POMS 20 th Annual Conference Orlando, Florida U.S.A. May 1 to May 4, 2009

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1 Abstract Number: - 3 Providing Optimal Management of Disruptions in Complex Supply Chains using Linear Programs by Frenck Waage, Professor University of Massachusetts Boston College of Management, Department of MSIS 5 / 35 McCormack Building Morrissey Boulevard Boston, Massachusetts Office Phone: frenck.waage@umb.edu frenckwaage@comcast.net POMS th Annual Conference Orlando, Florida U.S.A. May to May 4, 9. Abstract The literature on supply chain management frequently argues that the principles of demand pull should guide decision makers when they make the operating decisions for supply chains. Satisfying demand pull principles is a necessary condition for creating profitable growth. It is never stated, however, in the literature just how such decisions can in practice be developed. Nor is it stated anywhere just how a supply chain manager may optimally react to disruptions in the supply chain. This paper solves both of these problems. It shows how a supply chain manager can use linear programs to model the supply chain, and how to extract the optimal management decisions from the model in practice. It shows further how to react optimally to disturbances, changes and interruptions in the Supply Chain.. Optimal Management of a Supply Chain The objective of every company is to create sustained growth of its net income. To

2 achieve the desired growth requires superior management of the extended supply chain. The extended supply chain begins with the remotest supplier, runs through all suppliers, through own value chain and ends up with the customers (end users) at the end of the distribution chain. Superior management will lead to the competitive advantage that is needed to generate the income growth. The first step in modeling a supply chain by a linear program is to develop a linear program representation of the extended supply chain. Figure is an example of a flow network of an extended supply chain. Materials flow in the directions of the arrows. The information flows that control the material flows are discussed below. The variables shown in Figure will be explained shortly. Figure : An Extended Supply Chain Supplier Demand = z + z Z Supplier Ware House V V Manu facturer x x Distri bution Center y y y Market Z Z V x y 3 Market Supplier Demand = z + z Z Supplier Ware House V Manu facturer x Distri bution Center y Y 3 Market 3 Figure illustrates: () Customers orders arrive in the three markets on the right. () From there the orders are forwarded to a central supply chain manager. (3) The manager explodes the orders for the finished product into the component parts required to make the ordered quantities of the finished product. (4) He authorizes suppliers and on the left in Figure to produce the components in the correct quantities and ship them to one of two supplier warehouses. (5) He authorizes the two supplier warehouses to ship the

3 3 correct quantities of the components to each of two brand manufacturers. (6) He authorizes the two brand manufacturers to assemble these components into units of the finished brand in the quantities ordered. (7) He authorizes the two brand manufacturers to ship the correct quantities to two distribution centers. (8) He authorizes the two distribution centers to ship the quantities of the finished products that were ordered in each market to the correct markets. (9) The finished goods are sold in the markets, sales revenues are collected there and the books closed on these transactions. The supply chain manager s authorizations to the extended supply chain s partners constitute the supply chain s information flows. The resulting production and transportation activities constitute the material and information flows. The supply chain manager has modeled the extended supply chain in Figure by a Linear Program. The linear program minimizes the total cost of all activities and transports. The optimal (cost minimizing) solution of this Linear Program provides the Supply Chain manager with all the quantities he then authorizes the supply chain partners to realize. His authorizations deploy the cost minimizing optimal - activities of the supply chain. When changes or interruptions occur, the manager uses the same linear program to determine the optimal way to have the entire supply chain react. The supply chain will continue to operate optimally in a cost minimizing manner given that the disturbance occurred. The construction of the linear program is discussed in the next section. 3. A Linear Program Model of the Supply Chain in Figure Definitions used in the linear program are: D st = quantity of a brand demanded in market s =,, 3 in day t =,, 3,,T

