data uncertainty widely exists in realistic problems due to their 
 random nature , measurement errors , or other reasons . as a result , 

decision making inherently involves consideration of such uncertainties 
 since the solution of an optimization problem often exhibits high 
 sensitivity to data perturbations , and ignoring the uncertainty could 
 lead to suboptimal or even infeasible solutions . in past decades , 

developing optimization methods and tools to facilitate decision making 
 under uncertainty has become one of the most important topics in both 
 the operations research community and also the process systems engineering 
 community .
robust optimization belongs to an important methodology 
 for dealing with optimization problems with data uncertainty .
this 
 type of method enforces the constraint satisfaction for all possible 
 realizations of uncertain parameters inside a predefined uncertainty 
 set . comparing it to other methodologies that deal with uncertainty , 

one major motivation of robust optimization is that in many applications 
 the data set is an appropriate notion of parameter uncertainty , especially 
 for those cases that the parameter uncertainty is not stochastic , 
 or for instances where no distributional information is available .
one of the earliest papers on robust counterpart optimization is 
 the work of soyster , who considered simple 
 perturbations in the data and aimed to find a reformulation of the 
 original linear programming problem such that the resulting solution 
 would be feasible under all possible perturbations .
the approach admits 
 the highest protection and is the most conservative one since it ensures 
 feasibility against all potential realizations .
thus , it is highly 
 desirable to provide a mechanism to allow for the trade - off between 
 robustness and performance .
the work by ben - tal and nemirovski , el - ghaoui et al . , and bertsimas and sim investigated the framework of robust counterpart 
 optimization by introducing different types of uncertainty sets .
ben - tal 
 and nemirovski proposed the ellipsoidal 
 set based robust counterpart formulation .
el - ghaoui and lebret introduced a robust optimization approach for 
 least - squares problems with uncertain data .
bertsimas and sim studied robust linear programming with coefficient 
 uncertainty using an uncertainty set with budgets . in this robust 
 counterpart optimization formulation , a budget parameter ( which takes 
 a value between zero and the number of uncertain coefficient parameters 
 in the constraints and is not necessarily integer ) is introduced to 
 control the degree of conservatism of the solution .
developed 
 the theory of the robust optimization framework for general mixed - integer 
 linear programming problems and considered both bounded uncertainty 
 and several known probability distributions .
the robust optimization 
 framework was later extended by verderame and floudas and they studied both continuous ( general , bounded , uniform , 
 normal ) and discrete ( general , binomial , poisson ) uncertainty distributions 
 and applied the framework to operational planning problems .
the work 
 was further compared with the conditional value - at - risk based method 
 in verderame and floudas . in the first 
 two parts of this paper series
, we systematically studied 
 the set induced robust counterpart optimization technique for linear 
 and mixed integer linear optimization problems .
different uncertainty 
 sets were extensively studied , including those studied in literature 
 and novel ones were introduced in this work .
the relationship between 
 different representative uncertainty sets was discussed , and their 
 corresponding robust counterpart formulations for both linear optimization 
 ( lp ) and mixed integer linear optimization ( milp ) problems were derived . 

probabilistic guarantees on constraint satisfaction for the robust 
 solution of those different uncertainty set induced robust counterpart 
 optimization models were derived , for both bounded and unbounded uncertainty , 
 with and without detail probability distribution information . a key element in applying the robust optimization framework is the 
 selection of the type and the size of the uncertainty set , which is 
 strongly related to the desired reliability of the solution ( i.e. , 
 the probability of constraint satisfaction ) .
it is known that chance 
 constraint / probabilistic constraint is the most direct way to enforce 
 the reliability of the solution of an optimization problem , where the reliability is expressed as a minimum requirement on the 
 probability of satisfying constraints .
for example , 
 even evaluating the distribution of a sum of uniformly distributed 
 independent random variables is very difficult .
when the program has structural properties that allow for 
 an equivalent deterministic formulation , a chance constrained problem 
 can be converted to a deterministic problem and can be solved directly . however
, if the model does not admit sufficient 
 structure that can be exploited , an approximation method has to be 
 used .
the various approximation methods can be divided into sampling 
 based methods and analytical approximation based methods .
first , 
 sampling based methods are designed based on the assumption that it 
 is possible to draw observations from the distribution of the uncertainty . 
 sampling based methods fall broadly into two categories : scenario 
 approximation and sample average approximation . for scenario approximation , 
 it draws a finite number of samples from a given distribution , and 
 enforces all sampled constraints to hold .
sample average approximation refers to replacing the distribution 
 with another  easy - to - use  distribution , typically the 
 empirical distribution determined from a sample drawn from the original 
 distribution .
while solving the approximation 
 problem represents one aspect of complexity , the size of the sample 
 required to guarantee the quality of the approximation is another 
 important limitation .
second , analytical approximation methods 
 are based on either robust optimization or well - known probability inequalities . since the type and the size of the uncertainty set is determined 
 based on an initial assumption on the constraint satisfaction and 
 the a priori probability bound formulation , robust optimization provides a safe approximation of probabilistic 
 constraint .
in contrast to sampling based approximation , robust optimization 
 based approximation is a promising deterministic alternative for certain 
 classes of chance constrained problems .
in addition , other forms of 
 deterministic analytical approximation use probability inequalities , 
 such as the markov inequality , chebyshev s inequality , bernstein s 
 inequality , hoeffding s inequality , etc . although robust 
 optimization has been used widely in different areas to achieve solution 
 robustness / reliability , the quality of the solution is often ignored . 
 in other words , while the desired solution feasibility ( i.e. , desired 
 probability of constraint satisfaction ) is met , how far is the solution 
 from optimality ? in this work , we will first illustrate the above 
 issues and then propose an iterative strategy for improving the robust 
 solution . in the proposed method ,
the tight a posteriori probability 
 bounds are used to improve the robust solution within an iterative 
 framework .
compared to the single - step classical robust optimization 
 method , the quality of the robust solution can be improved .
on the 
 other hand , compared to the pure a posteriori probability bound based 
 methods , the proposed method has the advantage that it does not require 
 the global optimization of nonconvex problem .
we first present the problem of optimization with probabilistic 
 guarantee on constraint satisfaction , that is , probabilistically constrained 
 problem , and then introduce the traditional robust optimization based 
 approximation framework .
next , the a posteriori probability bound 
 based approximation framework is presented in section 3 .
, we present a novel iterative framework 
 which combines the advantage of the previous two different methods . 

