Multicut Benders Decomposition Algorithm for Process Supply Chain Planning under Uncertainty

Transcription

1 Multicut Bender Decompoition Algorithm for Proce Supply Chain Planning under Uncertainty Fengqi You 1,2 and Ignacio E. Gromann 3 1 Argonne National Laboratory, Argonne, IL Northwetern Univerity, Evanton, IL Carnegie Mellon Univerity, Pittburgh, PA Augut 21, 2011 Abtract In thi paper, we preent a multicut verion of the Bender decompoition method for olving two-tage tochatic linear programming problem, including tochatic mixedinteger program with only continuou recoure (two-tage) variable. The main idea i to add one cut per realization of uncertainty to the mater problem in each ation, that i, a many Bender cut a the number of cenario added to the mater problem in each ation. Two example are preented to illutrate the application of the propoed algorithm. One involve production-tranportation planning under demand uncertainty, and the other one involve multiperiod planning of global, multiproduct chemical upply chain under demand and freight rate uncertainty. Computational tudie how that while both the tandard and the multicut verion of the Bender decompoition method can olve large-cale tochatic programming problem with reaonable computational effort, ignificant aving in CPU time can be achieved by uing the propoed multicut algorithm. Keyword: Bender decompoition, tochatic programming, planning, upply chain

2 1. Introduction Many problem for upply chain planning under uncertainty can be formulated a two-tage tochatic programming problem with fixed recoure (Birge & Louveaux, 1997; Infanger, 1994; Shapiro, 2008). In the two-tage framework, the firt-tage deciion are made here-and-now prior to the reolution of uncertainty, while the econd-tage deciion are potponed in a wait-and-ee mode after the uncertaintie are revealed. The cenario planning approach i ued to repreent the uncertaintie through a number of dicrete realization of the tochatic quantitie, contituting ditinct cenario. The objective i to find a olution that perform well on average under all cenario. Thi approach provide a traightforward way to account for uncertainty, but the reulting tochatic programming model are often computationally demanding becaue their model ize increae exponentially a the number of cenario increae. In order to addre the computational challenge, a number of method have been propoed for the olution of two-tage tochatic programming problem (Ruzczyńki, 1997), uch a Bender decompoition (Bender, 1962; Van Slyke & Wet, 1969), tochatic decompoition (Higle & Sen, 1991), ubgradient decompoition (Sen, 1993; Sen and Huang, 2009), dijunctive decompoition (Ntaimo, 2010), and neted decompoition (Archibald et al., 1999). Among thee method, Bender decompoition (Bender, 1962), alo called the L-haped method, ha become the major approach to tackle tochatic programming problem becaue of it eae of implementation. Thi method take advantage of the pecial decompoable tructure of the two-tage tochatic programming model and generate duality cut baed on the ubgradient information atively. Since the tandard Bender decompoition return only one cut to the mater problem in each ation, it convergence might be low for ome computationally demanding problem (Birge & Louveaux, 1997). To addre thi iue, numerou reearcher have propoed variant to accelerate the algorithm (Bahn et al., 1995; Ecudero et al., 2007; Fragniere et al., 2000; Gerd Infanger, 1993; Latorre et al., 2009; Linderoth & Wright, 2003; Mulvey & Ruzczynki, 1995; Ruzczynki, 1993; Saharidi & Ierapetritou, 2010; Saharidi et al., 2010; Contrera, et al., 2010; Miller & Ruzczyńki, 2010; Trukhanov et al., 2010). In thi paper, we conider the olution method for tochatic linear programming problem and tochatic mixed-integer linear program with only continuou recoure. -2-

3 We firt decribe a multicut verion of the Bender decompoition, which i a variant of the tandard Bender decompoition method but converge fater in general cae. We dicu the theory behind thi algorithm and prove it convergence. Two application of thi algorithm are then preented to illutrate the effectivene of thi method. The firt application involve production-tranportation planning under demand uncertainty. Becaue of the relatively mall problem ize, the global optimal olution of thi problem can eaily be obtained to validate the propoed olution approach and illutrate it effectivene. The econd application involve global chemical upply chain planning under uncertainty, which originate from a real-world application in the Dow Chemical Company. The model wa taken from the author earlier work (You et al., 2009). Three teting data et with different ize are conidered. In both application, the reult how that the multicut verion of the Bender decompoition method require fewer ation and le computational time than doe the tandard verion to obtain a olution with a pecified optimality tolerance. The ret of thi paper i organized a follow. Section 2 preent the multicut Bender decompoition algorithm, for problem where the firt-tage deciion variable can include both dicrete and continuou variable, while the econd-tage deciion variable mut all be continuou variable. The problem tatement, model formulation, and computational reult for the two application are given in Section 3 and 4. In Section 5, we ummarize our concluion. 2. Multicut Bender Decompoition Algorithm Conider the following general form of the two-tage tochatic programming model (P0): (P0) min xy, (1) T T c x pq y S.t. Ax b, x 0 (2) Wy h T x, y(w ) 0, S (3) where x i a vector that tand for the firt-tage deciion variable, which may include 0-1 variable; y are the continuou econd-tage deciion for each cenario ; A and b are parameter matrice independent of the cenario; and W, h -3-