4 4 Y jst = the quantity of the brand that distribution center j =, ships to market s in day t X krt = the quantity that manufacturer k =, ships to distribution center r =, in day t VI mkt = quantity of component I =, that has to be shipped from warehouse m =, to manufacturer k =, in day t The brand customers buy in the three markets is assembled from two component parts α is the number of component part each unit of the brand requires for its completion. α is the number of component part each unit of the brand requires for its completion. ZI vjt = quantity of component I =, which supplier v =, must ship to supplier warehouse j in day t. The flows in Figure can now all be modeled by the following linear equations. The Constraint Equations of the Linear Program Distribution centers and are required by equation [] to supply market s with the quantities demanded D st in week t in order to satisfy demand exactly. [] y st + y st = D st three equations s =,,3 Manufacturers and are by equation [] required to ship to distribution center r =, exactly the quantities of the brand that distribution center r =, is required by equation [] to ship on to markets s =,, 3. [] x rt + x rt - y rt y rt y r3t = two equations r =, Manufacturer k will produce a total quantity ( X k + X k ) of the brand for k =,. For this production manufacturer k requires the correct quantities of two component parts. To produce complete units of the brand, manufacturer k requires ( V kt + V kt ) units of

5 5 component and ( V kt + V kt ) units of component shipped in from warehouses and. Equation [3.] requires that the two warehouses ship to manufacturer k =, exactly that quantity of component which is needed to produce the quantity ( X kt + X kt ) of the brand. Equation [3.] requires that the two warehouses ship to manufacturer k =, exactly that quantity of component which is needed to produce the quantity ( X kt + X kt ) of the brand. Both components are required in the correct quantities for a complete unit of the brand. [3.] V kt + V kt = α ( X kt + X kt ) Component, two equations k =, [3.] V kt + V kt = β ( X kt + X kt ) Component, two equations k =, The total quantity of component I =, that warehouse j =, must ship is therefore ( VI jt + VI jt ). This is identically the quantities that must be received from the suppliers who make the components. Therefore, the total supply of component that suppliers and must produce and ship to warehouse j is ( Z j + Z j ). The total supply of component that suppliers and must ship to ware house j is ( Z j + Z j ). That supply of component from the two suppliers equals demand for component at warehouse j is guaranteed by equation [4.] and of component by equation [4.]. [4.] z jt + z jt = V jt + V jt two equations j =, [4.] z jt + z jt = V jt + V jt two equations j =, The supply chain manager calculates in equation [5] the total cost to the supply chain from the activities generated by equations [] through [4.]. The Linear Program s Objective Function

6 6 [5] TC t = Σ { 6Z t +Z t +Z t +4Z t component from suppliers to warehouse + 6Z t +Z t +Z t +4Z t component from suppliers to warehouse +3V t +4V t +4V t +4V t component from warehouses to manufacturers + 3V t +4V t +4V t +4V t component from warehouses to manufacturers + X t + 5X t + 4X t + X t units of the finished product from manufacturers to two distribution centers + 3Y t + 4Y t + 5Y 3t + Y t + Y t + Y 3t } from distribution centers to markets The linear program has one objective function with 6 variables for each day t, 3 constraints from [] through [4.] plus 6 non-negativity conditions for each week t. The optimal solution is obtained by finding the values for the 6 variables (for each t) which minimize equation [5] while satisfying the constraints [] through [4.]. Microsoft s EXCEL will hold the data matrix, and the software SOLVER will solve the linear program. The optimal solution identifies, for the supply chain manager: () the cost minimizing material flows through the entire supply chain which will exactly satisfy given quantities demanded of the brand, () the optimal routes for the material flows through the supply chain, (3) the minimum cost of operating the supply chain, and (4) the optimal information flows in the supply chain. 4. Use of the Linear Program to determine the Optimal Supply Chain Information and Material Flows We now present an application. This application describes Figure by the linear program

7 7 described in section 3. The data used by the linear program is kept in an EXCEL spreadsheet and solved by SOLVER a software available in EXCEL. In this application we used α =. and β = 3. in equations [3.] and [3.]. The demand data for equation [] has been recorded in EXCEL as shown in Table. The objective function [5] uses the coefficients shown in [5]. Table Orders for the next 3 days received Orders received for the product Daily Demand Rate in Market Daily Demand Rate in Market Daily Demand Rate in Market 3 Total Daily Demand for the Brand Deliver in day 5 45 Deliver in days Deliver in 3 days 8 6 The optimal solution of each of the 6 variables is obtained with SOLVER. The 6 optimal variable values are then divided into subgroups, and a subgroup is recorded in one of tables through 6. Each table contains the optimal demand pull flow rates between the linked supply chain partners shown in Figure. Tables through 6 account for the entire optimal solution between them. 4. Information Flows at 8:AM on day from Supply Chain Manager To the two Suppliers and the two Supplier Warehouses The linear program is solved by 8:AM on day by the central manager. The optimal solution is also the optimal supply chain operating instructions. The Central Manager populates Table with the optimal linear program solution for the two suppliers and the two supplier warehouses. He populates Table 3 with the optimal linear program solution

