tarkan tan eindhoven university of technology osman alp bilkent university may 24, 2005
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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 - PowerPoint PPT PresentationTRANSCRIPT
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An Integrated Approach to Inventory and Flexible Capacity
Management under Non-stationary Stochastic Demand
and Set-up CostsTarkan Tan
Eindhoven University of Technology
Osman Alp Bilkent University
May 24, 2005FIFTH INTERNATIONAL CONFERENCE ON
"Analysis of Manufacturing Systems - Production Management"Zakynthos Island, Greece
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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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.
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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
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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Literature Review• Integrated Production/Capacity Management
Atamtürk & Hochbaum (MS 2001), Angelus & Porteus (MS 2002),Dellaert & de Kok (IJPE 2004)
• Workforce Planning and FlexibilityHolt et al. (1960), Wild & Schneeweiss (IJPE 1993), Milner & Pinker (MS 2001), Pinker & Larson (EJOR 2003)
• Capacitated Production/Inventory ModelsFedergruen & 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)
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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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 = cc
orig /
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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)
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Production Cost Structure
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Formulation
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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
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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Optimal Policy
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Optimal Control Parameters in Time
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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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
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Single Period Optimal Policy
otherwise
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Optimal order quantities for a single-period problem with set-up costs
0
50
100
150
200
0 20 40 60 80 100 120 140 160 180 200
x
s(x)
y*
Q*
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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
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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
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Optimal Order Quantities for a 3-period Problem with Set-up Costs
-20
0
20
40
60
80
100
-80 -60 -40 -20 0 20 40 60 80
Initial Inventory
y*Q*
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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Value of Flexible Capacity• VFC = ETCIC – ETCFC
• %VFC = VFC / ETCIC
• Value of Flexible Capacity increases as the contingent capacity becomes less costly to utilize
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%VFC versus Backorder and Permanent Capacity Costs
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%VFC versus Permanent Capacity Size and Coefficient of Variation
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%VFC versus Setup Costs
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Expected Production under Varying Setup Costs
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• Introduction• Related Literature• Model• Analysis with No Set-up Costs• Analysis with Set-up Costs• Value of Flexible Capacity• Conclusions and Future Work
Outline
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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
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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