Tan Tarkan

An Integrated Approach to Inventory and Flexible Capacity Management under Non-stationary Stochastic Demand and Set-up Costs: 

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

Outline: 

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

Introduction: 

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

Introduction: 

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.

Introduction: 

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

Outline: 

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

Literature Review: 

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)

Outline: 

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

Model: 

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

Model: 

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 / 

Model: 

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)

Production Cost Structure: 

Production Cost Structure

Formulation: 

Formulation CIMP:

Remark: 

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

Outline: 

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

Analysis with No Setup Costs: 

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

Optimal Policy: 

Optimal Policy

Optimal Control Parameters in Time: 

Optimal Control Parameters in Time

Outline: 

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

Analysis with Setup Costs: 

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

Single Period Optimal Policy: 

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

Single Period Optimal Policy: 

Single Period Optimal Policy

Multi-Period Problem: 

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

Actually, we expected this…: 

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

Outline: 

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

Value of Flexible Capacity: 

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

Value of Flexible Capacity: 

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

%VFC versus Backorder and Permanent Capacity Costs: 

%VFC versus Backorder and Permanent Capacity Costs

%VFC versus Permanent Capacity Size and Coefficient of Variation: 

%VFC versus Permanent Capacity Size and Coefficient of Variation

%VFC versus Setup Costs: 

%VFC versus Setup Costs

Expected Production under Varying Setup Costs: 

Expected Production under Varying Setup Costs

Outline: 

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

Conclusions: 

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

Future Research: 

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