Risk Management for a Global Supply Chain Planning under Uncertainty: Models and Algorithms

Transcription

1 Rik Management for a Global Supply Chain Planning under Uncertainty: Model and Algorithm Fengqi You 1, John M. Waick 2, Ignacio E. Gromann 1* 1 Dept. of Chemical Engineering, Carnegie Mellon Univerity, Pittburgh, PA The Dow Chemical Company, Midland, MI July 2008 Abtract In thi paper we conider the rik management for mid-term planning of a global multi-product chemical upply chain under demand and freight rate uncertainty. A two-tage linear tochatic programming approach i propoed within a multi-period planning model that take into account the production and inventory level, tranportation mode, time of hipment and cutomer ervice level. To invetigate the potential improvement by uing tochatic programming, we decribe a imulation framework that relie on a rolling horizon approach. The tudie ugget that at leat 5% aving in the total real cot can be achieved compared to the determinitic cae. In addition, an algorithm baed on the multi-cut L-haped method i propoed to effectively olve the reulting large cale indutrial ize problem. We alo introduce rik management model into the tochatic programming model, and multi-objective optimization cheme are implemented to etablih the tradeoff between cot and rik. To demontrate the effectivene of the propoed tochatic model and decompoition algorithm, a cae tudy of a realitic global chemical upply chain problem i preented. Keyword: Supply Chain Management, Rik Management, Stochatic Programming, Multicut L-haped Method, Simulation * To whom all correpondence hould be addreed. gromann@cmu.edu; Tel.: ; fax:

2 1. Introduction Global upply chain in the proce indutrie are uually very large cale ytem that can be compried of up to hundred of or even thouand of production facilitie, ditribution center and cutomer. Due to competition in the global marketplace, proce indutrie are facing increaing preure to manage their upply chain o a to reduce cot and rik. 1, 2 To achieve thi goal, effective mathematical tool for large-cale upply chain optimization, particularly for cot reduction and rik management, have drawn ignificant attention. 3 Thi paper i motivated by a real world application originating at The Dow Chemical Company, which ha everal global buine unit that upply multiple product to world wide cutomer. A large buine unit can pend well into the hundred of million of dollar every year on it upply chain to handle and ditribute the product. A determinitic planning model can be a ueful tool to help the buine unit reduce cot by taking into account the flexibility in the ytem to hift production, inventory, and hipping volume in uch a way that cutomer demand i met while cot are minimized. However, due to inaccurate forecat of cutomer demand and energy price, upply chain planning give rie to variou type of financial rik. Since addreing thee problem i a non-trivial tak, it i the objective of thi work to develop optimization model and olution algorithm for the rik management of large cale upply chain tactical planning under demand and freight rate uncertaintie. We conider in thi paper the problem of midterm planning for a large cale multiproduct upply chain under demand and freight rate uncertainty for which a two-tage tochatic linear programming approach i propoed, incorporating a multi-period 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 are made here-and-now prior to the reolution of uncertainty, while the deciion for the ret time period are potponed in a wait-and-ee mode. A reulting challenge i that a large number of cenario are required becaue the problem include a very large number of uncertain parameter due to the multi-period nature and the large ize of the upply chain network. To reduce the model ize and the number of cenario, we ue a Monte Carlo ampling approach to dicretize the continuou probability ditribution function and to generate the cenario. To quantify the cot aving -2-

3 achieved by modeling uncertainty in upply chain planning, we decribe a imulation framework that relie on a rolling horizon approach. Simulation tudie on the cae problem ugget that at leat 5% aving in the total cot can be achieved by uing the tochatic approach compared to the determinitic one. To olve the reulting large cale indutrial ize problem effectively, an algorithm baed on the multi-cut L-haped method i propoed. A an additional enhancement, we introduce four rik management model by incorporating different rik meaure into the propoed tochatic programming model. Different rik metric, including variance, variability index, probabilitic financial rik and downide rik are ued to explicitly meaure the rik ariing from uncertain cutomer demand and freight rate which allow managing thee rik according to the deciion maker preference. Multi-objective optimization cheme are alo developed to tradeoff the cot minimization and rik minimization objective for the global upply chain planning. To demontrate the effectivene of the propoed tochatic model and decompoition algorithm, a realitic cae tudy of a global chemical upply chain problem i preented. The problem addreed in thi paper ha a number of novel feature. Firt, we incorporate Monte Carlo ampling in a tochatic programming framework to reduce the number of cenario for a real world application. Secondly, we propoe a imulation framework baed on iteratively olving determinitic and tochatic programming problem o a to quantitatively ae the cot aving achieved by the ue of tochatic programming. The third feature i that we take into account the election of different tranportation mode with different tranportation time in the tochatic programming model and the imulation framework. To our knowledge, planning problem that conider tranportation time and tranportation mode under uncertainty have not been addreed in thi manner. An additional feature i that we implemented a multi-cut L-haped method to olve the large cale problem from a real world cae tudy. The propoed algorithm proved to be very effective for olving large-cale tochatic linear programming problem. Moreover, we preent a comprehenively comparion of everal rik management model for planning under uncertainty. The ret of thi paper i organized a follow. Section 2 review ome relevant literature on upply chain tactical planning under uncertainty and rik management. The general problem tatement i given in Section 3. Section 4 preent the two-tage tochatic programming model. A imulation framework to quantify the different cot by uing the tochatic and determinitic approache i propoed in Section 5. An -3-

