Multi source - Multiple Destination EOQ Model for Constant Deteriorating Items Incorporating Quantity and Freight Discount Policies

Proceedings of the 202 International Conference on Industrial Engineering and Operations Management Istanbul, Turkey, July 3 6, 202 Multi source - Multiple Destination EOQ Model for Constant Deteriorating Items Incorporating Quantity and Freight Discount Policies Kanika Gandhi and P. C. Jha Department of Operational Research Faculty of Mathematical Sciences University of Delhi, Delhi-0007, India Abstract Supply chain management is concerned with the flow of material and information linking across suppliers & users. The coordination among processes plays vital role when different types of products are moving from many suppliers to number of buyers and it turns into more difficult phase when products are deteriorating in nature. Current study employs the same bases to develop a model, where deteriorating natured products move from multi source to multi destination. An integrated procurement-distribution model is developed to minimize the cost associated with movement of goods from sources to destinations. The model is validated on a real life case. Keywords Inventory management, Discount model, Transportation management, Deteriorating Items, Supply chain coordination. Introduction Traditionally, marketing, planning, manufacturing, purchasing, and the distribution organizations along the supply chain operated independently. These organizations have their own objectives and these are often conflicting. Marketing's objective of high customer service and maximum sales conflict with manufacturing and distribution goals. Many manufacturing operations are designed to maximize output at lower costs with little consideration for the impact on inventory levels and distribution capabilities. Purchasing contracts are often negotiated with very little information beyond historical buying patterns. The result of these factors is that there is not a single integrated plan for the organization; rather there were as many plans as businesses. Clearly, there is a need for a mechanism through which these different functions can be integrated together. Supply chain management is a strategy through which such integration can be achieved. In this study a model has been developed to integrate the functions of purchasing, inventory, and transportation of deteriorating products from multi source to multi destination points. In the process, cost incurs during each function i.e. purchasing cost, holing cost and transportation cost. The objective of the model is to minimize the total cost incurred during all the three functions discussed above. And the decisions are inventory level at sources and destination; order size and weighted transport quantity. To boost the demand, source point offers quantity discounts on bulk purchase and freight discounts are offered by transporters on big weighted transport quantity. The unanticipated shortening of a product s life cycle can have a dramatic effect on inventory costs. Where the cause is physical decay or an unexpected decline in demand, producers can quickly find themselves quickly holding inventory that has degraded value or no value at all. Deterioration of products could be with fixed lifetime and random lifetime (Fink and Ferrell 2002). Fixed lifetime products have deterministic shelf life i.e. if a product remains unused up to its lifetime; it is considered to be out-dated and must be disposed off. Human blood is one of the examples of fixed lifetime deteriorating products. On the other hand, products whose exact lifetime cannot be determined in advance are known as random lifetime products. The analysis of such inventory is more difficult as compared to fixed lifetime products (Goyal and Giri 200). 787