4 and T are parameter matrice for each cenario S. The expanded verion of the general model (P0) i given in equation (4). The problem ha a pecial angular form, which can be decompoed into a mater problem and a number of cenario ubproblem. Mater problem Scenario ubproblem (4) The pecial decompoable tructure of (4) i uitable for Bender decompoition becaue it take advantage of ubgradient information to contruct convex etimate of the recoure function and atively generate a Bender cut to be added to the decompoed mater problem (Bender, 1962; Van Slyke & Wet, 1969). In the firt tep, a decompoed ubproblem with thoe contraint that do not include the econdtage variable i olved to obtain the value of the firt-tage deciion. Then we fix the firt-tage deciion and olve all the cenario ubproblem that include econdtage deciion, in order to obtain the optimal value of the econd-tage deciion. Let Q ( x ), the value function, be the objective function value of each cenario ubproblem. Q ( x) min q y T y.t. Wy h T x, y(w ) 0 (5) The reformulation of (P0) i then a follow. (P0) (6) T min c x pq( x) x S.t. Ax b, x 0 (7) To olve (P0), we can take advantage of the dual propertie of (6) by introducing a new variable for pq( x) and ating between the mater problem (P1) and S the cenario ubproblem (P2). The mater problem (P1) i given by -4-

5 (P1) min x, T c x.t. d x e, 1..N (8) Ax b, x 0 while the ubproblem (P2) for cenario i given by (P2) min y q T y.t. Wy h T x, y(w ) 0 (9) where the inequalitie in (P1) are the cut that link the mater problem and the cenario ubproblem. Here, d l and e l are coefficient for the Bender cut; they are given by (10) T d p, T S (11) T e p, h S where are the optimal dual vector of contraint (5) in the ubproblem (P2) for cenario. In thi paper, we aume that the problem (P0) ha complete recoure and that (P2) i alway feaible. We note that feaibility cut (Birge and Louveaux, 1997) can be added to the algorithmic framework to deal with problem with infeaible ubproblem, although in thi work we limit our cope on thoe that have complete recoure after introducing additional lack variable for hortfall or back order in upply chain planning Under thi aumption, feaibility cut are not preent in the mater problem (P1). Our algorithm and thi analyi can be generalized to handle ituation in which the aforementioned aumption doe not hold; but for the ake of implifying the analyi, we avoid dicuing thi more general cae here. The major tep for the tandard Bender decompoition algorithm are given in Figure 1. In thi algorithm, we firt olve the mater problem to obtain a lower bound of the objective value. We then fix all the firt-tage deciion and olve each cenario ubproblem to get an upper bound. If the lower bound and the upper bound are within a tolerance, then the algorithm top. Otherwie, we ue the dual of the cenario ubproblem to add a Bender cut and return to the mater problem. -5-

6 Figure 1 Algorithm for tandard Bender decompoition The tandard Bender decompoition algorithm only return one cut per ation to the mater problem. For large-cale problem, it convergence might be low and the algorithm might need many ation to reach a predefined optimality tolerance. To peed up the algorithm, we can decompoe the variable θ by cenario to return a many cut a the number of cenario at each ation. In thi variant, the mater problem i then given by (P3). (P3) min x, T c x S p.t. d x e, 1..N, 1..S (12) Ax b, x 0 The coefficient d l and e l for the cut (12) are updated a follow d e T (13) T, h (14) T, where are the optimal dual vector of contraint (5) in the ubproblem (P2) for cenario. -6-

7 Figure 2 Algorithm for multicut Bender method The algorithm framework for the multicut Bender algorithm i imilar to that for the tandard Bender algorithm (ee You et al, 2009) and i given a follow (ee alo Figure 2). Step 1 Set 1, LB, UB. Step 2 At ation, olve the mater problem (P3) with all the optimality cut generated in the previou ation. Denote the optimal objective function value a and the optimal olution of the firt-tage deciion variable x a X. If LB, et LB. Step 3: Solve all the cenario ubproblem (P2) with the value of firt-tage deciion variable fixed a variable y be x X. Let the optimal olution of the econd-tage deciion Y and the optimal dual vector of contraint (5) in the ubproblem (P2) for cenario be,. Compute the value of the objective of the original problem (P0); that i, et c X p q Y. If UB, then update T T S UB. -7-

