Tarkan Tan Eindhoven University of Technology Osman Alp Bilkent University May 24, 2005

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

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

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

9. 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. Model • The amount that each permanent worker produces per period is defined as 1 "unit" • cp 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 cc = ccorig/

11. 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. Production Cost Structure

13. Formulation CIMP:

14. Remark • When Kp= Kc= 0 and cc , CIMP boils down to a capacitated production/inventory problem • Similarly, when Kp > 0 and either Kc  or cc, CIMP boils down to a capacitated production/inventory problem with production setup cost

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

16. 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. Optimal Policy

19. Optimal Control Parameters in Time

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

21. 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. 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 sc(x), su(x), and sp(x) that can be characterized and s(x) takes the form of one these functions depending on the value of x

23. Single Period Optimal Policy

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 fT(x) is a quasi-convex function, summation of convex and quasi-convex functions is not necessarily quasi-convex

26. 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

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

29. 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, cc = 2.5, cp = 1.5, Kp = 40, Kc = 20,  = 0.99, and x1 = 0

30. Value of Flexible Capacity • VFC = ETCIC– ETCFC • %VFC = VFC / ETCIC • Value of Flexible Capacity increases as the contingent capacity becomes less costly to utilize

31. %VFC versus Backorder and Permanent Capacity Costs

32. %VFC versus Permanent Capacity Size and Coefficient of Variation

33. %VFC versus Setup Costs

34. Expected Production under Varying Setup Costs

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

36. 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. 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