the proposed method and the traditional methods are studied through 
 production planning and process scheduling problems in section 5 , and the paper is concluded in section 6 .
consider the following linear optimization 
 problem1where x    and xj ( j = 1 , ... , n ) can be either 
 continuous or integer variables , the left - hand - side ( lhs ) constraint 
 coefficients ij are subject to uncertainty , and ji represents the index subset that contains the variable indices 
 whose corresponding coefficients are subject to uncertainty .
 j  ji with aij being the nominal value and ij being a constant perturbation amplitude 
 ( ij > 0 ) , 
 { ij}jji are random variables which are subject 
 to uncertainty . with the above definition , the ith 
 constraint in problem 1 can be rewritten as 
 the follows:2 in many practical applications , enforcing constraint 
 satisfaction for all possible values of the uncertain parameters ( i.e. , 
 worst - case scenario ) can be too costly or even impossible .
probabilistic 
 constraint ( also called chance constraint ) provides a compromise to 
 avoid this situation and ensures that the constraints are satisfied 
 under certain given probability . a probabilistic version of the above 
 constraint is written as follows so that a probabilistic guarantee 
 on constraint satisfaction is applied:3or an upper bound on the probability 
 of constraint violation is applied4where  ( 0 <
for instance ,  = 0.05 means that the constraint 
 must be satisfied with a probability larger than 0.95 or the probability 
 of constraint violation must be less than 0.05 . while joint probabilistic 
 constraints are alternative for modeling solution reliability , individual 
 probabilistic constraints are investigated in this paper .
consider the following linear optimization problemand assume that the lhs constraint coefficients 
 of the constraints are uncertain and subject to uncertainty with 11 = 10 + 11 , 12 = 20 + 212 , 21 = 6 + 0.621 , and 22 = 8 + 0.822 and 
 11 , 12 , 21 , 
 22 are independent uncertain parameters uniformly 
 distributed in [ 1,1 ] . for the above problem , if we set the 
 allowed violation probability for each of the constraint as 0.05 , 
 then the probabilistic constrained version of the problem is in set induced robust optimization , the uncertain data is assumed 
 to be varying in a given uncertainty set and the aim is to choose 
 the best solution among those  immunized  against data 
 uncertainty .
for constraint 2 , the set induced 
 robust optimization method aims to find solutions that remain feasible 
 for any  in the given uncertainty set u so 
 as to immunize against infeasibility , that is5the corresponding robust optimization 
 problem is6for different uncertainty sets , 
 the robust counterpart formulation is distinct . furthermore , under 
 specific probability distribution assumption , the probabilistic guarantee 
 on the constraint satisfaction can be quantified using the size of 
 the uncertainty set . in our previous work
, we have systematically 
 derived the robust counterpart formulations under different uncertainty 
 sets and also derived their probability 
 bounds on constraint violation .
for example , 
 if the uncertainty set is given by a box7where  is the size 
 of the box , then the robust optimization counterpart constraint is8if the 
 uncertain parameters are subject to independent bounded symmetric 
 distribution , then the following a priori probability bound is valid9a priori probability bound means that if the 
 size of the box set is  , then the solution of the robust optimization 
 problem will ensure that the probability of constraint violation is 
 less than or equal to the following bound:10 in the literature , the traditional 
 way of applying robust optimization to solve the 
 probabilistically constrained problem is as follows .
first , the reliability 
 level  in the probabilistic constraint is set , and the type 
 of the robust optimization model ( i.e. , uncertainty set ) is selected 
 by the distribution of the uncertainty .
next , the size of the uncertainty 
 set is evaluated based on the a priori probability bounds . for example , 
 assuming that the box type uncertainty set is selected for applying 
 robust optimization , the size of the uncertainty set can be determined 
 by the following problem11using the size parameter value 
 determined from the above problem , the robust counterpart optimization 
 problem can be solved and the solution ensures that the constraint 
 is satisfied with the desired probability 1.12as a summary , the traditional framework of applying robust optimization 
 to address the probabilistic guarantee on constraint satisfaction 
 is shown in figure 1 .
the robust optimization problem provides a safe and conservative 
 approximation of the probabilistically constrained problem .
notice 
 that the minimum possible value for  is used to find the best 
 possible solution within this framework . in the following ,
we illustrate 
 the approximation of the probabilistic constrained problem using the 
 robust counterpart optimization through the motivating example .
uncertainty set induced robust counterpart optimization model is applied 
 to solve the motivating example problem .
the robust counterpart optimization 
 problem under the interval + ellipsoidal uncertainty set is as followswhere 1 and 2 are parameters determining the size of the interval + ellipsoidal 