4 efficient algorithm to olve the indutrial ize problem i preented in Section 6. Section 7 preent the rik management model, along with a comparion between different rik meaure and the olution quality. Numerical reult from a real world cae tudy of a large-cale global chemical upply chain problem are preented in Section 8. Finally, Section 9 conclude on the performance of the propoed tochatic model and decompoition algorithm. 2. Literature Review Tactical upply chain planning typically cover a midterm time horizon of between few month to one year, and deciion cover iue uch a production, inventory and ditribution. 4 Related work include, for intance, the one by Wilkinon et al., 5 who propoe an approach to integrate production and ditribution in multiite facilitie uing the reource tak network framework. Bok et al. 6 propoe a multiperiod upply chain optimization model for operational planning of continuou flexible proce network where ale, intermittent deliverie, production hortfall, delivery delay, inventory profile and changeover cot are taken into account. A bilevel decompoition algorithm wa propoed, which reduced the computational time ignificantly. Jackon and Gromann 7 preent a temporal decompoition cheme baed on Lagrangean decompoition for a nonlinear programming problem model for multi-ite production and ditribution planning, where nonlinear term arie from the relationhip between production and phyical propertie or blending ratio. Chen et al. 8 preent a multi-product, multitage and multiperiod production and ditribution planning model. They alo propoed a two-phae fuzzy deciion making method to obtain a compromie olution among all participant of the multi-enterprie upply chain. A multiproduct upply chain planning model with conideration of duty drawback i propoed by Oh and Karimi. 9 Recently, Guillen et al. 10 preent a mixed-integer linear programming model for tactical planning and operational cheduling of chemical upply chain with multi-product, multi-echelon ditribution network with conideration of financial management iue. All of thee model are determinitic upply chain planning model that do not take into account the uncertaintie or rik in the upply chain planning proce. A number of approache have been propoed in the chemical engineering literature for the quantitative treatment of uncertainty in the deign, planning and cheduling -4-

5 problem. A claification of different area of uncertainty for batch chemical plant deign i uggeted by Subrahmanyam et al., 11 where uncertainty in price and demand, equipment reliability and manufacturing are taken into account. The author ued a cenario-baed approach, which attempt to capture uncertainty by repreenting it in term of 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. The cenario-baed approach provide a traightforward way to implicitly account for uncertainty. It major drawback i that the problem ize increae exponentially a the number of cenario increae. Thi i particularly true when uing continuou multivariate probability ditribution with Gauian quadrature integration cheme. Thee difficultie can ometime be circumvented by analytically integrating continuou probability ditribution function for the random parameter. 12, 13 While thi approach can lead to a reaonable ize determinitic equivalent repreentation of the probabilitic model, thi i often at the expene of introducing nonlinearitie into the model. Furthermore, the nonlinear term in the reulting determinitic equivalent problem are often nonconvex requiring global optimization technique A recent popular method to addre the uncertainty i to ue Monte Carlo ampling in the cenario planning framework 18, 19 and then combine it with tatitical method to determinate the number of required cenario o a to achieve a deired level of accuracy. 20 By uing thi method, the required number of cenario in the tochatic program can be ignificantly reduced, while the olution quality can be guaranteed at the deired level. 21 In thi work, we ue the Monte Carlo ampling method to deal with large cale upply chain planning problem under uncertainty. In the tochatic programming model, the total expected performance meaure i optimized o a to obtain optimal olution that perform well on average for all the cenario. However, tandard tochatic programming method uually do not provide any control on the olution variability over the different cenario. In other word, the deciion maker are aumed to be rik-neutral. One may have different attitude toward the rik, thu the upply chain rik hould be controlled and managed baed on the deciion maker preference. Related work about rik management include, for intance, Eppen and Martin 22 who propoe the downide rik a a rik meaure and incorporated it into a two-tage tochatic programming model for the production capacity planning under demand uncertainty in auto indutry. Later Mulvey et al. 23 decribe a robut optimization model to control the mean value and variance of the -5-

6 objective function in tochatic program. Ahmed and Sahinidi 24 propoe the upper partial mean a a meaure of rik and apply it in the long-term chemical proce planning. Applequit et al. 25 dicu rik premium a a meaure that provide the bai for a rational balance between expected value of invetment performance and variance. Recently, Barbaro and Bagajewicz 26 introduced the probabilitic financial rik a a metric of rik for planning under uncertainty problem. Similar technique are preented by Bonfill et al. 27 for managing financial rik in cheduling problem. The probabilitic financial rik meaure i alo ued for refinery planning 28 and hort term cheduling with pricing policie Problem Statement The problem addreed in thi paper 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 world wide 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 1. 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, one i from a facility to another facility (plant or ditribution center), and the other one i from a facility to a cutomer. Beide the upply chain network topology, we are alo 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. -6-