8 Step 4 If UB LB (e.g., 10-3 ), top and output the optimal olution ( X, Y ); T otherwie, compute the coefficient of the optimality cut d, T and e h, and add the optimality cut d xe,, to the mater T, problem (P2). Then, et 1, and go to Step 2. Convergence i guaranteed in thi algorithm by the following propoition. Propoition 1. The recoure function R x p Q x i a convex piecewie linear function. Proof: The proof of thi propoition i given in (Birge & Louveaux, 1997). S Propoition 2. Each optimality cut (12) upport the recoure function R x and Q x from below. Proof: The proof of thi propoition i given by (Birge & Louveaux, 1997). Propoition 3. Given ome ( X, X i an optimal olution of the original problem (P0). Proof: ) uch that d X e,, then The original problem (P0) i equivalent to the following problem (P4). cx p T min x S.t. Ax b, x 0 Q x Baed on Propoition 2 and the duality property of (P2), we have e X d Q X,. Therefore, ( X, ) i a feaible olution of (P4). Since ( X, ) i the optimal olution of (P2), it i alo an optimal olution of (P4), which i equivalent to the original problem (P0). Propoition 4. (Convergence) Since the algorithm generate a finite equence of -8-

9 ( X, ) and ince ( X, ) i the limit of thi equence, and lim d x e 0, where M i a ufficiently large integer, then M optimal olution of the original problem (P0). Proof: From Propoition 2 and lim d x e 0 M Thu, ( X, function R x p Q x Propoition 1, ( S, we have Q X. X i an ) i a feaible olution of problem (P4). Becaue i a convex piecewie linear function a hown in X, the original problem (P0). ) i alo an optimal olution of (P4), which i equivalent to We note that while the multicut L-haped method can provide more cut to upport the recoure function from below and mot likely reduce the number of ation, it introduce more variable and contraint in the mater problem, which may potentially low the computation. Thi algorithm would benefit from olving it with parallel computing, which could ignificantly reduce the wall-clock time. 3. Production-Tranportation Planning under Uncertainty The firt application of the propoed algorithm i about a ingle product, ingleperiod production-tranportation planning under demand uncertainty. Thi problem can be formally tated a follow. We are given a et of plant i I with production capacity cap i and a et of demand zone l L. The elling price at demand zone l i price l, the unit tranportation cot from plant i to demand zone l i ctr i,l, the unit production cot at plant i i cpd i, and the unit wate dipoal cot in demand zone l i cu l. Here, S i the et of cenario, p i the cenario probability, and demand l, i the demand at demand zone l of cenario. The major deciion include the production level (prod i ), tranportation amount (hip i,l ), ale amount (ale l, ), and unold product amount (unold, ). We note that in the two-tage tochatic linear programming framework, the production and tranportation deciion are made here and now prior to the reolution of demand uncertainty, wherea the ale and wate dipoal deciion are -9-

10 potponed in a wait-and-ee mode after the uncertaintie are revealed. Thu, the production and tranportation deciion are independent of the cenario, wherea the ale deciion are made for each cenario. The objective of thi problem i to maximize the total expected profit (E[profit]) by optimizing the aforementioned deciion. Baed on the problem tatement, a two-tage tochatic linear programming model can be formulated a follow. E profit p price ale ctr hip max [ ] l l, i, l i, l ll S ii ll.t. prod prod i i i cpd prod p cu unold i i l l, ii ll S (15) cap, i I (16) hip, i I (17) ll i, l hipil, alel, unoldl,, l L ii ale l, l,, S (18) demand, l L, S (19) prod 0, hip, 0, ale, 0, unold, 0 i il l l Table 1 Probability ditribution of demand realization for the productiontranportation planning problem Demand Demand Realization (ton) Probability Zone Low Medium High Low Medium High Table 2 Unit tranportation cot for the production-tranportation planning problem ($/ton) Plant/Demand Zone P P P In thi cae tudy we conider a production-tranportation network with three -10-