 uncertainty set . using the probability bound on constraint violation 
 for this type of robust counterpart optimization model ( under the 
 assumption that the uncertainty is bounded and symmetric which is 
 satisfied in this example)we obtained the smallest possible value : 1 = 2 = 2.4477 .
the above robust optimization problem is convex 
 and can be efficiently solved using convex nonlinear optimization 
 solvers . with this value ,
the robust counterpart optimization problem 
 can be solved , the optimal objective value is obj * = 90.9091 and the 
 robust solution is x = ( 7.2727 , 2.7273 ) .
this 
 solution ensures that the constraints are satisfied with the desired 
 probability 0.95 on constraint satisfaction .
once a robust solution 
 is obtained , the probability of constraint violation can also be quantified 
 by a posteriori probability bound . in our previous paper
if the probabilistic distribution information on the uncertain parameters 
 is known , then the following relationship holds:13 in the 
 above derivation ,  is an arbitrary positive number .
as studied 
 in our previous work , with the above 
 probability inequality , we can evaluate the a posteriori probability 
 bound on constraint violation as follows once we have a set of solution x , ( i.e. , we have xj as the solution):14notice that in the above equation , 
 a minimization with respect to  ( i.e. , only one variable ) is 
 performed to find the tightest probability bound . for the traditional 
 robust counterpart optimization
based framework , the adjustable parameter 
 defining the size of the uncertainty set is initially selected based 
 on the a priori probability bound which is a function of the adjustable 
 parameter .
however , usually , the resulting solution could be too conservative , 
 since the actual probability of constraint violation is much smaller 
 than the bound . for example , with the robust solution x = ( 7.2727 , 2.7273 ) obtained for the motivating example , the probability 
 of constraint violation can be evaluated using the above a posteriori 
 probability bound 14 , and the following upper 
 bounds on constraint violation for the two constraints can be calculated : which are far less than the desired violation 
 probability 0.05 .
this implies that the obtained robust solution is 
 conservative and there is room for improvement .
while the a posteriori probability bound can be used to check the 
 probability of constraint satisfaction with a given solution , it can 
 also be used in another way to formulate a safe approximation of the 
 probabilistic constraint . using inequality 13 ,
the following safe approximation of 4 is 
 obtained:15because for any feasible solution satisfying 15 , it also satisfies the constraint 4 .
constraint 15 can be further rewritten 
 as16and finally the following safe approximation of the probabilistic 
 constrained problem is obtained:17note that while 
 for any fixed value of i > 0 , 15 is an approximation of the original probabilistic 
 constrained problem , here i is 
 set as a decision variables in problem 17 so 
 as to find the tightest possible approximation and to seek the best 
 possible solution .
the a posteriori probability bound based 
 framework addressing the probabilistic constraint can be represented 
 using figure 2 .
it is seen that the approximation 
 optimization problem is constructed based on the selected a posteriori 
 bound , in comparison to the selection of uncertainty set in the traditional 
 robust optimization framework .
, then the explicit 
 formulation of above approximation problem depends on the moment generating 
 function e[e ] . for several
known distributions , their probability density functions 
 and moment generating functions are listed in table 1 , which can be substituted into 17 and 
 the resulting problem can be solved as a deterministic optimization 
 problem .
following the derivation 
 in this section , the probability inequality based safe approximation 
 of the probabilistic constrained problem of the motivating example 
 is obtained : since the random 
 variable 11 is subject to uniform distribution in 
 [ 1 , 1 ] , we haveevaluate the expectation terms in 
 the similar way and finally the following problem is obtained : the above problem 
 is a nonconvex optimization problem , which can be solved through a 
 deterministic global optimization approach .
we solve the above problem 
 through global optimization solver antigone 1.1 in gams 24.2.2 ( with relative optimality gap tolerance optcr = 0 and resource limit reslim = 10000 ) 
 and obtain the following solution after 10  000 s ( with a relative 
 gap 0.11% to the upper bound 92.33):comparing the above solution with the solution 
 from the traditional robust optimization framework , it is observed 
 that while both solutions ensure the desired probability on constraint 
 satisfaction , the a posteriori probability bound based method generates 
 a solution which is better than the classical method .
comparing the previous two methods , the following observations 
 can be made:(1)in terms of the information needed , the a posteriori probability 
 bound based approximation method needs the exact probability distribution 
 function while the robust optimization method only needs partial information . 