7 Plant Ditribution Center Cutomer Figure 1 Global chemical upply chain The uncertaintie arie from the cutomer demand and freight rate. The value of thee uncertain parameter follow ome probability ditribution (uch a, but not retricted to, normal ditribution) with 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 baed on expert opinion. The mean value of thee uncertain parameter typically come from forecating, and the variance come from hitorical data. 12 It i important to note that 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 due to more accurate forecating and information. Different level of uncertaintie are very important for the operation of indutrial upply chain, and hould be taken into account in the model. The problem i to determine the monthly production I 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. I The model could be alo eaily extended to deal with weekly production planning (or even horter time interval) by changing the length of time period. -7-

8 4. Stochatic Programming Model 4.1. Two-tage Approach We conider a two-tage tochatic programming 30 approach to deal with different level of uncertaintie and incorporate it into a multi-period planning model that take into account the production and inventory level, tranportation mode and time of hipment and the cutomer ervice level. II In the two-tage framework, the production, ditribution and inventory deciion for the current time period 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 due to the multi-period nature of the model and the large ize of 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. Each cenario i then aigned the ame probability with the ummation of the probabilitie for all the cenario equal to 1. 18, 19 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 20, 21 to obtain olution within pecific confidence interval for a deired level of accuracy. Thi method i very 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% Mathematical Formulation In thi work, we ue a multi-period formulation to allow the cot and ourcing deciion to change with time while taking into account the tranportation time for each hipment. The model include five type of contraint. They are ma balance contraint for the production plant, ditribution center and cutomer, together with the contraint for production capacity and minimum inventory level. The definition of et, variable and parameter of the model are given at the end of thi paper. Note that exchange rate, taxe, tariff and duty drawback ue linear approximation 9, 31 and II In principle, the problem can be formulated a a multi-tage tochatic programming model, to reduce the computational effort we only conider a two-tage approach. -8-

9 are taken into account in the parameter of freight rate and facility throughput cot. The mathematical formulation of the multi-period linear programming planning model i given in the following ection Ma balance for plant Let u conider the ma balance for plant k product j at the firt time period ( t = 1). At the firt time period ( t = 1), all the deciion are aumed to be independent of the future cenario. The ma balance for the plant period ( t = 1) i then given a follow. k K product j at the firt time F + S = I I + W + F λ, 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 m M r R m M k' K m M P j, k K, t = 1 (1) P Equation (1) tate that the total freight hipped from plant k K to other facilitie and cutomer with all the tranportation mode m M hould be equal to the change in inventory plu the production amount and the volume hipped to plant k K P from other facilitie. Becaue we need to conider the tranportation time during the hipping proce, the input freight coming from other facilitie hould tart at the time period of t λ k ', k, j, m o that the freight can arrive at the detination at time P period t, where λ k', k, j, m i the hipping time from facility k ' to plant k K of P product j with tranportation mode m. For the remaining time period ( t 2 ) contained in the econd tage time period, mot of the deciion will be the econd tage deciion. So for plant k K product j at time period t 2 for cenario, the ma balance can be expreed a follow. F + S = I I + W + F λ, k, k', j, m, t, k, r, j, m, t, k, j, t 1, k, j, t, k, j, t, k', k, j, m, t k', k, j, m, k' K m M r R m M k' K m M j,, k KP, t 2 (2) Equation (2) i imilar to Eq. (1), but all the variable are replaced by the econd tage variable, i.e. related to cenario. It i important to note that for the econd time period ( t = 2), the term Ik, j, t 1, in Eq. (2) refer to the ending inventory level of the firt time period, which i a firt tage deciion independent of cenario. Similarly, if the freight F tart from the firt time period, i.e. t = λ k ', k, j, m + 1, then the k', k, j, m, t λk', k, j, m, freight i alo a firt tage deciion independent of the cenario. P -9-