The organization of this paper is as follows: Section 2 review the previous literature of related work, Section 3 discusses the problem statement & assumptions for model development, Section 4 present the proposed mathematical model for the problem, which includes sets, parameters, decision variables, formulated model, analysis of the model s equation and discount breaks. Section 5 discusses the case study of a retail store, and finally the study is concluded in section 6. 2. Review of Literature Discounts are a primary marketing mechanism to increase the size of purchases. Quantity discounts from suppliers and freight discounts from transporter are commonly encountered by organizations. Tersine and Barman (99) structures quantity and freight discounts into the order size decision in a deterministic economic order quantity system. Ghare and Schrader (963) presented the EOQ model by considering the combined effects of demand, usage and linear decay. Covert and Philip (973) used the variable deterioration rate of the two-parameter Weibull distribution, to formulate the inventory decision model under the assumptions of a constant demand rate, with no shortages allowed. Philip (974) modified this model by using the deterioration rate of the three-parameter Weibull distribution. Tadikamalla (978) adopted gamma distributed deterioration under constant demand over time, without shortages. Moon, I., Lee (2000) presented the EOQ model with a normally distributed deterioration rate. Dave and Patel (98) proposed an EOQ model under time-proportional demand, with no shortages allowed. Sachan (984) extended their model by considering shortages. Bahari-Kashani (989) generalized the problem by permitting variations in both replenishment cycle length and order quantity. Liu and Lian(999) analyzed an (s,s) continuous review perishable inventory system with a general renewal demand process and instantaneous replenishment. Moorthy et al. (992) proposed an inventory in which an item is put on display and becomes available only after the presently displayed item is sold or expired. The aging process of an item is assumed to begin after it is put on display. Wee (999) developed a deterministic inventory model with quantity discount, pricing and partial backordering when the product in stock deteriorates with time. Wang et al. (20) considered three-echelon (one producer, one distributor, and one retailer) optimal integrated inventory policy for time-sensitive deteriorating products. In the study author has empirically investigated how different deterioration rates in each echelon affect performances of individuals and integrated inventory policies. Yu and Luo (2008) studies the retailer's ordering size for deteriorating products under the condition of trade credit offered by the supplier, determinate demand and determinate deteriorating rate. Sahoo et al. (200) developed a deterministic inventory model when the deterioration is constant. Demand rate is a function of selling price and holding cost is taken as time dependent. The model is solved allowing shortage in inventory. The current study develops a model for zero shortages, which shows flow of deteriorating type multi products from multi sources to multi demand points (destinations). In the above coordination, cost associated at every stage like purchase cost, distribution cost, inspection cost and inventory carrying cost. Objective of the current study is to minimize the total costs discussed above by integrating procurement and distribution phase with incorporating quantity discount policies at the time of purchase and freight discount policies at the time of transportation. 3. Problem Statement & Assumptions The current study develops a model which shows flow of deteriorating type multi products from multi sources to multi demand points (destinations). In this coordination, the cost is associated at every stage like purchase cost, distribution cost, inspection cost and inventory carrying cost. Objective of the current study is to minimize the total costs discussed above by integrating procurement & distribution phase with incorporating quantity discount policies at the time of purchase and freight discount at the time of transportation. At the time of model development following assumptions was taken: Finite planning horizon Deterministic demand Supply is instantaneous Lead time is zero Variant prices of one type product at different destinations All products are available at all the sources Initial inventory at the planning horizon is zero 788