for instance , in the studied robust optimization formulations , the 
 assumptions on uncertainty are only bounded and symmetric so that 
 a probabilistic guarantee is valid.(2)in terms of the solution complexity , the probability 
 inequality based approximation problem can be nonconvex and global 
 optimization is necessary ( i.e. , higher computation complexity ) .
the 
 robust optimization based approximation leads to convex problem which 
 can be solved very efficiently.(3)in terms of the quality of the solution , the a posteriori probability 
 bound based approximation method leads to less conservative solution 
 because it is tighter than the a priori probability bound as illustrated 
 in previous work . in terms of the information needed
, the a posteriori probability 
 bound based approximation method needs the exact probability distribution 
 function while the robust optimization method only needs partial information . 

for instance , in the studied robust optimization formulations , the 
 assumptions on uncertainty are only bounded and symmetric so that 
 a probabilistic guarantee is valid . in terms of the solution complexity , the probability 
 inequality based approximation problem can be nonconvex and global 
 optimization is necessary ( i.e. , higher computation complexity ) .
the 
 robust optimization based approximation leads to convex problem which 
 can be solved very efficiently . in terms of the quality of the solution , the a posteriori probability 
 bound based approximation method leads to less conservative solution 
 because it is tighter than the a priori probability bound as illustrated 
 in previous work .
the aforementioned observations show that there is a trade - off 
 between the two different types of approximations . to fully take advantage 
 of both of them , an iterative solution framework , which is also the 
 major contribution of this paper ,
is proposed to combine the use of 
 the traditional robust optimization approximation and the a posteriori 
 probability bound .
the objective is to improve the quality of robust 
 solution while still ensure the probabilistic guarantee of the robust 
 solution . at the same time , the computational complexity is decreased 
 comparing to the a posteriori probability bound based approximation 
 method .
the proposed solution framework is shown in figure 3 , which can be detailed as follows .
initially , the 
 type of the uncertainty set ( i.e. , the robust optimization formulation ) 
 is selected .
based on the desired degree of constraint satisfaction , 
 the smallest possible size of the uncertainty set is determined using 
 the specific a priori probability bounds ( e.g.,exp(/2 ) , exp(/2|ji| ) , etc . ) of that type of robust formulation . 
 with the determined size of the uncertainty set , the robust counterpart 
 optimization problem is solved and the robust solution is obtained . 

then , an upper bound on the constraint violation is evaluated using 
 the derived solution and the a posteriori probability bound .
if the gap between them is larger than a certain predefined 
 tolerance , the size of the uncertainty set is adjusted and the robust 
 optimization problem is solved again . the size adjusting is 
 the important step in the algorithm .
based on the fact that the a 
 posteriori probability upper bound is monotonically decreasing function 
 of the set size , this adjustment can 
 be made heuristically : if the probability upper bound exceeds the 
 desired level , the set size should be decreased ; if the bound is below 
 the desired level , the set size should be increased .
the above procedure 
 is repeated until the gap between the desired degree of constraint 
 violation and the computed upper bound on constraint violation is 
 less than the tolerance . finally , the solution from the last round 
 robust optimization step is reported .
a pseudo code of the iterative 
 algorithm is given as follows : notice 
 that in the proposed iterative framework , the problem of evaluating 
 the a posteriori probability bound using the right - hand side ( rhs ) 
 of 14 is an optimization problem . since any 
 feasible solution of the rhs of 14 will be a 
 valid a posteriori upper bound on the probability of constraint violation , 

furthermore , 
 since x is a known solution and taken fixed value , 
 the rhs of 14 is a single variable optimization 
 problem , which can be solved relatively efficiently .
proposed iterative solution 
 framework to improve the quality of robust solution . applying the proposed 
 framework and using the interval + ellipsoidal uncertainty set based 
 robust counterpart optimization formulation , we obtain the following 
 results for the motivating example as shown in table 2 for different iterations ( i.e. , k stands 
 for iteration ) :
the detailed solution 
 procedure is explained as follows : step 1 : initialize 
 1satisfy = 2.4477 , 1violate = 0 , 2satisfy = 2.4477 , 2violate = 0 using the a priori 
 bound . set tolerance parameter  = 0.01.step
2 : set 1 = 1satisfy = 2.477 , 
 2 = 2satisfy = 2.477.iteration 1step 3 : solve the robust optimization 
 problem and obtain solution obj * = 90.91 , x1 = 7.2727 , x2 = 2.7273.step 4 : compute the a posteriori probability bound from 
 solution p1 = 2.51  10 , p2 = 3.46  10.step 5 : since the probability violation 
 upper bound is less than 0.05 for both constraints , there is room 
 to contract the uncertainty set and improve the solution .
so decrease 
 the size of both uncertainty sets based on the value in iteration 
 1 : 1 = 2 = ( 0 + 2.4477)/2 = 1.2238.iteration 2step 3 : solve the robust model and obtain obj * = 91.807 , x1 = 7.2745 , x2 = 
 2.8009.step 4 : compute the a posteriori 
 probability bound p1 = 0.0305 , p2 = 0.0205.step 
 5 : since the probability violation upper bound is still less than 
 0.05 for both constraints , it is necessary to further decrease the 
 size of both uncertainty sets based on the size value in iteration 
 1 1 = 2 = ( 0 + 1.2238)/2 = 0.6119.iteration 3step 3 : solve the robust model and get obj * = 95.6954 , x1 = 7.6045 , x2 = 
 2.9049.step 4 : compute the a posteriori 
 probability bound p1 = 0.5486 , p2 = 0.5426.step 
 5 : since the probability violation upper bound becomes larger than 
 0.05 for both constraints , this means the uncertainty set should be 
 enlarged to satisfy chance constraint .
so we adjust the uncertainty 
 sets based on smallest size leading to constraint satisfaction so 
 far ( 1.2238 in iteration 2 ) and the size that leads to violation ( 0.6119 
 in iteration 3 ) : 1 = 2 = ( 0.6119 
 + 1.2238)/2 = 0.9179.iteration 4step 3 : solve the robust model 
 and obtain obj * = 93.685 , x1 = 7.422 , x2 = 2.859.step 
 4 : compute the a posteriori probability bound p1 = 0.222 , p2 = 0.205.step 5 : since the probability violation 
 is still larger than 0.05 for both constraints by using size 0.9179 , 
 we need to further enlarge the uncertainty set to satisfy chance constraint . 