8 8 for the two warehouses and the two brand manufacturers. He populates Table 4 with the optimal linear program solution for the two brand manufacturers and the two distribution centers. He populates Table 5 with the optimal linear program solution for the two distribution centers and the three markets where the demand originated. These sets cover the complete supply chain. They represent the information flows. Material Flows from Suppliers to Warehouses on day At 8:AM on day, the supply chain manager s Table, to the two suppliers and to the two warehouses. Upon receipt of Table supplier immediately starts producing the quantities in Table, namely units of component and 3 units of component. When completed he ships these quantities to warehouse. Supplier also starts producing the other quantities in Table, namely 7 units of component and,5 units of component. Production takes 4 hours to complete. The products leave the supplier s facilities at :Noon. They will reach the two warehouses in 4 hours at 4:PM. Supplier is instructed by Table to produce zero components on day. Table : The Optimal Linear Program Solution for the two Suppliers and the two Warehouses day Component To Warehouse Component To Warehouse Component To Warehouse Component To Warehouse Supplier will produce and Ship to warehouses units of Components Supplier will produce and Ship to warehouses units of Components 3 The suppliers have a preview of demand for the future days and 3. These quantities

9 9 are for planning. Suppliers and will produce absolutely nothing on day for days and 3. The preview of demand for days and 3 will be used for analysis, for tooling up, for getting capacity on line, and for getting ready. After loading the trucks, the two suppliers will carry zero inventories. They will produce nothing till they receive the next production authorization from the supply chain manager. 4. Information Flows at 8:AM on day from Supply Chain Manager to the two Warehouses and the two Manufacturers At 8:AM the Supply Chain Manager populates Table 3 with the optimal linear program solution for the activities the two warehouses and the two brand manufacturers must undertake. Table 3 is then ed to the two warehouses and to the two manufacturers. Table 3: The Optimal Linear Program Solution for the two Warehouses and Manufacturers day Component shipped to Manufacturer Component shipped to Manufacturer Component shipped to Manufacturer Component shipped to Manufacturer Warehouse will Ship required units of Components and to Warehouse will Ship required units of Components and to Material Flows from the two Warehouses to the two Manufacturers on day Ware house receives the units of component and 3 units of component at 4:PM, when the trucks arrive there. Ware house receives the 7 units of component and,5 units of component at 3:PM, when the trucks arrive there in

10 accordance with the Table instructions. These are exactly the quantities that Table 3 calls for. Warehouse immediately ships units of component and 3 units of component to manufacturer and zero units to manufacturer. Transportation requires 3 minutes, and manufacturer will have the components at 4:3PM. Warehouse immediately ships 7 units of component and,5 units of component to manufacturer. Transportation requires.5 hours, and manufacturer will have the components at 5:3PM. After these shipments, the warehouses will carry zero inventories. 4.3 Information Flows at 8:AM on day from Supply Chain Manager to the two Manufacturers and the two Distribution Centers At 8:AM the Supply Chain Manager populates Table 4 with the optimal linear program solution for the activities the two brand manufacturers and the two distribution centers must undertake. Table 4 is then ed to the two brand manufacturers and to the two distribution centers. Table 4 instructs manufacturer to produce units of the brand and ship the units to distribution center and zero units to distribution center. Manufacturer is instructed to produce 35 units of the finished product and ship 35 units to distribution center and zero units to center. Production commence immediately at 6:3PM for both manufacturers. At 4:3PM manufacturer received units of component and 3 units of component from warehouse. These quantities enable manufacturer to assemble