10 Ma balance for ditribution center For the ditribution center, the ma balance equation i very imilar to that for a plant; only the production term, i miing. So for ditribution center k KDC W i j, t and product j in the firt time period ( t = 1), the ma balance equation i given by: F + S = I I + F λ, 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 m M r R m M k' K m M j, k K, t = 1 (3) For the remaining time period ( t 2 ) and cenario, the ma balance i given by: F + S = I I + F λ, k, k', j, m, t, k, r, j, m, t, k, j, t 1, k, j, t, k', k, j, m, t k', k, j, m, k' K m M r R m M k' K m M DC j,, k KDC, t 2 (4) Similarly, for the econd time period ( t = 2 ), the term Ik, j, t 1, in Equation (4) refer to the ending inventory level of the firt time period, which i a firt tage deciion. Thu Ik, j, t 1, hould be replaced by Ik, j, t 1 when t = 2. Similarly, hipment F that originate in the firt time period, i.e. t = λ k ', k, j, m + 1, are firt tage k', k, j, m, t λk', k, j, m, deciion independent of the cenario and hould be replaced by F k', k, j, m, t λ. k', k, j, m Ma balance for cutomer To atify the demand of product j at cutomer r in time period t, the um of all hipment from other facilitie (plant and ditribution center) via all the hipping mode m tarting at time period t λ kr,, jm, (and arriving at cutomer r at time period t) hould be no le than the demand ( d r j, t,, ). To atify certain ervice level and to enure the contraint i feaible, we introduce a poitive lack variable SF r, j, t, to quantify the unmet demand. Hence, the ma balance for product j at cutomer r in the firt time period ( t = 1) and cenario can be formulated a follow: Skr,, jmt,, λ + SF kr,, jm, r, jt,, dr, jt,,,,, k K m M r j, t = 1 (5) For the remaining time period ( t 2 ) and cenario, the ma balance for product j at cutomer r i given a: Sk, r, j, m, t λkr,, jm,, + SFr, j, t, dr, j, t,, r, j,, t 2 (6) k K m M Note that in contraint (6), we conider cutomer demand a the lower bound of the ale. One could alo enforce the ale to be equal to the demand by changing contraint -10-

11 (6) a an equality Capacity contraint The production amount ( W k, j, t, W k, j, t, ) of each plant ( k KP ) hould not exceed the capacity ( Q k, j, k KP ). W Q, j, t = 1, k, j, t k, j k K (7) P W Q, j,, t 2, k KP (8) k, j, t, k, j Minimum inventory contraint The minimum inventory of product j in facility k at each time period t hould be atified. Equation (9) and (10) model thi contraint.,, k, j, t = 1 (9) m I k j, t I k, j, t,, k, j,, t 2 (10) m I k j, t, I k, j, t Objective function: expected total cot The objective function of thi tochatic linear programming model i to minimize the total expected cot that include the firt tage cot, Cot 1, plu 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, econd tage cenario cot, Cot2, ummed over all the cenario. p, and the aociated E[ Cot] = Cot1+ p Cot2 (11) S Both the firt tage cot and the econd tage cenario cot 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 inter-facility 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 inter-facility 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 -11-

12 Penalty cot of all the product for lot unmet demand of all the cutomer in all the time period Thu, the firt tage cot i given a, Cot h I F S 1 = k, jt, k, jt, + γkk, ', jmt,, kk, ', jmt,, + γkr,, jmt,, kr,, jmt,, k K j J t= 1 k K k' K j J m M t= 1 k K r R j J m M t= 1 + δ F + k, j, t k, k', j, m, t k, j, t k, r, j, m, t k K k' K j J m M t= 1 k K r R j J m M t= 1 (12) The cot of each cenario i equal to, Cot h I F S δ 2 = k, jt, k, jt,, + γ kk, ', jmt,,, kk, ', jmt,,, + γ kr,, jmt,,, kr,, jmt,,, k K j J t 2 k K k' K j J m M t 2 k K r R j J m M t 2 + δ F + δ S + ηr, j, tsf,,, k, j, t k, k', j, m, t, k, j, t k, r, j, m, t, k K k' K j J m M t 2 k K r R j J m M t 2 r R j J t T S, (13) Minimizing the objective function in (11) (13), ubject to the contraint in (1) (10), we can obtain the olution for the two-tage tochatic programming model. However, ince the number of cenario may be too large we ue a ampling cheme a dicued in the next ection. r j t 4.3. Calculation of Confidence Interval The number of cenario i determined by the deired level of accuracy of the olution, which can be meaured by the confidence interval of the expected total cot. The confidence interval can be calculated a follow. The Monte Carlo ampling variance etimator of the reult for a tochatic programming problem, which i 18, 19 independent of the probability ditribution of the uncertain parameter, i given by, n ( [ ] ) 2 ECot Cot = 1 (14) Sn ( ) = n 1 where n i the number of cenario and Cot i the total cot of cenario. Then the confidence interval of 1 α i given a: /2 ( ) /2 ( ) ECot [ ] z S n, ECot [ ] z S n α α + (15) n n where z α /2 i the tandard normal deviate uch that 1 α / 2 atifie for a tandard normal ditributed variable z ~ N(0, 1), Pr( z z α /2) = 1 α / 2. For example, for 95% confidence interval (i.e. 1 α = 95% ), we have z α /2 =