we adjust the parameter toward the smallest size leading to constraint 
 satisfaction ( 1.2238 in iteration 2 ) : 1 = 2 = ( 0.9179 + 1.2238)/2 = 1.0709.iteration 5step 
 3 : solve the robust model and obtain obj * = 92.712 , x1 = 7.335 , x2 = 2.836.step 4 : compute the a posteriori probability 
 bound p1 = 0.105 , p2 = 0.086.step 5 : since the probability 
 violation is still larger than 0.05 for both constraints in the previous 
 iteration , we need to further enlarge the uncertainty sets : 1 = 2 = ( 1.0709 + 1.2238)/2 = 1.1474.iteration 6step 3 : solve the robust model and obtain obj * = 92.236 , x1 = 7.293 , x2 = 
 2.824.step 4 : compute the a posteriori 
 probability p1 = 0.062 , p2 = 0.045.step 5 : since 
 the probability violation is larger than 0.05 for the first constraints 
 and the second constraint is satisfied , we keep 2 unchanged and further increase 1 as 1 = ( 1.1474 + 1.2238)/2 = 1.1856.iteration 7step 
 3 : solve the robust model and obtain obj * = 92.153 , x1 = 7.354 , x2 = 2.777.step 4 : compute the a posteriori probability p1 = 0.045 , p2 = 
 0.045 ; both are less than 0.05 and the gap is smaller than  
 = 0.01 , so the iteration stops.step 6 : return the final solution obj * = 92.153 , x1 = 7.354 , x2 = 
 2.777 .
step 1 : initialize 
 1satisfy = 2.4477 , 1violate = 0 , 2satisfy = 2.4477 , 2violate = 0 using the a priori 
 bound .
step 2 : set 1 = 1satisfy = 2.477 , 
 2 = 2satisfy = 2.477 .
step 3 : solve the robust optimization 
 problem and obtain solution obj * = 90.91 , x1 = 7.2727 , x2 = 2.7273 .
step 4 : compute the a posteriori probability bound from 
 solution p1 = 2.51  10 , p2 = 3.46  10 .
step 5 : since the probability violation 
 upper bound is less than 0.05 for both constraints , there is room 
 to contract the uncertainty set and improve the solution .
so decrease 
 the size of both uncertainty sets based on the value in iteration 
 1 : 1 = 2 = ( 0 + 2.4477)/2 = 1.2238 .
step 3 : solve the robust model and obtain obj * = 91.807 , x1 = 7.2745 , x2 = 
 2.8009 .
step 4 : compute the a posteriori 
 probability bound p1 = 0.0305 , p2 = 0.0205 .
step 
 5 : since the probability violation upper bound is still less than 
 0.05 for both constraints , it is necessary to further decrease the 
 size of both uncertainty sets based on the size value in iteration 
 1 1 = 2 = ( 0 + 1.2238)/2 = 0.6119 .
step 3 : solve the robust model and get obj * = 95.6954 , x1 = 7.6045 , x2 = 
 2.9049 .
step 4 : compute the a posteriori 
 probability bound p1 = 0.5486 , p2 = 0.5426 .
step 
 5 : since the probability violation upper bound becomes larger than 
 0.05 for both constraints , this means the uncertainty set should be 
 enlarged to satisfy chance constraint .
so we adjust the uncertainty 
 sets based on smallest size leading to constraint satisfaction so 
 far ( 1.2238 in iteration 2 ) and the size that leads to violation ( 0.6119 
 in iteration 3 ) : 1 = 2 = ( 0.6119 
 + 1.2238)/2 = 0.9179 .
step 3 : solve the robust model 
 and obtain obj * = 93.685 , x1 = 7.422 , x2 = 2.859 .
step 
 4 : compute the a posteriori probability bound p1 = 0.222 , p2 = 0.205 .
step 5 : since the probability violation 
 is still larger than 0.05 for both constraints by using size 0.9179 , 
 we need to further enlarge the uncertainty set to satisfy chance constraint . 