4. Proposed Model Formulation In this section we developed an integrated procurement transportation model to find economic ordered quantity and to minimize total cost obtained at the time of holding ending inventory at sources and destinations, procurement from the sources and transportation to various destinations. The formulation is as follows: 4. Sets Product set with cardinality P and indexed by i. Period set with cardinality T and indexed by t. Price discount break point set with cardinality L and indexed by l. Freight discount break point with cardinality K and indexed by k. Source set with cardinality J indexed by j. Destination set with cardinality M indexed by m 4.2 Parameters h ijt Inventory carrying cost per unit of item i for tth period at source j φ ijmt Unit purchase cost from jth source to mth destination for ith item in tth period d ijmlt Discount factor that is valid if more than aijmlt unit are purchased 0< d ijmlt < β jmt Weight freight cost in tth period from source j to destination m f jmkt Transportation freight discount factor from source j to destination m in period t at freight break k h imt Inventory carrying cost per unit of item i for tth period at destination m D imt Demand for item i in period t from mth destination CR imt Consumption rate at destination m of product i in period t a ijmlt Limit beyond which a price break becomes valid for mth destination availed from jth source in period t for item i for lth price break b jmkt Limit beyond which a freight break becomes from jth source to mth destination valid at period t for kth freight break w i Per unit weight of ith product IN ij Initial inventory of the planning horizon at source j for product i IN im Initial inventory of the planning horizon at destination m for product i λ i Rate of inspection of ith item in terms of quantity ordered i.e. λ i = f(xijmt) η Percentage of defective items of the stored units 4.3 Decision Variables X ijmt Amount of item i ordered in period t from source j for destination m R ijmlt If the order size for all item types for period t is greater than a ijmlt, the discount is applied aijmlt Xijmt < aijm( l+ ) t Rijmlt = 0 otherwise I ijt Inventory level at jth source for product i at the end of period t Z jmkt If the order size for all item types for period t is greater than b jmkt, the freight discount is applied, if bjmkt Ljmt < bjm( k + ) t Z jmkt = 0, otherwise L jmt Total weighted quantity transported in period t from source j to destination m I imt Inventory level at mth destination for product i at the end of period t 789

4.4 The Mathematical Model s.t. Min T P L C = h I + + R d X t= i= ijmt l= J M M ijt ijt λimt Xijmt ijmlt ijmltφ ijmt j= m= m= T J M K T M P + Z jmkt f jmkt β jmt L + himt I jmt imt t= j= m= k = t= m= i= M I ijt = I + X ijmt D ijt imt m= m= M, i=,..., Pj ; =,..., Jt ; = 2,..., T M M = ij +, i =,..., P; j =,..., J m= m= Iij IN Xijm Dim Iimt = Iimt + Dimt CRimt ηiimt, i =,..., P; m =,..., M ; t = 2,..., T (5) () (2) (3) T M T M T (4) I + X D, i =,..., P, j =,..., J ijt ijmt imt t= m= t= m= t= I = IN + D CR ηi, i =,..., P; m =,..., M im im im im im T T T ( η) I + D CR, m =,..., M ; i =,..., P imt imt imt t= t= t= L Xijmt aijmlt Rijmlt i=... P, t=... T, j=,..., Jm, =,..., M l= L Rijmlt = i =,..., P, t =,..., T, j =,..., J, m=,..., M l= P L Ljmt = wx i ijmt Rijmlt, t=,..., T, j=,..., J, m=,..., M i= l= K Ljmt bjmkt Zjmkt, j =,..., J; m=,... M; t =,..., T k= K Zjmkt =, j =,..., J; m=,..., M; t =,..., T k= Xijmt, Iijt, Iimt, L jmt 0 Rijmlt, Z jmkt = either 0 or (6) (7) (8) (9) (0) () (2) 4.5 Analysis of equations Equation () is the objective function to minimize the cost incurred in holding ending inventory at source, cost of purchasing the items, and cost of inspection on ordered quantity by destination m in period t reflected (where the inspection rate can be taken same at all the destination in all period for product i) by the first term of the objective function; the combination of transportation cost from the source to the destination, and holding cost at destination is the second term. The cost is calculated for the duration of the planning horizon. The ordering cost is a fixed cost not affected by the ordering quantities and therefore is not the part of objective function. Equation (2) (7) are the balancing equations for sources and destinations where equation (2) finds total ending inventory at each source of ith product in tth period is found by reducing the demand of all the destinations and fraction of deteriorated ending inventory in tth period from total of ending inventory of previous period and ordered quantity at tth period of all the destinations. Equation (3) finds total ending inventory at each source of ith product in first period is found by 790

