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

/faculteit technologie management

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


• 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• 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• 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• 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


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)



Outline


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• 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 /


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


Formulation

TtWyfE

dwwgywbdwwgwyh

cxyxyKxyKUcxf

ttt

y t

y

t

cUtt

Uttcttpyxptt

t

t

tt

,...,2,1for

min

1

0

CIMP:


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


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

tptt

ptt

pt

pt

ptt

ctt

ctt

ct

tt

xyx

yxUyy

UyxUyUx

Uyxy

xy

if

if

if

if

*

bhcbGy

bhbGy c

TcTT

pT

11



Optimal Control Parameters in Time



Outline


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• 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

otherwise

if*

xxsxxS

xyT

xUyxs

UyxUyxs

Uyxxsxs

xspT

p

pT

cT

u

cT

cu

if

if

if,max

xsxy

xsxUx

xsxy

xSpp

T

u

ccT

s if

s if

s if


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*


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…• 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


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*



Outline


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• 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 Permanent Capacity Size and Coefficient of Variation


%VFC versus Setup Costs


Expected Production under Varying Setup Costs



Outline


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• 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

Documents

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