13 On the other hand, if we are given the ampling etimator Sn ( ) and the deired confidence interval H, the minimum number of cenario required can be determined by, 2 zα /2 S( n) N = H (16) Therefore, to determine the number of cenario N, we firt olve the tochatic programming model with a mall number of cenario n (uch a ), to etimate the value of ampling etimator Sn ( ) by uing Eq. (14). Then uing Eq. (15) and (16), we can determine the required number of cenario for a deired confidence interval. 20, Simulation Framework To ae the impact of uing the tochatic programming approach, we developed a imulation framework. The baic idea i to compare the imulated operation of two planner, one uing a determinitic model for planning and the other one uing a tochatic model for planning (Figure 2). At the beginning of each time period, the tochatic planner will run the two-tage tochatic programming model with the current time period (for intance, month) a the firt tage time period in the model and the remaining time period a the econd tage time period. After a olution i returned, the tochatic planner will execute the deciion for the current time period. Similar action will be taken by the determinitic planner uing the determinitic model. After both planner execute their deciion, the ytem randomly generate the information for demand and freight rate. Thee include the realization of the uncertain demand for the current time period, and the forecating value of demand and freight rate for the future time period. Both planner then update their information. The tochatic planner ue the information for both mean value and variance of the uncertain parameter (including the demand and freight rate), while the determinitic planner only ue the mean value of the uncertain parameter. Once the information i updated both planner move on to the next time period. The entire ytem operate under a rolling horizon approach a hown in Figure 3. For example, if the planning horizon i 12 month and January i the current time period, the tochatic planner will run the tochatic model with the deciion for January a the firt tage deciion and the deciion for thi February to December a -13-

14 the econd tage deciion. After the problem i olved, the planner will execute the deciion for January only. Then, in the next iteration the deciion for February are conidered a the firt tage deciion, and the deciion for thi March to the following January are treated a the econd tage deciion. The proce continue until deciion are executed for December. The determinitic planner follow a imilar proce uing the determinitic model and mean value of the uncertain parameter. Stochatic Planner Solve Stochatic model and execute deciion for period t Update information on the uncertain parameter (mean and variance) period t-1 Randomly generate demand and freight rate period t+1 Determinitic Planer Solve Determinitic model and execute deciion for period t Update information on the uncertain parameter (only mean value) period t-1 period t period t+1 Figure 2 Simulation framework Thi proce give rie to a rolling horizon where the firt time period of the model i moving forward but the length of the planning horizon i unchanged. The proce once initiated continue until an entire year deciion have been made. In thi way we imulate the typical planning cycle carried on in an indutrial etting. -14-

15 Figure 3 Rolling horizon trategy There are everal iue that require pecial attention in thi imulation framework. The firt i initial condition. The previou time period ending inventory repreent the initial inventory of the new time period. For example, in the econd iteration (for the deciion of February), the ending inventory of January hould be treated a the initial inventory of February. Another iue we need to take into account i the tranportation time. All inter-facility hipment initiated in previou time period hould be treated a pipeline inventorie and conidered a part of the initial inventorie at the detination in the arrival time period. For example, if in the firt iteration, the planner decide to hip ome product from one facility to another and the hipment take two time period to arrive, then the amount of thi hipment hould be conidered a part of the initial inventory of the detination for March in the next two iteration, i.e. the econd time period in the econd iteration and the firt time period in the third iteration (Figure 3). Similarly, for a hipment from a facility to a cutomer, the amount of the hipment hould be conidered a part of the demand realization in future time period if the tranportation time exceed one time period. The third iue i the difference in demand and freight rate uncertainty depending on the length of the forecat. A the rolling horizon move forward, the variance of the uncertain demand for a particular time period will change becaue we conider different level of uncertainty for different forecating horizon. Thi mut be taken into account in the imulation. For example in Figure 3, the demand for March ha a tandard deviation of 10% of the mean value in the firt and econd iteration, but in the third iteration, March become the firt time period and the tandard deviation of demand i reduced to 5% of the mean value. A final iue that mut be dealt with i the variation in the reult that are obtained for a imulated year due to the random cutomer demand driving the optimization. In other word, the difference between the tochatic cae and the determinitic will vary in different imulation cycle becaue the demand encountered may be different. To addre thi variation the imulation ytem iterate through a elected number of imulation cycle to produce data that can be ued to report tatitic of the difference between the annual performance of the tochatic cae and the determinitic cae. Thi proce give rie to an inner loop tepping though the time period (for intance, -15-

16 month) and an outer loop iterating through imulation cycle (for intance, year). A flow chart of the whole imulation framework i hown in Figure 4. Calculate the real cot, tore data, Set iter = iter+1, t =1 Solve the S/D model and implement the deciion for current time period Next iteration Move to next time period t = t+1 Randomly generate demand and freight rate information Update information No No t = t_max? Ye Reach iteration limit? Ye STOP Figure 4 Simulation flowchart 6. Solution Algorithm Stochatic programming model that rely on cenario are often computationally very demanding becaue their model ize increae exponentially a the number of cenario increae. In particular, the determinitic equivalent of the problem addreed in thi paper cannot be olved directly due to it very large ize (ee Section 8 for detail). Therefore, we need an effective algorithm to overcome the computational challenge. A popular method for olving tochatic programming model i the L-haped method, 30, 32 which take advantage of the pecial decompoable tructure of the two-tage tochatic programming model. Conider the following general form of the two-tage tochatic programming model (P0). (P0) min xy, + (17) T T c x pq y S.t. Ax = b, x > 0 (18) Wy = h T x, y(w ) 0, S (19) where x i the vector that tand for the firt tage deciion variable, and y are the econd tage deciion for each cenario. Equation (17) tand for the objective function given in (11)-(13). Equation (18) tand for the contraint without econd tage deciion, i.e. contraint (1), (3), (7) and (9) in the tochatic programming model. -16-