reducing the demand of all the destinations and fraction of deteriorated ending inventory of the same period from total of initial inventory if the planning horizon and ordered quantity at first period of all the destinations. Equation (4) shows that total demand in all the periods from all destinations is less than or equal to total of ending inventory and ordered quantity at all the sources in all the periods i.e. shortages are not allowed. Equation (5) calculates ending inventory at mth destination for tth period by reducing consumption rate of the same destination from the combination of ending inventory of previous period and demand of at mth destination. Equation (6) calculates ending inventory for the first period at mth destination by reducing consumption rate of the same destination from the combination of initial inventory of planning horizon and demand of at mth destination. Equation (7) shows that total consumption in all the periods at mth destination is less than or equal to total of ending inventory and demand at destination m in all the periods i.e. shortages are not allowed. Equation (8) & (9) find out the order quantity of all products in period t which may exceed the quantity break threshold, and avails discount on ordered quantity at exactly one quantity discount level. Equation (0) is the integrator for procurement equations (2-9) and transportation equations ( - 2), which calculates weighted quantity to be transported from source j to destination m according to weights per product. Equation () & (2) find out the weighted transport quantity of all products in period t which may exceed the freight break threshold, and avails discount on transportation quantity at exactly one freight discount break. 4.6 Price Breaks As discussed above, variable R ijmlt specifies the fact that when the order size at period t is larger than a ijmlt it results in discounted prices for the ordered items for which the price breaks are defined as: Price breaks for ordering quantity are: d f dijmlt aijmlt Xijmt aijm( l+ ) t = dijmlt Xijmt aijmlt i =,..., P; t =,..., T; l =,..., L; j =,..., J; m =,..., M Freight breaks for transporting quantity are: Here b is the minimum required quantity to be transported. jmkt d f fjmkt bjmkt Ljmt bjm( k+ ) t = fjmkt L jmt b jmkt t =,..., T; j =,... J; m=,... M 5. Case Study Real Fresh is the name of a retail outlet who deals in FMCG products including vegetables. Real Fresh purchases goods directly from companies and nearer farmers to sell them on cheaper rates. Real Fresh is running more than 400 retail stores in metro cities of India. The major problem is with vegetables as it deteriorates very fast. Real Fresh is not concerned for all the stores but some areas are matter of concern for them like store at Somajiguda (Hyderabad), Chambur (Mumbai), Delhi Cantt. (Delhi). In the current discussion we are considering only Delhi region, with three outlets (Kirti Nagar (KT), Model Town (MT) and Inderpuri (INP)) and two farmer sites (Narela, Azadpur). Working policies are very different in Real Fresh stores. They check the vegetables lot at store and generally finds it 5% of the available lot is deteriorated and is discarded too. 5% is the highest deterioration rate of Mushroom, Tomato, Spring as per Real Fresh. The inspection cost at store is Rs. 8 per Kg Mushroom, Rs. 7 per Kg Spring and Rs. 9 per kg Tomato. Same are the packet sizes made by Real Fresh to sell them so weight (in Kg) of Mushroom s packet is o.5 Kg, Spring packet is weighing Kg and Tomato s packet weight is.5 Kg. Stores are availed with all purchasing cost, holding cost at stores transportation cost and holding cost at farmer site. Stores are bearing holding cost at farmer site otherwise deterioration may hike. In return farmers are providing discounts on bulk purchase. Transportation of vegetables is outsourced to a transported that also provides discount after a certain weighted quantity. Stores have idea for consumption at the stores and accordingly they are fixing stores demand. Stores consider three months period (May, June & July, 20). Real Fresh is concerned to find out 79