11 plant and five demand zone. The probability ditribution of the demand realization i given in Table 1. In each demand zone there are three poible demand realization. We aume thee probabilitie are independent. By conidering the joint probability ditribution, we generate a total of 3 5 =243 cenario for thi problem. The unit production cot i $14/ton, the ale price i $24/ton, and cot of removal of unold product i $4/ton. The unit tranportation cot between plant and demand zone i given in Table 2. The determinitic equivalent of the reulting two-tage tochatic linear program include 2,448 continuou variable and 2,436 contraint. Le than one econd wa needed to obtain the optimal olution ($10,793) with 0% gap uing GAMS /CPLEX 12 (Roenthal, 2010), on an IBM T400 laptop with an Intel 2.53 GHz CPU and 2 GB RAM. To illutrate the application of the propoed multicut algorithm and compare it performance with that of the tandard Bender decompoition, we olved thi problem with both algorithm. The reult are hown in Figure 3 and 4. A can be een from Figure 3, the upper bound decreae and the lower bound increae a the number of ation increae. However, wherea the tandard Bender method require 22 ation to reach the optimality tolerance, the multicut verion require only 6 ation. The reult in Figure 3 clearly how that the multicut verion converge much fater than doe the tandard Bender method. The reaon ret mainly with the improved approximation of the value function in (5), ince a larger number of Bender cut are added to the mater problem at each ation. The computational time for both algorithm how little difference for thi cae tudy (0.15 CPU econd for the ingle cut verion and 0.13 CPU econd for the multicut verion), although the multicut verion require far fewer ation than doe the tandard Bender decompoition. The main reaon i that the mater problem of the multicut Bender method include more variable and contraint (Bender cut) than doe the tandard verion and thu require longer computational time per ation. Another reaon i that the computational time for thi cae tudy are o hort that the caling effect and the advantage of the multicut verion cannot be fully illutrated. -11-

12 Figure 3 Comparion between the tandard Bender method and the multicut verion in term of number of ation for the production-tranportation planning problem 4. Global Chemical Supply Chain Planning under Uncertainty The econd cae tudy conidered in thi work i baed on the problem decribed by You et al. (2009), which originate from a real-world application in the Dow Chemical Company. Global upply chain in the proce indutrie are uually largecale ytem that can comprie hundred or even thouand of production facilitie, ditribution center, and cutomer (Waick, 2009). Thi cae tudy addree the midterm planning for a global multiproduct chemical upply chain under demand and freight rate uncertainty. A two-tage tochatic mixed-integer linear programming model i ued, incorporating a multiperiod planning model that take into account the production and inventory level, tranportation mode, time of hipment, and cutomer ervice level. In the two-tage framework, the production, ditribution, and inventory deciion for the current time period, which include 0-1 variable, are made here and now prior to the reolution of uncertainty, while the deciion for the remaining time period, which only involve continuou variable, are potponed in a wait-and-ee mode. The problem include a large number of uncertain parameter a a reult of the multiperiod nature and the large ize of the upply chain network. A Monte Carlo ampling approach i ued to dicretize the continuou probability -12-

13 ditribution function and to generate the cenario. To demontrate the effectivene of the propoed decompoition algorithm, we olve three intance for mall, medium, and large upply chain network, uing both the tandard Bender decompoition method and the multicut verion. We preent the problem tatement, model formulation, and computational reult in the following ubection Problem tatement Thi cae tudy can be tated a follow. We are given a midterm planning horizon (for intance, one year), which can be ubdivided into a number of time period (for intance, one month a a time period). A et of product are manufactured and ditributed through a given global upply chain that include a large number of worldwide cutomer and a number of geographically ditributed plant and ditribution center. All the facilitie (plant and ditribution center) can hold inventory and are connected to each other by an aociated tranportation link. Each cutomer i erved by one or more facilitie with pecified tranportation link. A implified verion of the network i hown in Figure 4. The network ha multiple echelon whereby material may flow from the manufacturing plant through everal ditribution center on it way to the final cutomer. Freight rate are pecific to the tranportation link involved and depend on ditance and mode of tranport. Generally, the tranportation link are claified into two type: from one facility to another facility (plant or ditribution center) and from a facility to a cutomer. Some tranportation link with certain tranportation mode are managed by third-party logitic companie; thee require either that no product be hipped through thee link with the correponding tranportation mode or that a minimum quantity be hipped in each time period. Beide the upply chain network topology, we are given the minimum and initial inventory of each facility. The inventory holding cot and the facility throughput cot are already known, together with future monthly demand of each product by each cutomer. The tranportation time of each hipping lane i known and hould be taken into account. The uncertaintie arie from the cutomer demand and freight rate. The value of thee uncertain parameter follow ome probability ditribution (uch a normal -13-