17 Equation (19) i for the econd tage contraint, i.e. contraint (2), (4), (5), (6), (8), (10) in the tochatic programming model. c and q are the vector of coefficient for the firt and econd tage deciion in the objective function, i.e. the unit inventory, hipping, throughput, and penalty cot. A and b are parameter matrix independent of the cenario, while W, h and T are parameter matrix for each cenario S. The expanded verion of the general model (P0) i given in equation (20). We can ee that the model ha a pecial angular form, which can be decompoed into a mater problem and a number of cenario ubproblem. Mater problem Scenario ubproblem (20) The baic idea of the tandard L-haped method i to firt olve the model with thoe contraint that do not include the econd tage variable to obtain the value of firt tage deciion. Then we fix the firt tage deciion and olve all the cenario ub-problem that include econd tage deciion to obtain the optimal value of the econd tage deciion., If we define Q ( x ) a the objective function value of each cenario ubproblem Q ( x) = min q y T y.t. Wy = h T x, y(w ) 0 (21) then the reformulation of (P0) i a follow, (P0) c x+ p Q x (22) T min ( ) x S.t. Ax = b, x > 0 (18) To olve (P0), we can take advantage of the dual propertie of (21) by introducing a new variable θ for p Q ( x), and iterate between the mater problem (P1) and the S -17-

18 cenario ubproblem (P2). The mater problem (P1) i given by, (P1) min x, θ T c x + θ.t. θ ex+ d, l = 1..N (23) l Ax = b, x > 0 while the ubproblem (P2) for cenario i given by, l (P2) min y q T y.t. Wy = h T x, y(w ) 0 (21) where the inequalitie in (P1) are the cut that link the mater problem and the cenario ubproblem. e l and d l are coefficient for the Bender cut, and they are given by, e = pπ T (24) T l S d = pπ h (25) T l S where π are the optimal dual vector of contraint (21) in the ubproblem (P2) for cenario. The major tep for the L-haped method are given in Figure 5. 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 ub-problem to add a cut and return to the mater problem. -18-

19 Solve mater problem to get a lower bound (LB) Add cut Solve the ub problem to get an upper bound (UB) No UB LB < Tol? Ye STOP Figure 5 Algorithm for tandard L-haped method The tandard L-haped method only return one cut to the mater problem during each iteration. In order to peed up the algorithm we can decompoe the variable θ by cenario to return a many cut a the number of cenario in each iteration. The mater problem i then given by (P3). (P3) min x, θ T c x + S pθ.t. θ e x + d, l = 1..N (26) l l Ax = b, x > 0 where the coefficient e l and d l for the cut (26) are updated a follow e = pπ T (27) T l d = pπ h (28) T l The algorithm framework for multi-cut L-haped method i imilar to the tandard L-haped method, and i given in Figure

20 Solve mater problem to get a lower bound (LB) Add cut Solve the ub problem to get an upper bound (UB) No UB LB < Tol? Ye STOP Figure 6 Algorithm for multi-cut L-haped method Although the multi-cut L-haped method can provide tronger cut to the mater problem and reduce the number of iteration, it introduce more variable in the objective function of the mater problem, which may potentially low down the computation. Computational reult for comparing thee two algorithm are preented 30, 32 in Section 8. We hould alo note that convergence i guaranteed in both cae. 7. Rik Management Model In the tochatic programming model we optimize the total expected cot to obtain the optimal olution that are optimal on average for all the cenario. However, the expected total cot i a rik-neutral objective that cannot manage the rik explicitly. On the other hand, ome deciion maker are rik-avere and would like to manage the rik and improve the economic objective imultaneouly. Thi require extending the aforementioned tochatic programming for rik management. To manage the rik, we need firt to define a metric for rik. For comparion purpoe, we conider in thi work four popular rik meaure including variance, 23 variability index, 24 probabilitic financial rik 26 and downide rik Managing the Variance Due to the uncertain environment the total realized cot i alo uncertain (ee Figure 7). Thi cot ha a mean value and a variance. Our objective in the tochatic programming model i to minimize the expected value of the total cot, while the -20-

21 variance of the total cot i not addreed. Thu, it i poible that an optimum olution may have low expected cot but a large variance. Application of uch a olution would therefore involve a high amount of rik in that the poibility exit for the realized cot to be far higher than the expected value. If a deciion maker i rik advere uch a olution would not be atifactory. Therefore, we may want to find a robut olution that would yield imilar reult but alo conider the variance of the olution. The rik management by variance i alo called robut optimization in mot of the Operation Reearch literature. 23 In robut optimization, we not only minimize the expected total cot, but alo minimize the variance of the total cot. Since the original tochatic programming olution i the minimum expected cot olution the olution from robut optimization uually reult in higher expected cot but with le variance. Probability 0.9 Expected Cot Deirable Penalty Undeirable Penalty Cot Figure 7 Robut optimization Since rik i meaured by variance, a traightforward extenion to reduce it value i to add a variance term to the objective function of the tochatic program. Thi yield a goal programming formulation to reduce both the expected cot and the variance. The new objective function i: min ECot [ ] + ρ VCot [ ] 2 = Cot1+ p Cot2 + ρ p [( p Cot2 ) Cot2 ] (29) ' ' S S ' S where the expected cot term i equal to the firt tage cot ( Cot 1) plu the expected econd tage cot ( Cot 2 ). The variance term ( VCot) [ ] i equal to the mean p -21-