11 exactly the requested units of the brand with no components left over. At 5:3PM manufacturer received 7 units of component and,5 units of component from warehouse. The 7 units of component and the,5 units of component enable manufacturer to assemble exactly the requested 35 units of the finished product with no components left over. Table 4: The Optimal Linear Program Solution for the two Manufacturers & Distribution Centers Weeks To Distribution Center To Distribution Center Totals Production Ship units of the BRAND from Manufacturer Ship units of the BRAND from Manufacturer Material Flows from the two Manufacturers to the two Distribution Centers on day Production of the units of the brand are completed at 6:3PM and immediately shipped from manufacturer to distribution center according to the Table 4 instructions. The units arrive by truck at at Distribution Center at 7:3PM. Production of the 35 units of the brand are completed at :3PM and immediately shipped to distribution center according to the Table 4 instructions. They arrive at Distribution Center by truck at :3 AM next day. After shipments the two manufacturers carry zero inventories. 4.4 Information Flows at 8:AM on day from Supply Chain Manager to the two Distribution Centers and the three Markets

12 At 8:AM the Supply Chain Manager populates Table 5 with the optimal linear program solution for the activities the two Distribution Centers and the three markets must undertake. Table 5 is then ed to the two brand manufacturers and to the two distribution centers. Ship a total of units). Table 5 instructs Distribution Center to ship units of the brand to Market and zero units to Markets and 3. Distribution Center is instructed to ship zero units to Market, units of the brand to Market, and 5 units to Market 3. (Ship a total of 3,5 units). Table 5: The Optimal Linear Program Solution for the Distribution Centers & the 3 Markets Distribution Center Ships units of the BRAND to Distribution Center Ships units of the BRAND to day To Market To Market To Market 3 Totals Received Material Flows from the two Distribution Centers to the two markets on day Distribution Center received units of the brand at 7:3PM from manufacturer. Distribution Center received 35 units of the brand at :3 AM next day. In accordance with the Table 5 instructions Distribution Center immediately ships the units of the brand to market ( hour away by truck), zero units to markets ( hours away) and zero units to market 3 (3 hours away)

13 3 Distribution center immediately ships units to market (4 hours away by truck), units of the finished product to market ( hours away), and 5 units to market 3 ( hour away). 4.5 Closing the Books on these orders The three markets start the next day 8:AM with the precise quantities that have been ordered on their shelves. Customers collect precisely what they ordered. Payment is collected from the customers upon delivery and the cases are closed. This application illustrates how a linear program can be used to manage pure demand pull material flows through a supply chain. 5. Optimal Reactions to Disturbances and Interruptions in Supply Chains The linear program is used when the need arises to calculate the best (optimal) way to align all the partners activities in the supply chain in response to an interruption, or a disruption. Examples are:. There has been a fire in Supplier s facilities. His capacity to produce and deliver components has been reduced to 3% of normal. To calculate the optimal response for everyone in the supply chain to this supplier limitation, a constraint is introduced that measures this limitation. The program is solved with the constraint active, and this new solution is the optimal actions by everyone.. The demand forecast is wrong. New and improved demand forecast figures are introduced in Table, and in equations [] in the linear program. Solve the Linear Program with the new demand forecast. The optimal solution identifies how everyone in the supply chain must act (react) to this change for the entire supply

14 4 chain to perform optimally. 3. Some of the objective function coefficients change. Change the corresponding coefficients in equation [5] and solve. The new optimal solution is the optimal response for the entire supply chain to these changes. 4. In general, incorporate any interruption to the supply chain into the linear program (as a change in an equation, as an added constraint, as a changed coefficient, etc). Resolve the linear program. The new optimal solution is the optimal way to react to the interruption. 6. Conclusions and Further Research We have shown that it is possible to manage a supply chain with the optimal solution of a linear program. The decisions we have explained are consistent with pure demand-pull principles. There is no better way to manage a supply chain. Much added experience will be gained with these models if more companies apply them to solve the supply chain problems of real companies. The size of Linear Programs is not as constricting as it once was. Linear programs with more than one million variables and more than one million constraints can be solved in practice. 8. Related Literature Using linear programs to characterize complex supply networks is not new. See (Dantzig 963), (Johnson et al. 973), (Chopra et al. 3), (Ragsdale 4), (Simchi-Levi et al., 3). What is new in this paper is the use of linear programs to generate the optimal supply chain operating doctrine for pure demand-pull systems. Relevant references for supply chain management include (Arntzen 995), (Beyer ), (Tayur 999), and for