quantity which may be ordered to different farmers to keep minimum holding, procurement and transportation cost. The data is follows: Table : Holding cost at all the famer sites (INR) Month Mushroom Spring Tomato May 2.4 3. 4.9 June 2.6 3.5 5. July 2.8 3.3 4.7 Table 2: Holding cost at stores (INR) Month Mushroom Spring Tomato May.4 2..9 June.6 2.5 2. July.8 2.3.7 Table 3: Demand from different stores Products Mushroom Spring Tomato Destinations May June July May June July May June July KT 300 20 350 500 580 330 260 230 90 MT 430 450 360 490 320 480 290 240 80 INP 330 350 360 390 420 380 390 340 380 Table 4: Consumption rate at stores Products Mushroom Spring Tomato Destinations Period I Period 2 Period 3 Period I Period 2 Period 3 Period I Period 2 Period 3 KT 298 2 349 489 583 33 255 232 90 MT 429 450 358 487 320 480 288 240 80 INP 327 35 360 387 42 450 388 340 378 Table 5: Quantity discount from farmer sites to stores in all periods for Mushrooms Quantity Breaks Narela Quantity Breaks Azadpur 0 00 0 20 00 200 0.97 20 250 0.90 200 320 0.96 250 360 0.84 320 & above 0.95 360 & above 0.79 Table 6: Quantity discount from farmer sites to stores in all periods for Spring Quantity Breaks Narela Quantity Breaks Azadpur 0 200 0 00 200 320 0.98 00 25 0.90 320 480 0.96 25 300 0.86 480 & above 0.94 300 & above 0.80 Table 7: Quantity discount from farmer sites to stores in all periods for Tomato Quantity Breaks Narela Quantity Breaks Azadpur 0 60 0 60 60 05 0.92 60 00 0.95 05 229 0.87 00 227 0.88 229 & above 0.82 227 & above 0.80 792

Table 8: Freight discounts breaks in May from each farmer site to the stores Transportation Breaks Narela Azadpur 500 000 000 200 0.94 0.97 200 & above 0.90 0.9 Table 9: Freight discounts breaks in June from each farmer site to the stores Transportation Breaks Narela Azadpur 600 900 900 400 0.9 0.95 400 & above 0.87 0.87 Table 0: Freight discounts breaks in July from each farmer site to the stores Transportation Breaks Narela Azadpur 600 050 050 500 0.90 0.9 500 & above 0.85 0.86 Table : Transportation cost (INR) per weight from farmer sites to stores May June July KT MT INP KT MT INP KT MT INP Narela 4 7 4 5 7 4 3 5 5 Azadpur 7 5 6 5 6 5 4 4 4 The data above is employed in the developed model and solved with the help of Lingo.0. Stores are able to manage minimum inventory at farmer site, as keeping more inventory at famer site is additional cost for the stores. In the month of May there was no inventory kept at any of the farmer site. For other two months the ending inventory is mentioned in Appendix A, Table 2. Ending inventory at stores is as per consumption rate i.e. at KT store 2 packets of Mushroom, packets of Spring and 5 packets of Tomato. At MT store packets of Mushroom, 3 packets of Spring and 2 packets of Tomato, at INP store 3 packets of Mushroom, 3 packets of Spring and 2 packets of Tomato. For remaining periods ending inventory is mention in Table 3 of Appendix A. Order quantity of Mushrooms from Narela to KT store is 740 packets, to MT store is 320 packets and 0 packets to INP store with quantity discount of 5%, 5% and 0% respectively. From Azadpur to KT store 0 packets, to MT store is 060 packets and 0 packets to INP store with quantity discount of 0%, 2% and 0% respectively. All the other periods have discussed in Appendix A, Table 4. And for other two products one can refer Table 5 and 6 in Appendix A. The weighted quantity transported from Narela to KT store is 20 Kg, which is combination of all the three products, 000 Kg is transported to MT store and 200 Kg to INP store. Other stores and periods are mentioned in