14 ditribution) with a given mean and variance. Uually, the probability ditribution of the uncertain parameter can be obtained by fitting the hitorical data for different probability ditribution or can be baed on expert opinion. The mean value of thee uncertain parameter typically come from forecating, and the variance come from hitorical data. We allow the demand and freight rate to have different level of uncertaintie changing with time. For example, in January the uncertain demand of May ha a tandard deviation a much a 20% of the mean value, but in April the tandard deviation of that demand of May reduce to 5% of the mean value a a reult of more accurate forecating and information. Different level of uncertaintie are important for the operation of indutrial upply chain and hould be taken into account in the model. The problem i to determine the monthly or weekly production and inventory level of each facility, and the monthly hipping quantitie between network node uch that the total expected cot and the total rik of the global upply chain are minimized, while atifying cutomer demand over the pecified planning horizon. Figure 4 Global chemical upply chain 4.2. Two-tage tochatic programming model We conider a two-tage tochatic mixed-integer programming approach to deal with different level of uncertaintie. We incorporate thi approach into a multiperiod planning model that take into account the production and inventory level, tranportation mode, time of hipment, and cutomer ervice level. In principle, the problem can be formulated a a multitage tochatic programming model. To -14-

15 reduce the computational effort, we conider only a two-tage approach. In thi twotage framework, the production, ditribution, and inventory deciion for the current time period and the tranportation mode election deciion are made here and now prior to the reolution of uncertainty, while the deciion for the ret of the time period are potponed in a wait-and-ee mode after the uncertaintie are revealed. The cenario planning approach i ued to repreent the uncertaintie. A reulting challenge i that a large number of cenario are required becaue the problem include a very large number of uncertain parameter a a reult of the multiperiod nature of the model and the large ize of the global upply chain network. To reduce the model ize and the number of cenario, we ue a Monte Carlo ampling approach to generate the cenario (Linderoth et al., 2006; Shapiro, 2000; A. Shapiro & Homem-de-Mello, 1998). Each cenario i then aigned the ame probability, with the ummation of the probabilitie for all the cenario equal to 1. For example, if we ue Monte Carlo ampling to generate 100 cenario, the probability of each cenario i given a The number of cenario i determined by uing a tatitical method to obtain olution within pecific confidence interval for a deired level of accuracy. Thi method i effective for cenario reduction, particularly for large-cale problem. A an example, for a problem with cenario, a ample ize of around 400 can find the true optimal olution with probability 95%. The proce of generating cenario by Monte Carlo ampling i illutrated through Figure 5. A the tatitical analyi method for determining the required number of cenario i not the focu of thi paper, we do not introduce the detail here and the reader can refer to our earlier work for detail (You et al., 2009). Figure 5 Dicretization of the continuou probability ditribution by uing Monte Carlo ampling for cenario generation In thi work, we ue a multiperiod formulation to allow the cot and ourcing deciion to change with time while taking into account the tranportation time for each hipment. Set, variable, and parameter of the model are defined at the end of -15-

16 thi paper. The mathematical formulation of the multiperiod mixed-integer linear programming planning model i given below. min : E[ Cot] Cot1 p Cot2 (20).t. Cot h I 1 k, j, t k, j, t kk jj t1 kk k' K jj mm t1 kk rr jj mm t1 kk k' K jj mm t1 kk rr jj mm t1 Cot h I S F kk, ', jmt,, kk, ', jmt,, S kr,, jmt,, kr,, jmt,, 2 k, j, t k, j, t, kk jj t2 kk k' K jj mm t2 kk rr jj mm t2 kk k' K jj mm t2 kk rr jj mm t2 r j tsf rr jj tt F k, j, t k, k', j, m, t S k, j, t k, r, j, mt, F kk, ', jmt,,, kk, ', jmt,,, S kr,, jmt,,, kr,, jmt,,,,, r, j, t, F k, j, t k, k', j, m, t, S k, j, t k, r, j, m, t, (21), (22) F S I I W F, 0 k, k', j, m, t k, r, j, m, t k, j k, j, t k, j, t k', k, j, m, t k', k, j, m k' K mm rr mm k' K mm j, k K, t 1 (23) F S I I W F, k, k', j, m, t, k, r, j, m, t, k, j, t1, k, j, t, k, j, t, k', k, j, m, t k', k, j, m, k' K mm rr mm k' K mm 0 k, k', j, m, t k, r, j, m, t k, j k, j, t k', k, j, m, t k', k, j, m k' K mm rr mm k' K mm P j,, k KP, t 2 (24) F S I I F, j, k K, t 1 (25) F S I I F, k, k', j, m, t, k, r, j, m, t, k, j, t1, k, j, t, k', k, j, m, t k', k, j, m, k' K mm rr mm k' K mm Skr,, jmt,, SF kr,, jm, r, jt,, dr, jt,,,,, kk mm DC j,, k KDC, t 2 (26) r j, t 1 (27) -16-

