1. /faculteit technologie management An Integrated Approach to Inventory and Flexible Capacity Management under Non-stationary Stochastic Demand and Set-up Costs Tarkan Tan Eindhoven University of Technology Osman Alp Bilkent University May 24, 2005 FIFTH INTERNATIONAL CONFERENCE ON "Analysis of Manufacturing Systems - Production Management" Zakynthos Island, Greece

2. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

3. /faculteit technologie management Introduction Make-to-stock production Coping with fluctuating demand –Holding inventory –Changing capacity by utilizing flexible resources Capacity: Total productive capability of all productive resources utilized

4. /faculteit technologie management Introduction Permanent Capacity: maximum amount of production possible in regular work time by utilizing internal resources This can be increased temporarily by acquiring contingent resources – called as the contingent capacity Human workforce jargon is used but our model may also apply to different forms of capacity; e.g. subcontracting, hiring machinery, etc.

5. /faculteit technologie management Introduction Change of permanent capacity level is a tactical decision, not to be made frequently Therefore, for a given permanent capacity level we focus on operational decisions on increasing the total capacity level by use of contingent labor Decisions to be made: –How much capacity to have –How much to produce for a given permanent capacity and a finite planning horizon

6. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

7. /faculteit technologie management Literature Review Integrated Production/Capacity Management Atamtürk & Hochbaum (MS 2001), Angelus & Porteus (MS 2002), Dellaert & de Kok (IJPE 2004) Workforce Planning and Flexibility Holt et al. (1960), Wild & Schneeweiss (IJPE 1993), Milner & Pinker (MS 2001), Pinker & Larson (EJOR 2003) Capacitated Production/Inventory Models Federgruen & Zipkin (MOR 1986), Kapuscinski & Tayur (OR 1998), Gallego & Scheller-Wolf (EJOR 2000) Strategic Capacity Management: van Mieghem (MSOM 2003) Continuous Review: Hu et al (AOR 2004), Tan & Gershwin (AOR 2004)

8. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

9. /faculteit technologie management Model Finite horizon DP Relevant Costs –Inventory holding –backorder –permanent labor –contingent labor –set-up for production –set-up for ordering contingent labor Simplifying assumptions: –Infinite supply of contingent labor –Zero lead time

10. /faculteit technologie management Model The amount that each permanent worker produces per period is defined as 1 "unit" c p is the unit cost of permanent capacity Productivity of contingent resources may be different than the productivity of permanent resources, let  denote this ratio Cost of contingent workers is adjusted to reflect the cost per item produced, that is c c = c c orig / 

11. /faculteit technologie management Model Observation: –permanent labor cost does not affect the decision on the number of contingent workers to be ordered each period (for a given number of permanent workers) –production quantity is sufficient to determine the number of contingent workers to be ordered Under these conditions, the problem (for a given permanent workforce size) translates into a prod/inv model with piecewise linear (non-convex / non- concave) unit production cost (convex under zero set- up costs)

12. /faculteit technologie management Production Cost Structure

13. /faculteit technologie management Formulation CIMP:

14. /faculteit technologie management Remark When K p = K c = 0 and c c  , CIMP boils down to a capacitated production/inventory problem Similarly, when K p > 0 and either K c   or c c  , CIMP boils down to a capacitated production/inventory problem with production setup cost

15. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

16. /faculteit technologie management Analysis with No Setup Costs The problem translates into a typical production/inventory problem with piecewise convex production costs Karlin (1958) shows that for multi-period problem with strictly convex production cost, optimal policy is of order-up-to type

17. /faculteit technologie management Optimal Policy

18. /faculteit technologie management

19. Optimal Control Parameters in Time

20. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

21. /faculteit technologie management Analysis with Setup Costs When we introduce setup costs of production and/or of ordering contingent capacity, the problem becomes much more complicated We first analyze the optimal policy of a single period problem

22. /faculteit technologie management Single Period Optimal Policy Optimal policy for a single period problem is a state dependent (s, S) policy We represent it as an (s(x), S(x)) policy where x is the starting inventory level There are three critical functions s c (x), s u (x), and s p (x) that can be characterized and s(x) takes the form of one these functions depending on the value of x

23. /faculteit technologie management Single Period Optimal Policy

24. /faculteit technologie management

25. Multi-Period Problem This single period policy cannot be generalized to multiple periods One possible way of generalizing this policy requires the expressions in the “min” function of CIMP to be either convex or quasi-convex However, this requirement is not satisfied even for period T – 1 While f T (x) is a quasi-convex function, summation of convex and quasi-convex functions is not necessarily quasi-convex

26. /faculteit technologie management Actually, we expected this… The characterization of the optimal policy of capacitated production/inventory problems under setup costs is still an open question Gallego and Scheller-Wolf (1990) characterize the optimal policy to a limited extent and discuss the difficulties in achieving this We conjecture that the optimal ordering policy of CIMP to be even more complicated

27. /faculteit technologie management

28. Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

29. /faculteit technologie management Value of Flexible Capacity We conducted a computational study to reveal the importance of utilizing value of flexible capacity We consider a seasonal Poisson or Gamma Demand with a cycle of 4 periods where expected demand in each period are 10, 15, 10, and 5 respectively T = 12, U = 10, b = 5, h = 1, c c = 2.5, c p = 1.5, K p = 40, K c = 20,  = 0.99, and x 1 = 0

30. /faculteit technologie management Value of Flexible Capacity VFC = ETC IC – ETC FC %VFC = VFC / ETC IC Value of Flexible Capacity increases as the contingent capacity becomes less costly to utilize

31. /faculteit technologie management %VFC versus Backorder and Permanent Capacity Costs

32. /faculteit technologie management %VFC versus Permanent Capacity Size and Coefficient of Variation

33. /faculteit technologie management %VFC versus Setup Costs

34. /faculteit technologie management Expected Production under Varying Setup Costs

35. /faculteit technologie management Introduction Related Literature Model Analysis with No Set-up Costs Analysis with Set-up Costs Value of Flexible Capacity Conclusions and Future Work Outline

36. /faculteit technologie management Conclusions Flexibility is very important under – lower costs of contingent capacity – higher setup costs of production – lower levels of permanent capacity, and – higher costs of backordering For businesses with high demand volatility, the value of flexibility is extremely high even under abundant permanent capacity levels –long-term contractual relations with third-party contingent capacity providers would be suggested

37. /faculteit technologie management Future Research Relaxing some of the assumptions: –Upper limit on contigent capacity –Uncertainty on capacity –Positive lead times Incorporating tactical level changes in permanent capacity Developing an efficient heuristic for the multi- period problem with set-up costs