely and accurate review of tax provision. Certainly it takes time to design and implement a robust value stream flow, but

3. Constrained resource allocation--- Many project management techniques assume that a project team has the necessary resources to complete each activity within the specified function and budget. In reality, this might not be the case. Consequently, a number of supplementary techniques allocate resources to different project activities effectively. A resource breakdown structure and resource leveling are two commonly used resource management techniques. A **resource breakdown structure (RBS)** is a standardized list of personnel required to complete various activities in a project. This technique is often used in combination with the work breakdown structure. **Resource leveling** is an approach used to reduce the number of fluctuations in day-to-day resource requirements within an organization. This approach is especially useful when employees in an organization work on multiple projects simultaneously. Each project might have separate deadlines and a different set of resource requirements during different time periods. The number of employees typically remains constant. Therefore, the resource leveling approach is used to adjust the project schedule so that almost the same amount of personnel time is required every day to work on different projects. Two problems arise in the deployment of scarce resources: the activity analysis problem and

The Activity Analysis Problem. A company can employ n activities, A1, A2. .. . ,An , using the available supply of m resources, R1, R2. . . . , Rm (labor hours, computing capability, tax software availability, etc.). Let bi be the available supply of resource Ri. Let aij be the amount of resource Ri used in operating activity Aj at unit intensity. Let cj be the net value to the company of operating activity Aj at unit intensity. Choose the intensities with which the various activities are to be operated to maximize the value of the output to the company

Let xj be the intensity at which Aj is to be operated. The value of such an activity allocation

The amount of resource Ri used in this activity allocation must be no greater than the

It is assumed that an activity at negative intensity cannot be operated; that is,

(8)

(9)

(10)