17 Sk, r, j, m, tkr,, jm,, SFr, j, t, dr, j, t,,,, kk mm r j, t 2 (28) W Q, j, t 1, k, j, t k, j k K (29) P W Q, j,, t 2, k KP (30) k, j, t, k, j,, k, j, t 1 (31) m I k j, t I k, j, t,, k, j,, t 2 (32) m I k j, t, I k, j, t F ZF F, ',, L k, k', j, m, t k, k', j, m k, k', j, m F ZF F, ',, L kk, ', jmt,,, kk, ', jm, kk, ', jm, S ZS S kr,, jm, L k, r, j, m, t k, r, j, m k, r, j, m S ZS S kr,, jm, L kr,, jmt,,, kr,, jm, kr,, jm, ZF, ZS kk, ', jm, 0,1 kr,, jm, 0,1 kk jm KKJM, t 1 (33) kk jm KKJM,, t 2 (34) KRJM, t 1 (35) KRJM,, t 2 (36) Cot1 0, Cot2 0, F, ',,, 0, F, ',,,, 0, I,, 0, I,,, 0, S,,,, 0, kr,, jmt,,, 0 kk jmt kk jmt S, SF,,, 0, W,, 0, W,,, 0 r j t k j t k j t k j t k j t kr jmt The objective function of thi tochatic mixed-integer linear programming model i to minimize the total expected cot given in (20), which include the firt-tage cot, Cot 1, and the expected econd-tage cot. Since the cenario follow dicrete ditribution, the expected econd-tage cot i equal to the product of the cenario probability, p, and the aociated econd-tage cenario cot, Cot2, ummed over all the cenario. Both the firt-tage cot given in (21) and the econd-tage cenario cot given in (22) are equal to the um of the following item: Inventory holding cot for all product at all facilitie for all time period Freight cot for interfacility freight hipment in all the hipping lane of all the product in all time period Freight cot for facility-cutomer hipment in all the hipping lane of all the product in all the time period Facility throughput cot for interfacility hipment for all the hipping lane of all the product in all the time period Facility throughput cot for facility-cutomer hipment for all the hipping lane of all the product in all the firt-tage time period -17-

18 Penalty cot of all the product for unmet demand of all the cutomer in all the time period Six type of contraint are included in the model. The ma balance relationhip for the plant are given in contraint (23) and (24), the ma balance for ditribution center are given in contraint (25) and (26), the demand balance for cutomer are given in contraint (27) and (28), production capacity contraint are given in (29) and (30), and minimum inventory level contraint are given in (31) and (32); contraint (33) (36) are minimum tranportation level contraint for elected tranportation link/mode managed by third-party logitic companie. Contraint (23), (25), (29), (31), (33), and (35) are firt-tage contraint that do not include any cenario-dependent (econd-tage) variable, while the remaining contraint are econd-tage contraint for each cenario. The firt-tage contraint are for the production, inventory, and tranportation planning of the firt time period ( t 1), except for the demand balance contraint (27) that account for the uncertain demand realization. Binary variable ZF k,k,j,m and ZS k,r,j,m are introduced to model the emicontinuou tranportation level for elected tranportation link or mode. A lack variable SF r,j,t, i ued to model the hortfall and avoid infeaibility of the planning problem. An additional feature of thi model i that the tranportation time are taken into account through the multiperiod formulation, where hipment acro multiple time period are explicitly modeled. Minimizing the objective function in (20), ubject to the contraint in (21) (36), we can obtain the olution for the two-tage tochatic programming model. Computational reult for olving thi model with the tandard and the multicut verion of the Bender decompoition method are preented in the next ection Computational reult The problem i baed on the global upply chain of a major commodity chemical producer. We conider a planning horizon of one year, which i ubdivided into 12 time period, one month a a time period. Two product are produced and ditributed in the global upply chain. The cutomer demand and freight rate, which are uncertain, follow normal ditribution, with the forecat a the mean value and the variance coming from the hitorical record. The demand uncertainty ha three level -18-