Table 7 Appendix A. Holding cost at farmer sites is Rs.,449.7, holding cost at stores is Rs.53.9322, Further procurement cost is Rs.4,54,408.4 and transportation cost is Rs.8,527.3. The combination of all the costs is Rs.5,37,539.3, which the minimum optimum cost. 6. Conclusion The integrated procurement distribution model explains dual benefit from quantity and freight discount. From the results, the same has been validated. As one can see, in almost every period, discounts facility on ordered and transported quantity is availed by the buyer, which provides benefits to buyer, transporter as well as to the supplier. Model is successfully managing the ending inventory and keeping it at minimum level at sources and destinations. The model coordinates the best minimum cost incurred through holding at source & destination, procurement and transportation of deteriorating natured products. The model is suitable for the problems, where manager needs to find coordination between demand and supply points. The model can be extended for uncertain demand, which could make the environment fuzzy. 793

Acknowledgements Kanika Gandhi (Lecturer, Quantitative Techniques & Operations) is thankful to her organization Bharatiya Vidya Bhavan s Usha & Lakshmi Mittal Institute of Management to provider her opportunity for carrying research work. References Bahari-Kashani, H., Replenishment schedule for deteriorating items with time-proportional demand, Journal of Operational Research Society, vol. 40, pp. 75-8, 989. Covert, R. P., Philip, G. C., An EOQ model for items with Weibull distribution deterioration, AIIE Transactions, vol. 5, pp. 323-326, 973. Dave, U., Patel L. K., 98, (T, Si) policy inventory model for deterioration items with time proportional demand, Journal of the Operational Research Society, vol. 32, pp. 37-42, 98. Fink, M. M., and Ferrell, W. G., Jr., Inventory policy for products with short life cycles, Proceeding of the th Industrial Engineering Research Conference, Orlando, 2002. Ghare, P. N., Schrader, G. F., A model for exponentially decaying inventories, Journal of Industry Engineering, vol. 5, pp. 238-243, 963. Goyal, S.K., and Giri, B.C., Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, vol. 34, pp. -6, 200. Liu, L., Lian, Z., (s,s) continuous review models for inventory with fixed lifetimes, Operations Research, vol. 47, no., pp. 50-58, 999. Moon, I., Lee, S., The effects of inflation and time value of money on an economic order quantity model with a random product life cycle, European Journal of Operational Research, vol. 25, pp. 588-60, 2000. Moorthy, K. A., Narasimhulu, Y. C., and Basha, I. R., On perishable inventory with Markov chain demand quantities, International Journal of Information Management Science, vol. 3, pp. 29-37, 992. Philip, G. C., A generalized EOQ model for itmes with Weibull distribution deterioration, AIIE Transactions, vol. 6, pp. 59-62, 974. Sachan, R. S., On (T, Si) policy inventory model deterioration items with time proportional demand, Journal of the Operational Research Society, vol. 35, pp. 03-09, 984. Sahoo, N. K., Sahoo, C. K., and Sahoo, S.K., An Inventory Model for Constant deteriorating Items with Price Dependent Demand and Time-varying Holding Cost, International Journal of Computer Science & Communication, vol., no., pp. 267-27, 200. Tadikamalla,P. R., 978, An EOQ inventory model for items with Gamma distributed deterioration. AIIE Transactions, 0, 78. Tersine, R.J., Barman, S., Economic Inventory/Transport Lot, Sizing with Quantity and Freight Rate Discounts, Decision Sciences, vol. 22, no. 5, pp. 7 79, 99. Wang, K. J., Lin, Y. S., and Yu, J. C. P., Optimizing inventory policy for products with time-sensitive deteriorating rates in a multi-echelon supply chain, International Journal of Production Economics, vol. 30, pp., pp. 66-76, 20. Wee, H. M., Deteriorating inventory model with quantity discount, pricing and partial backordering, International Journal of Production Economies, vol. 59, no. -3, pp. 5-58, 999. Yu, D., and Luo, J., Determining Optimal Ordering Policy for Deteriorating Products Under the Trade Credit Incentives, 4 th International Conference on Wireless Communications, Networking and Mobile Computing, pp. -4, 2008. 794