22 quare error between the expected econd tage cot ( p' Cot2' ) and the econd tage cenario cot ( Cot2 ). The coefficient ρ in the objective function i the weight coefficient for the variance. For different value of ρ we can tradeoff lower expected cot with lower cot variance. In ummary, the variance management model include the objective function given in (29), (12), (13), ubject to contraint (1)-(10). ' S 7.2. Managing the Variability Index The variance management model i a traightforward approach to reduce both expected value and variance of cot, but it include a quadratic term in the objective function (29) that make the optimization problem difficult to olve for large cale problem. To circumvent thi problem an alternative i to ue the poitive deviation between the cenario cot ( Cot1+ Cot2 ) and the expected cot Cot + P Cot ( 1 ' ' 2 ' ). Ahmed and Sahinidi 24 defined the variability index (or called upper partial mean ) a a non-negative continuou variable Δ for each cenario that i defined by the following contraint: Δ 2 ( 2 ) Cot P ' ' Cot ', Δ 0, S (30) Equation (30) tate that if the cenario cot ( Cot ) i le than the expected cot ( E [Cot] ), Δ would be 0; If the Cot i greater than the E [Cot], Δ would be equal to their poitive difference. Thi reformulation, which can be interpreted a a 1-norm meaure of the variance, yield a linear programming problem which can be olved more efficiently. The objective function of the variability index management model i to minimize the weighted um between total expected cot and the expected variability index. Thu, the variability index management model i a follow: min ECot [ ] + ρ p Δ (31) S.t. Δ Cot 2 ( P Cot 2 ), Δ 0, S (30) ' ' ' Contraint (1)-(10) Similarly, for different value of the weighted parameter ρ, we can tradeoff the expected cot and the cot variability index. -22-

23 7.3. Managing the Probabilitic Financial Rik Sometime deciion maker are not atified with a robut olution in which the variance of the cot i limited. Intead they are more concerned with the extreme of the cot pread. For example they may want a lower probability of high cot or a higher probability of low cot. In thi cae, we can ue another rik meaure, the o called probabilitic financial rik. 26 Thi meaure i defined a the probability that the real cot i higher than a certain threhold or target Ω (Figure 8). By reducing the probabilitic financial rik for threhold or target Ω, we can reduce the rik of having high cot. Figure 8 Probabilitic financial rik Uing the target Ω, rik can be defined a the probability of the cot being greater than Ω. For a cenario planning model we can introduce a binary variable Z equal to 1 if Cot Ω, otherwie equal to 0. To define the value of the binary variable propoed the following Big-M contraint: Cot Z Z, uch that Z, Barbaro and Bagajewicz 26 Ω + M, S (32) Cot Ω M (1 Z ), S (33) where M i a ufficient large poitive parameter. Contraint (32) and (33) tate that if -23-

24 the cenario cot Cot i greater than the target Ω, Z mut be 1 or ele contraint (32) will be violated; if the cenario cot Cot i le than the target Ω, Z mut be 0 or ele contraint (33) will be violated. By doing thi, we define Z a an indicator for each cenario. Thu, the probabilitic financial rik i equal to the ummation over all the cenario for the product of the cenario probability and the binary variable Z. Rik( x, Ω ) = Pr[ Cot( x) Ω ] = p Z (34) S The probabilitic financial rik management model i then follow: min E[ Cot] = Cot1+ p Cot2 (11) S min Rik( x, Ω ) = p Z (34) S.t. Cot Ω+ M Z, S (32) Cot Ω M (1 Z ), S (33) Contraint (1)-(10) Thi model ha two objective function: to minimize the probabilitic financial rik in (34) and minimize the expected total cot in (11), (12), (13), ubject to the contraint (1)-(10), (32)-(33). A there are two conflicting objective function, the correponding problem yield an infinite et of Pareto-optimal olution for which it i not poible to improve both objective function imultaneouly. 34 In order to obtain the Pareto-optimal curve for the bi-criterion optimization problem, one of the objective i pecified a an inequality with a fixed value for the bound which i treated a a parameter. There are two major approache to olve the problem in term of thi parameter. One i to imply olve it for a pecified number of point to obtain an approximation of the Pareto optimal curve, which i the ε-contraint method. 34 The other i to olve it a a parametric programming problem, 35 which yield the exact olution for the Pareto optimal curve. While the latter provide a rigorou olution approach, the former i impler to implement. For thi reaon we have elected the firt approach. The procedure include the following three tep. The firt one i to minimize the expected cot ECot [ ] to obtain the minimum expected cot, which in turn yield the larget Pareto optimal rik Rik( x, Ω ). The econd tep i to minimize Rik( x, Ω ) that yield the mallet Pareto optimal expected rik Rik( x, Ω ). The lat -24-