19 of tandard deviation. For the current month the tandard deviation of demand i 5% of the mean value; in the coming three month, the tandard deviation i 10% of the mean value; and for the remaining eight month, the demand ha a tandard deviation of 20% of the mean value. Similarly, the freight rate ha two level of uncertainty. For the current month, the variance i 0 (i.e., determinitic cae); for the remaining 11 month, the freight rate ha a tandard deviation of 10% of the mean value. Three makeup intance are conidered, repreenting three upply chain network. The firt intance i for a mall network with 2 plant, 4 ditribution center, 2 cutomer, 1 tranportation mode and 9 tranportation link. The econd intance i for a medium ize upply chain network with 5 plant, 17 ditribution center, 46 cutomer, 4 tranportation mode and 75 tranportation link. The third intance i for a large network with 14 plant, 70 ditribution center, 126 cutomer, 14 tranportation mode, and 328 tranportation link. Although the ize of the tochatic programming problem exponentially increae a the number of cenario increae, we found that at leat 1,000 cenario are required in order to achieve reaonable confidence interval. Thu, we conider 1000 cenario for each of the three intance. The problem ize of the determinitic equivalent for three intance are given in Table 3, and the ize of the firt-tage and econd-tage ubproblem are lited in Table 4-5. All the intance are modeled with GAMS and olved with the CPLEX 12 olver on an IBM T400 laptop with an Intel Core Duo 2.53 GHz CPU and 2 GB RAM. We note that none of thee intance can be olved directly becaue of their large ize. Thu, the tandard and multicut verion of the Bender decompoition method are ued. The optimality tolerance for both method are et to 0.001%. Table 3 Problem ize of the determinitic equivalent of the numerical example Problem Size Intance 1 Intance 2 Intance 3 No. of Binary Variable No. of Continuou Variable 423,036 3,703,384 75,356,014 No. of Contraint 201,018 1,301,189 52,684,

20 Table 4 Problem ize of the firt-tage problem of the numerical example Problem Size Intance 1 Intance 2 Intance 3 No. of Binary Variable No. of Continuou ,014 Variable No. of Contraint ,187 Table 5 Problem ize of the econd-tage problem of the numerical example Problem Size Intance 1 Intance 2 Intance 3 No. of Continuou 423 3,703 75,352 Variable No. of Contraint 201 1,301 52,682 The computational performance of the tandard and multicut verion of the Bender algorithm are hown in Figure We can ee that how the upper bound decreae and the lower bound increae with the number of ation, and how the computational time increae for both olution method in all thee figure. For Intance 1, the mall-cale problem (reult hown in Figure 6 and 7), the tandard Bender method require 21 ation (around 12 CPU-econd) to converge, while the multicut verion can reach the ame optimality gap in 6 ation (4 CPUecond). Similarly, for Intance 2 with a medium-ize upply chain network (reult hown in Figure 8 and 9), the multicut method require only 45 ation (around 5 CPU-minute) to converge, while the tandard Bender method take 534 ation (around 45 CPU-minute) to reach to the ame optimality tolerance. A the problem ize become larger, the multicut Bender method i computationally much more efficient than the tandard method. For Intance 3, the larget problem (reult hown in Figure 10 and 11), the multicut verion need only 47 ation (around 11 CPUminute), while the tandard Bender method require 564 ation (about 3.5 CPUhour). The high computational efficiency of the multicut Bender method i becaue it mater problem require relatively mall olution time depite it large ize, and the number of ation i ignificantly reduced a a reult of the multiple cut. In contrat, while the mater problem in the tandard Bender method i maller in ize and fater to olve, it alo require a ignificantly larger number of ation. Note that both algorithm would benefit from olving the cenario ubproblem with parallel computing and coordinate through a mater-worker computational framework -20-

21 (Linderoth & Wright, 2003), which could ignificantly reduce the computational time. Figure 6 Comparion between the tandard Bender method and the multicut verion in term of number of ation for the firt intance of the global upply chain planning problem Figure 7 Comparion between the tandard Bender method and the multicut verion in term of CPU-econd for the firt intance of the global upply chain -21-

22 planning problem Figure 8 Comparion between the tandard Bender method and the multicut verion in term of number of ation for the econd intance of the global upply chain planning problem Figure 9 Comparion between the tandard Bender method and the multicut verion in term of CPU-econd for the econd intance of the global upply chain planning problem -22-

23 Figure 10 Comparion between the tandard Bender method and the multicut verion in term of number of ation for the third intance of the global upply chain planning problem Figure 11 Comparion between the tandard Bender method and the multicut verion in term of CPU-econd for the third intance of the global upply chain planning problem -23-