we adjust the parameter toward the smallest size leading to constraint 
 satisfaction ( 1.2238 in iteration 2 ) : 1 = 2 = ( 0.9179 + 1.2238)/2 = 1.0709 . step 
 3 : solve the robust model and obtain obj * = 92.712 , x1 = 7.335 , x2 = 2.836 .
step 4 : compute the a posteriori probability 
 bound p1 = 0.105 , p2 = 0.086 .
step 5 : since the probability 
 violation is still larger than 0.05 for both constraints in the previous 
 iteration , we need to further enlarge the uncertainty sets : 1 = 2 = ( 1.0709 + 1.2238)/2 = 1.1474 .
step 3 : solve the robust model and obtain obj * = 92.236 , x1 = 7.293 , x2 = 
 2.824 .
step 4 : compute the a posteriori 
 probability p1 = 0.062 , p2 = 0.045 .
step 5 : since 
 the probability violation is larger than 0.05 for the first constraints 
 and the second constraint is satisfied , we keep 2 unchanged and further increase 1 as 1 = ( 1.1474 + 1.2238)/2 = 1.1856 . step 
 3 : solve the robust model and obtain obj * = 92.153 , x1 = 7.354 , x2 = 2.777 .
step 4 : compute the a posteriori probability p1 = 0.045 , p2 = 
 0.045 ; both are less than 0.05 and the gap is smaller than  
 = 0.01 , so the iteration stops .
step 6 : return the final solution obj * = 92.153 , x1 = 7.354 , x2 = 
 2.777 .
the solution procedure of the 
 above iterative method is also illustrated in figure 4 , which shows how the constraint satisfaction probability 
 converges to the desired level and how the quality of the robust solution 
 is eventually improved comparing to the traditional framework .
iterative solution 
 procedure for the motivating example : ( upper ) a posteriori probability 
 bound ; ( lower ) optimal objective value of robust solution .
finally , solutions from three different methods 
 for the motivating example are summarized in table 3 .
the columns  traditional  ,  a posteriori  , 
 and  iterative  represent the traditional robust optimization 
 framework , the a posteriori probability bound based method , and the 
 iterative method , respectively .
the row  obj *  and  probviolationposterioriub  represent the optimal objective value of the robust optimization 
 problem and the a posteriori probability bound based on the robust 
 solution obtained .
the percentage numbers represent the gaps between 
 the traditional / iterative method s solutions and the a posteriori 
 method s solution . comparing 
 the traditional robust optimization based approximation framework and the proposed iterative method
, we observed that it improves 
 the quality of the solution while still ensures the degree of constraint 
 satisfaction .
the percentage gap to the a posteriori solution 92.2292 
 has been decreased from 1.43% to 0.08% as shown table 3 . comparing the pure a posteriori probability bound based 
 approximation method and the proposed iterative method , we observed 
 that its computational complexity is decreased , since only a set of 
 convex robust optimization problem is solved .
in addition , when the gap 
 tolerance is defined small enough , the solution of the proposed method 
 will be close to the solution of the probability inequality based 
 method .
in 
 this section , we apply the different methods to solve a production 
 planning problem and a process scheduling problem to compare their 
 performances and illustrate the effectiveness of the proposed iterative 
 method .
all the optimization problems are solved on a unix workstation 
 with 3.40 ghz intel core i7 - 2600 cpu and 8 gb memory .
the related global 
 optimization problems are solved via antigone 1.1 and the ( mixed integer ) linear optimization problems are 
 solved using cplex 12.0 in gams 24.2.2 .
this 
 example was introduced by li et al . , which 
 addresses the problem of planning the production , storage and marketing 
 of a product for a company .
it is assumed that the company needs to 
 make a production plan for the coming year , divided into six periods 
 of 2 months each , to maximize the sales with a given cost budget . 

the production cost includes the cost of raw material , labor , machine 
 time , etc . , and the cost fluctuates from period to period
. the product 
 manufactured during a period can be sold in the same period , or stored 
 and sold later on .
operations begin in period 1 with an initial stock 
 of 500 tons of the product in storage , and the company would like 
 to end up with the same amount of the product in storage at the end 
 of period 6 . this problem can be formulated as a linear optimization 
 problem as follows:18a18b18c18d18e18f18gin this example , it 
 is assumed that the production costs cj are subject to independent 
 uncertainty distributions .
the uncertainty is normalized using 50% 
 of the nominal value cj as the base perturbation amplitude .
then the original constraint 18b can be rewritten aswhere j are independent random variables . to ensure 
 the reliability of the solution , the minimum probability for constraint 18b to be satisfied is set as 0.85 ( i.e. , the upper 
 bound on the probability of constraint violation
is set to 0.15 ) , 
 then the probabilistic constrained version for this constraint isin the sequel , this example is studied under different assumptions 
 on the uncertainty distributions . in this case , it is assumed that the production costs are subject 
 to uniform uncertainty , that is , j are random variables that uniformly distributes in [ 1 , 1 ] . 

the traditional robust optimization method is applied first to solve 
 the probabilistically constrained optimization problem . using the 
 interval + ellipsoidal type uncertainty set
, the following robust 
 counterpart optimization constraints can be formulatedthe 
 robust optimization problem is obtained by replacing the original 
 deterministic constraint 18b with the above 
 constraints . using the a priori probability bound for the interval 
 + ellipsoidal set induced robust optimization model , the size of the 
 uncertainty set is computed as  = 1.9479
. then the robust optimization 
 problem is solved and the corresponding objective is 2  356  977 . 
 the corresponding robust planning solution is shown in figure 5 .
the a posteriori probability bound 
 based method is applied next and the following constraint is applied 
 to replace the original constraint 18b : for the a posteriori probability bound based method , 
 the objective value is 2  550  538
. it is seen that this 
 solution is better ( higher sales ) than that of the classical robust 
 optimization method .
finally , the iterative framework 
 is applied to solve the problem , the solution procedure is shown in 
 table 5 and the robust solution is shown in 
 figure 7 . in the above solution procedure , the parameter 
  is adjusted as follows : 2 = 0.51 , 3 = 0.5(1 + 2 ) , 4 = 0.5(2 + 3 ) . notice that although the parameter value is decreased in 
 the fourth step , it is still a conservative solution ( 0.0714 < 
 0.15 ) , so we do not adjust the parameter using 5 = 0.5(3 + 4 ) , but rather 5 = 0.5(2 + 4 ) .
it is seen from table 6 that 
 while all the solutions satisfy the probabilistic requirement on constraint 
 satisfaction , the solution of the a posteriori probability bound method 
 ( with relative optimality gap tolerance optcr = 0 
 and resource limit reslim = 10  000 s ) is the 
 best although global optimization is necessary .
if we compare the 
 classical robust optimization method and the iterative robust optimization 
 method and consider the difference between their solution and the 
 best solution ( i.e. , the a posteriori bound based solution ) , the gap 
 has been decreased from 193  561 to 13  679 ( a percentage 
 of 92.93% ) , and the solution has been greatly improved through the 
 iterative framework .
comparing the solution of the a posteriori probability 
 bound based method ( figure 6 ) and the iterative 
 method ( figure 7 ) , it is seen that the difference 
 between the solutions is very small .
note that global optimization 
 is not needed in the iterative framework and only five convex robust 
 counterpart optimization problems are solved . in this case , it is assumed that the random 
 variables j are subject to symmetric 
 triangular distribution with support on [ 1,1 ] .
notice that 
 this type of distribution is bounded and symmetric , so we can still 
 apply the a priori probability bounds to determine the size of the 
 uncertainty set and then apply the traditional robust optimization 
 framework . under the interval + ellipsoidal
set induced robust optimization 
 model , the solution will be the same as the previous uniform distribution 
 since the a priori probability bound does not depend on the distribution . 