25 tep i to fix the rik Rik( x, Ω ) to dicrete value between the mallet and greatet value, and optimize the model by minimizing ECot [ ] at each elected point. In thi way we can obtain an approximation to the Pareto-optimal curve, together with the optimal planning deciion for different value of probabilitic financial rik Managing the Downide Rik In the aforementioned probabilitic rik management method, a binary variable i required for each cenario to calculate the probabilitic financial rik. Thu, the rik management model ize will be very large a the number of cenario increae. To avoid the integer variable, we can ue downide rik 22 intead of probabilitic rik for financial rik management. The baic idea i to introduce a poitive deviation variableψ defined a the variability index of cenario. The variable the poitive deviation between the target Ω and the cenario cot cot Cot i le than the target Ω, ψ i defined a Cot. If the cenario ψ i equal to 0. If the cenario cot Cot i greater than the target Ω, ψ i equal to their difference. Thee condition can be enforced with the following inequalitie: ψ Cot Ω, ψ 0, S (35) Then the downide rik aociated with target Ω, i defined a follow, DRik( x, Ω) = ψ (36) P Thu, we have the downide rik management model a follow: min E[ Cot] = Cot1+ p Cot2 (11) S min DRik( x, Ω ) = P ψ (36).t. ψ Cot Ω, ψ 0, S (35) Contraint (1)-(10) Similar to the probabilitic financial rik management model, a downide rik management model alo ha two objective function: to minimize the total expected cot in (11), (12), (13) and to minimize the downide rik in (36), ubject to the contraint (1)-(10) and (35). The optimal olution of thi multi-objective optimization model alo yield a Pareto curve, which can alo be obtained by uing the ε-contraint method. -25-

26 8. Cae Study In thi ection, we preent a cae tudy to demontrate the effectivene of the propoed model and algorithm. The problem i baed on the global upply chain of a major commodity chemical producer. Some baic information about the global upply chain i dicued. The reult for the tochatic programming model, imulation framework, decompoition algorithm, together with the reult for different rik management model are preented and dicued. All the intance are modeled with GAMS 36 and olved with CPLEX olver on an IBM T60 laptop with an Intel Core Duo 1.83 GHz CPU and 1GB RAM Baic Information of the Cae Study In the cae tudy we conider a planning horizon of one year, which i ubdivided into 12 time period, i.e. one month a a time period. Two product are produced and ditributed in a global upply chain, coniting of a global upply chain with 5 plant, 13 ditribution center, 121 tranportation link and 46 cutomer. 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 of tandard deviation (ee Figure 3). For the current month the tandard deviation of demand i 5% of the mean value, in the coming three month (i.e. 2-4 month), the tandard deviation i 10% of the mean value; for the remaining 8 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); in the remaining 11 month, the freight rate ha a tandard deviation of 10% of the mean value. All the other data about the upply chain, uch a the unit cot coefficient, capacitie, minimum inventory level, are omitted due to confidentiality reaon Reult for Stochatic Programming Model and Simulation We olve the cae tudy with a ampling ize of 600 cenario. The reult are given in Figure 9 and Figure 10. The minimum total expected cot i $ MM. The 95% confidence interval of the expected cot i given $0.37MM above and below thi -26-

27 value, which i relatively mall compared to the expected cot Cumulative Probability Ditribution E(Cot) = ± Cot ($MM) Figure 9 Cumulative probability ditribution for the two-tage tochatic programming model with 600 cenario 0.27 E[Cot] = $182.32± 0.37 MM (600 cenario) Probability Cot ($ MM) Figure 10 Hitogram of the reult for the two-tage tochatic programming model with 600 cenario To quantify the cot aving by uing tochatic programming, we implement the imulation framework for the cae problem. We ued a ampling ize of 1000 cenario and imulated 100 iteration. Year-by-year reult are given in Figure 11. The operational cot from tochatic planning i alway le than the operational cot from -27-

28 determinitic planning. On average, 5.70% cot aving wa achieved by uing the tochatic programming approach. Figure 12 how the component of the average operational cot for both approache. Figure 13 and 14 are the comparion on the inventory level and ourcing for one of the production facilitie. A can be een, by uing the tochatic programming approach, the production facility hold le inventorie compared to the one with determinitic approach (Figure 13), and thu the ourcing amount for thi facility by tochatic approach are fewer (Figure 14) Cot ($MM) Stochatic Soln Determinitic Soln Iteration Figure 11 Simulation reult for the real cot of one year planning 6 5 Stochatic Approach Determinitic Approach 4 Cot ($MM) Freight Cot Throughput Cot Inventory Cot Shortfall Penalty Figure 12 Average component real cot for two planner -28-