24 5. Concluion In thi work, we decribed a multicut verion of the Bender decompoition method for the olution of two-tage tochatic programming problem. We dicued the theory behind thi algorithm and proved it convergence property. Two example were preented to illutrate the application of the propoed olution method. The firt example involve production-tranportation planning under demand uncertainty. A mall example, for which the global optimal olution can be eaily obtained by olving it determinitic equivalent, wa olved with both the tandard and the multicut verion of the Bender decompoition method. The reult illutrated the effectivene of the multicut method. The econd example involved a global chemical upply chain planning under demand and freight rate uncertainty. The decompoition method wa teted on three large-cale intance, which cannot be olved directly with a regular peronal computer. Computational tudie howed that although both verion of the Bender decompoition method can olve large-cale tochatic programming problem with reaonable computational effort, ignificant aving in CPU time can be achieved by uing the propoed multicut algorithm. Future work will focu on invetigating valid inequalitie, uch a the one propoed by Georgio et al. (2011) and Santoo et al (2005) that can be ued to initialize the decompoed problem and improve the efficiency of the propoed algorithm. Another future reearch direction i to invetigate how to accelerate the Bender decompoition algorithm, uch a developing efficient cut bundle generation method (Saharidi, et al. 2010). Acknowledgment We gratefully acknowledge financial upport from the Dow Chemical Company, the Pennylvania Infratructure Technology Alliance (PITA), the National Science Foundation under Grant No. CMMI , and the U.S. Department of Energy under contract DE-AC02-06CH Nomenclature for Section 3 Set/Indice I Set of production plant indexed by i -24-

25 L S Set of demand zone indexed by l Set of cenario indexed by Deciion Variable (value: 0 to ) E[ profit ] Total expected profit prod i Production amount at plant i ale l, Total amount of the product old to demand zone l of cenario hip il, Tranportation amount from plant i to demand zone l unold l, Unold amount at demand zone l of cenario Parameter cap i Production capacity of plant i cpd i ctr il, cu l demand l Unit production cot at plant i Unit tranportation cot from plant i to demand zone l Unit unold product cot in demand zone l Demand in demand zone l of cenario demand l, Demand in demand zone l of cenario p price l Probability of cenario Sale price at demand zone l Nomenclature for Section 4 Set, Subet, and Indice J Set of product indexed by j K Set of facilitie (including plant and ditribution center) indexed by k K DC Set of ditribution center indexed by k K P M R S Set of manufacturing plant indexed by k Set of tranportation mode indexed by m Set of cutomer indexed by r Set of cenario indexed by -25-

26 T Set of time period indexed by t KKJM Subet of the combination of (k, k, j, m) that ha a minimum tranportation level requirement if elected KRJM Subet of the combination of (k, r, j, m) that ha a minimum tranportation level requirement if elected Deciion Variable (value: 0 or 1) ZF kk, ', jm, ZS kr,, jm, Binary variable, equal to 1 if tranportation mode m for interfacility freight of product j from facility k to k i elected Binary variable, equal to 1 if tranportation mode m for facility-cutomer freight of product j from facility k to cutomer r i elected Deciion Variable (value: 0 to ) Cot 1 Firt-tage cot Cot2 Second-tage cot of cenario E[ Cot ] Total expected cot F kk, ', jmt,, Interfacility freight of product j from facility k to k with mode m at time period t F kk, ', jmt,,, Interfacility freight of product j from facility k to k with mode m at time period t of cenario I k j, t, Inventory level of product j at facility k at time period t I k j, t,, Inventory level of product j at facility k at the end of time period t of cenario S kr,, jmt,, S kr,, jmt,,, Facility-cutomer freight of product j from facility k to cutomer r with mode m at time period t Facility-cutomer freight of product j from facility k to cutomer r with mode m at time period t of cenario SF r, j, t, Unmet demand of product j in cutomer r at time period t of cenario W k, j, t Production amount of product j at plant k at time period t, k KP W k, j, t, Production amount of product j at plant k at time period t of cenario, k KP Parameter d r j, t,, Demand of product j in cutomer r at time period t of cenario h k, j, t Unit inventory cot of product j in facility k at time period t F L kk, ', jm, Minimum tranportation amount of product j from facility k to k with mode m at each time period if thi tranportation link/mode i elected -26-

27 I Initial inventory level of product j at facility k 0 k, j m I k j, t, Minimum inventory of product j at facility k at time period t p Probability of cenario Q k, j Capacity of plant k for product j, k KP S L kr,, jm, Minimum tranportation amount of product j from facility k to cutomer r with mode m at each time period if thi tranportation link/mode i elected kk, ', mjt,, Freight rate of product j from facility k to k with mode m at time period t kr,, jmt,, Freight rate of product j from facility k to cutomer r with mode m at time period t Freight rate of product j from facility k to k with mode m at time t of cenario kk, ', jmt,,, kr,, jmt,,, Freight rate of product j from facility k to cutomer r with mode m at time period t of cenario k, j, t Unit throughput cot of product j in facility k at time period t r, j, t Unit penalty cot of product j for lot unmet demand in cutomer r at time period t Shipping time of product j from facility k to facility k with mode m kk, ', jm, kr,, jm, Shipping time of product j from facility k to cutomer r with mode m -27-

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