on the other hand , the solution of the other two methods will change 
 since they depend on the distribution of the uncertainty . specifically , 
 while the a posteriori probability bound based method is applied , 
 the solution is 2  626  457 after a 10  000 s resource 
 limit reaches in gams ( with a relative gap of 7.52% to the upper bound 
 2  840  000 ) .
when the iterative framework is applied , 
 the final solution is 2  619  188 and the solution procedure 
 is shown in table 7 .
the results of three different 
 methods are 
 compared in table 8 . while the traditional 
 framework leads to an 11.4% difference to the a posteriori method 
 solution , the iterative method s solution only has 0.28% difference . 

this shows that the iterative framework significantly improve the 
 quality of the robust solution while the reliability of the solution 
 is satisfied . in the 
 above studies
, it is assumed that the uncertainty distribution is 
 independent , bounded and symmetric , such that we can apply the traditional 
 robust optimization framework based on only the a priori probability 
 bounds .
however , when the uncertainty distribution does not fall into 
 this characteristic , there is no basis for determining the size of 
 the uncertainty set .
consequently , the traditional robust optimization 
 framework can not be directly applied ( i.e. , if we still use the a 
 priori bound to determine the size , the solution will not ensure the 
 probabilistic guarantee ) . on the other hand , with the proposed iterative 
 framework
next , we study two cases where the distribution does not 
 satisfy the bounded or symmetric condition . in this case
, we assume the random variables are subject to exponential 
 distribution with rate parameter  = 1 .
notice that this distribution 
 is unbounded , so we apply the  ellipsoidal  type uncertainty 
 set induced robust optimization model rather than interval + ellipsoidal 
 type in this study . with the a posteriori probability bound based 
 method
, the final solution is 1  818  269 . with the iterative 
 solution framework ,
the solution is 1  803  310 and the 
 corresponding a posteriori probability upper bound of constraint violation 
 is 0.1474 .
notice that we do not list the a priori bound value here , 
 since it is not applicable for the asymmetric distribution in this 
 case . it is assumed 
 that each j is subject to normal 
 distribution n(0,0.5 ) in this case .
although for 
 the case of the normal distribution , it is not necessary to apply 
 an approximation scheme to solve the probabilistically constrained 
 problem since analytical deterministic equivalent problem can be formulated 
 and solved , we study the robust optimization approximation based method 
 here to compare the solution quality .
type uncertainty set is also used in this case to deal with the unbounded 
 distribution .
the a posteriori method lead to solution of 2  569  004 , 
 and the iterative method leads to 2  563  734 , with the 
 constraint violation probability less than 0.1394 as shown in table 10 . in this example , a process scheduling problem is studied .
this example involves the scheduling 
 of a batch chemical process related to the production of two chemical 
 products using three raw materials .
the mixed integer linear optimization 
 model for the scheduling problem is formulated as follows , and the 
 readers are directed to the paper for 
 the detailed mixed integer linear optimization formulation and problem 
 data19a19b19c19d19e19f19g19h19i19j19k19l19m19nin this example , we consider 
 the demand uncertainty only and the following constraints are affected : we assume independent uncertainty distributions on 
 the product demand parameters rs and assign a base perturbation of 20% of the nominal 
 demand data ( rp1 = 50 , rp2 = 75 ) : rs = rs(1 + 0.2s ) .
we set the expected minimum probability 
 level on constraint satisfaction to 0.5 ( i.e. , set the upper bound 
 on constraint violation to 0.5 ) .
then the probabilistic constrained 
 version isseveral different type of uncertainty distributions 
 are considered to study the proposed method .
specifically , 
 we assume a uniform distribution on the product demand parameters rs ( i.e. , s is uniformly distributed in [ 1,1 ] ) . 

based on the traditional robust optimization method , the robust counterpart 
 optimization constraint is formulated as follows : which is equivalent for box , ellipsoidal , 
 and polyhedral type of uncertainty sets since the number of uncertain 
 parameter in each demand constraint is 1 . from the desired bound on 
 constraint violation 0.5 , we get the uncertainty set size value 1.1774 . 