15 5 the management of demand-pull systems read (Cheng et al. ), (Feitzinger et al. 997), (Song ). Section 5 of this paper introduced the dynamic smoothing decision problem that faces every supply chain partner. The epochal reference is (Holt et al. 96). Good updated readings include (Blinder 986), (Eichenbaum 989), (Kaminsky 4), (Lawrence et al. 984). Though not addressed in this paper, the natural expansion of the focus of this paper is into the subjects of lean process management, and six Sigma quality in supply chains. References to these two subjects are found in (Askin et al., ), (Conner ), (Jordan et al., ). Bibliography. Arntzen, B. C., Brown G. G., Harrison T. P., and Trafton, L. L. (995). Global Supply Chain Management at Digital Equipment Corporation. Interfaces, 5 (), Askin, R., and J. Goldberg,, Design and Analysis of Lean Production Systems, New York, Wiley 3. Beyer, D. And Ward, J. (). Network Server Supply Chain at HP: A Case Study. In Song, J and Yao, D (eds) Supply Chain Structures: Coordination, Information and Optimization. Kluwer 4. Blinder, A., (986). Can the production smoothing model of inventory behavior be saved? The Quarterly Journal of Economics, CI (August), Cheng, F., Ettl, M., Lin, G. Y., and Yao, D. D. (), Inventory-Service Optimization in Configurer-to-Order Systems, Manufacturing and Service Operations management 4, 4 3

16 6 6. Chopra, Sunil and Meindl, Peter, Supply Chain Management, Pearson Prentice Hall, Upper Saddle River, New Jersey, 3 7. Connor, G.,, Lean manufacturing for a Small Shop, Dearborn, MI., Society of Manufacturing Engineering 8. Dantzig, George B, Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey Eichenbaum, M., (989) Some empirical evidence on the production level and production cost smoothing models of inventory investment, American Economic Review 79 (4), Feitzinger, E. And Lee, H. (997), Mass Customization at Hewlett Packard: The Power of Postponement., Harvard Business Review, 75,, 6. Glasserman, P. And Tayur, S. (995). Sensitivity Analysis for Base-Stock Levels in Multi-Echelon Production-Inventory Systems, Manqagement Science, 4(5), Holt, C.C., Modigliani, F., Muth, J.F., Simon, H.A., (96), Planning Production, Inventories, and Work Force, Prentice Hall, New Jersey. 3. Johnson, Richard, William Newell, Roger Vergin, Operations Management A Systems Concept, Houghton-Mifflin, Boston, Massachusetts, Jordan J., and F. Michael,, The Lean Company: Making the Right Choices, Dearborn, MI., Society of Manufacturing Engineers 5. Kaminsky, P. And Swaminathan, J. M. (4). Effective Heuristics for Capacitated Production Planning with Multiperiod Production and Demand with

17 7 Forecast Band Refinement, Manufacturing and Service Operations Management 6, Lawrence, K and Stelios Zanakis, Production Planning and Scheduling (eds), (984) Industrial Engineering and Management Press, Institute of Industrial Engineers, Norcross, Georgia 7. Lu, Y., Song, J. S. And Yao, D. D. (3). Order Fill Rate, Lead Time Variability, and Advance Demand Information in an Assemble-to-order System. Operations Research 5, Ragsdale, Cliff T, Spreadsheet Modeling & Decision Analysis, Thompson South Western, Mason, Ohio, 4 9. Simchi-Levi D., P. Kaminski, and E. Simchi-Levi, 3, Designing and Managing the Supply Chain, ( nd ed.) New York, McGraw-Hill. Song, J. S. And Yao, D. D. (). Performance Analysis and Optimization of Assemble-to-Order Systems with Random lead Times. Operations Research, 5, Song, J. S. And Yao, D. D. (eds) () Supply Chain Structures: Coordination, Information, and Optimization, Kluwer. Tayur, S, Ram Ganeshan, and Michael Magazine (eds), (999) Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Boston, Massachusetts

18 8 Gains from switching to Demand Pull Management of the supply chain EXPENSES AND INCOME Expenses 6 Incomes 6 Expenses 7 Incomes 7 Sales $ $ Fixed Costs Facilities, Equipment Direct Cost of Goods Sold $ 4. $ 3. Costs of Inventories $. $ 4. Costs of Transportation $ 6. $ 6. Operating Expenses $ 56. $ 4. Depreciation $ 3. $ 3. Selling and Admin Expenses $ 34. $ 3. Total Operating Expenses $ 93. $ 73. Operating Profit $ 7. $ 47.