Appendix A: Solution Tables Table 2: Ending inventory at the farmer site Product Mushroom Spring Tomato Mushroom Spring Tomato Mushroom Spring Tomato Destination Narela 0 0 0 0 0 0 0 2 52 Azadpur 0 0 0 0 0 53 Table 3: Ending inventory at Stores Product Mushroom Spring Tomato Mushroom Spring Tomato Mushroom Spring Tomato Destination KT 2 5 7 3 2 6 3 MT 3 2 3 2 3 3 2 INP 3 3 2 2 2 2 2 2 4 Table 4: Ordered quantity & Quantity discounts from farmer site to stores Mushroom Farmer Site Destination KT MT INP KT MT INP KT MT INP Narela Order 740 320 0 0 0 00 0 070 0 Discount 5% 5% 0% 0% 0% 5% 0% 5% 0% Azadpur Order 0 060 0 0 0 00 0 7 360 Discount 0% 2% 0% 0% 0% 2% 0% 2% 2% Table 5: Ordered quantity & Quantity discounts from farmer site to stores for Spring Farmer Site Destination KT MT INP KT MT INP KT MT INP Narela Order 750 630 0 0 480 840 0 55 747 Discount 6% 6% 0% 0% 6% 6% 0% 6% 6% Azadpur Order 300 080 0 600 26 695 0 0 260 Discount 20% 20% 0% 20% 0% 0% 0% 0% 20% Table 6: Ordered quantity & Quantity discounts from farmer site to stores for Tomato Farmer Site Destination KT MT INP KT MT INP KT MT INP Narela Order 0 40 800 400 280 30 400 0 502 Discount 0% 3% 8% 8% 8% 3% 8% 0% 8% Azadpur Order 34 6 800 0 583 227 400 463 40 Discount 2% 0% 20% 0% 20% 20% 20% 20% 0% Table 7: Weighted transport quantity & Freight discounts from farmer sites to stores Farmer Site Destination KT MT INP KT MT INP KT MT INP Narela Transport 20 000 200 600 900 540 600 050 500 Discount 6% 6% 0% 0% 9% 3% 0% 0% 5% Azadpur Transport 50 69 200 600 900 540 600 050 500 Discount 0% 9% 9% 0% 5% 3% 0% 9% 4% 795

Appendix B: Lingo Program SETS: PRODUCT/P,P2,P3/: Wi, INSi; PERIOD/T,T2,T3/; SOURCE/J,J2/; DESTN/M,M2,M3/; QTY_LEVEL/L,L2,L3,L4/; TPT_LEVEL/K,K2,K3/; PURCH_SOURCE_TIME(PRODUCT, SOURCE, PERIOD): Hijt, Iijt, INij; PURCH_DESTN_TIME(PRODUCT, DESTN, PERIOD):Himt, Dimt, CRimt, INim, Iimt; PURCH_SOURCE_DESTN_TIME(PRODUCT, SOURCE, DESTN, PERIOD):P_COST, Xijmt; PURCH_SOURCE_DESTN_QTY_LEVELTIME(PRODUCT, SOURCE, DESTN, QTY_LEVEL, PERIOD): Dijmlt, Aijmlt, Rijmlt; SOURCE_DESTN_TPT_LEVEL_TIME(SOURCE, DESTN, TPT_LEVEL, PERIOD):Fjmkt, Bjmkt, Zjmkt; SOURCE_DESTN_TIME(SOURCE, DESTN, PERIOD):BETA, Ljmt; ENDSETS Main Code MIN = @SUM(PURCH_SOURCE_TIME(I, J, T): Hijt(I,J,T)*Iijt(I,J,T)) +@SUM(PURCH_SOURCE_DESTN_TIME(I, J, M, T):Xijmt(I,J,M,T)*P_COST(I,J,M,T) *@SUM(QTY_LEVEL(L):Rijmlt(I,J,M,L,T)*Dijmlt(I,J,M,L,T))+INSi(I)*Xijmt(I,J,M,T)) + @SUM(SOURCE_DESTN_TIME(J,M,T): BETA(J,M,T)*Ljmt(J,M,T)*@SUM(TPT_LEVEL(K): Zjmkt(J,M,K,T)*Fjmkt(J,M,K,T))) +@SUM(PURCH_DESTN_TIME(I,M,T):Himt(I,M,T)*Iimt(I,M,T)); @FOR(PURCH_SOURCE_TIME(I,J,T) T#EQ#:INij(I,J,T)=0); @FOR(PURCH_SOURCE_TIME(I,J,T) T#EQ#:Iijt(I,J,T)=INij(I,J,T)+@SUM(DESTN(M):Xijmt(I,J,M,T)- Dimt(I,M,T))-ETA*Iijt(I,J,T)); @FOR(PURCH_SOURCE_TIME(I,J,T) T#GT#:Iijt(I,J,T)=Iijt(I,J,T-)+@SUM(DESTN(M):Xijmt(I,J,M,T)- Dimt(I,M,T))-ETA*Iijt(I,J,T)); @FOR(PRODUCT(I):@FOR(SOURCE(J):@SUM(PERIOD(T):(- ETA)*Iijt(I,J,T)+@SUM(DESTN(M):Xijmt(I,J,M,T))) >=@SUM(PERIOD(T):@SUM(DESTN(M):Dimt(I,M,T))))); @FOR(PURCH_DESTN_TIME(I,M,T) T#EQ#:INim(I,M,T)=0); @FOR(PURCH_DESTN_TIME(I,M,T) T#EQ#:Iimt(I,M,T)=INim(I,M,T)+ Dimt(I,M,T)-CRimt(I,M,T)- ETA*Iimt(I,M,T)); @FOR(PURCH_DESTN_TIME(I,M,T) T#GT#:Iimt(I,M,T)=Iimt(I,M,T-)+Dimt(I,M,T)-CRimt(I,M,T)- ETA*Iimt(I,M,T)); @FOR(PRODUCT(I):@FOR(DESTN(M):@SUM(PERIOD(T):(- ETA)*Iimt(I,M,T)+Dimt(I,M,T))>=@SUM(PERIOD(T):CRimt(I,M,T)))); @FOR(PURCH_SOURCE_DESTN_TIME(I,J,M,T):Xijmt(I,J,M,T)>=@SUM(QTY_LEVEL(L):Aijmlt(I,J,M,L,T) *Rijmlt(I,J,M,L,T))); @FOR(PURCH_SOURCE_DESTN_TIME(I,J,M,T):@SUM(QTY_LEVEL(L):Rijmlt(I,J,M,L,T))=); @FOR(SOURCE_DESTN_TIME(J,M,T):Ljmt(J,M,T)=@SUM(PRODUCT(I):Wi(I)*Xijmt(I,J,M,T)*@SUM(QTY _LEVEL(L):Rijmlt(I,J,M,L,T)))); @FOR(SOURCE_DESTN_TIME(J,M,T):Ljmt(J,M,T)>=@SUM(TPT_LEVEL(K):Bjmkt(J,K,M,T)*Zjmkt(J,M,K, T))); @FOR(SOURCE_DESTN_TIME(J,M,T):@SUM(TPT_LEVEL(K):Zjmkt(J,M,K,T))=); @FOR(PURCH_SOURCE_DESTN_TIME:@GIN(Xijmt)); @FOR(PURCH_SOURCE_TIME:Iijt>=0); @FOR(PURCH_DESTN_TIME:Iimt>=0); @FOR(SOURCE_DESTN_TIME:Ljmt>=0); @FOR(PURCH_SOURCE_DESTN_QTY_LEVELTIME:@BIN(Rijmlt)); @FOR(SOURCE_DESTN_TPT_LEVEL_TIME:@BIN(Zjmkt)); END 796