since the uncertainty is bounded , a set with size value 1 will cover 
 the whole uncertainty space .
the a posteriori probability 
 bound based method is applied next and the following constraint is 
 applied : the resulting nonconvex mixed integer nonlinear optimization 
 problem is solved using antigone 1.1 ( with 
 tolerance parameter optcr = 0.01 and reslim = 10  000 ) and the objective is 1070.04 ( with relative optimality 
 gap 1.82% to the upper bound 1089.47 after 10  000 s ) .
since 
 set size value 1 makes the robust optimization problem infeasible , 
 in the iterative framework , we start from 0.5 for the parameter s to make the problem feasible .
notice that 
 the adjustments of the parameter values are based on the change of 
 the a posteriori bounds .
for example , we realize that for the first 
 demand constraint is 0.6646 in the 2nd iteration and 0.3397 in the 
 third iteration .
to move the bound close to less than 0.5 , we set 
 the new parameter value p1 in the 
 fourth iteration as ( 0.5 + 0.75)/2 = 0.625 , and the resulting solution 
 lead to a new probability bound 0.507 .
notice in the sixth step , there 
 is no change on the objective solution , so the solution procedure 
 is stopped .
the final optimal objective value is 1064.29 , and the 
 corresponding schedule is shown in figure 9 .
this solution has only 0.54% difference to the a posteriori solution 
 as shown in table 12 .
schedule obtained from the iterative method . comparing the solution from all the three different 
 methods shown in table 12 , the following observations 
 can be made .
first , the traditional robust solution framework is conservative 
 and even leads to an infeasible problem .
however , the iterative framework 
 successfully addresses the same problem and finds feasible solution . 

the reason is that the iterative framework utilizes not only the a 
 priori probability bound but also the a posteriori probability bound , 
 thus avoids the conservative solution .
second , comparing the a posteriori 
 probability bound based method and the iterative method , it is seen 
 that the optimal solutions are very close . however ,
with the iterative 
 framework , we obtain a solution with almost same quality but far less 
 computational efforts .
this further validates the effectiveness of 
 the proposed iterative method . for this scheduling example , two 
 unbounded distributions are also studied .
the traditional framework 
 is not applicable in this situation because the a priori probability 
 bound is based on the bounded distribution assumption .
box , ellipsoidal , 
 and polyhedral type of uncertainty set lead to same robust optimization 
 formulations here .
the results of the different methods are summarized 
 in table 13 , and the solution procedure of 
 iterative method is given in table 14 .
the 
 iterative method s solution has only 0.6% difference to the 
 a posteriori solution in this case .
the results are given 
 in table 15 , and the solution procedure of 
 iterative method is summarized 
 in table 16 .
the a posteriori solution 1074.22 
 has a relative gap of 1.33% to the upper bound 1088.75 after 10  000 
 s. iterative method s solution has only 0.1% difference when 
 compared to the a posteriori solution . from the above results ,
it 
 is observed that 
 while the traditional method is not applicable to the unbounded distribution 
 cases , the iterative method applies the robust optimization approximation 
 and uses the a posteriori bound to check the solution reliability . 

the solutions of iterative method are consistently very close to the 
 a posteriori bound based method in terms of the optimal objective 
 values while the computational complexity is greatly reduced since 
 only convex robust optimization problem is solved in several iterations .
the traditional robust optimization 
 framework can be used to approximate probabilistic constraints and 
 provide safe solution .
when a detailed probability distribution on uncertainty is available , 
 the a posteriori probability bound based method leads to less conservative 
 approximation , but the trade - off is that the resulting nonconvex problem 
 needs to be solved via a deterministic global optimization approach . 
 a novel solution framework combining the robust optimization approximation 
 and the a posteriori probability bound evaluation is proposed to improve 
 the solution quality of traditional robust optimization framework 
 without significant computation effort . the effectiveness of the proposed 
 method has been illustrated through a motivating example , as well 
 as planning and scheduling problems .
furthermore , while the traditional 
 robust optimization method requires information on certain probability 
 distribution on the uncertainty such that the a priori probability 
 bound is valid , the proposed iterative framework extends the application 
 to general distributions since we can always use the a posteriori 
 probability bound to ensure the constraint is satisfied within desired 
 probability .
finally , it is worth mentioning that the probability 
 bounds used in this work are derived based on the assumption of independence 
 on the uncertain parameters .
one of the future research directions 
 will be to investigate the correlation between uncertain parameters .
 in 
 this paper , we study the solution quality of robust optimization problems 
 when they are used to approximate probabilistic constraints and propose 
 a novel method to improve the quality . 
 two solution frameworks are 
 first compared : ( 1 ) the traditional robust optimization framework 
 which only uses the a priori probability bounds and ( 3 ) the approximation 
 framework which uses the a posteriori probability bound . 
 we illustrate 
 that the traditional robust optimization method is computationally 
 efficient but its solution is in general conservative . on the other 
 hand , 
 the a posteriori probability bound based method provides less 
 conservative solution but it is computationally more difficult because 
 a nonconvex optimization problem is solved . 
 based on the comparative 
 study of the two methods , we propose a novel iterative solution framework 
 which combines the advantage of the a priori bound and the a posteriori 
 probability bound . 
 the proposed method can improve the solution quality 
 of traditional robust optimization framework without significantly 
 increasing the computational effort . 
 the effectiveness of the proposed 
 method is illustrated through numerical examples and applications 
 in planning and scheduling problems . 
Introduction
Framework for Robust Optimization
A Posteriori Probability Bound Based Solution Method
Iterative Solution Strategy
Case Studies
Conclusion