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
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Outline
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•
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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
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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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Outline
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•
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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
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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 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)
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Outline
•
•
•
•
•
•
•
Introduction
Related Literature
Model
Analysis with No Set-up Costs
Analysis with Set-up Costs
Value of Flexible Capacity
Conclusions and Future Work
/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
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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 =
ccorig / 
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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 / nonconcave) unit production cost (convex under zero setup costs)
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Production Cost Structure
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Formulation
CIMP:
U
U 
f t  xt   Uc p  min K p  yt  xt   K c  yt  xt   yt  xt  cc
x y
t
t


yt  wg wdw  b  w  yt g wdw
0
y
 E  f t 1  yt  Wt  for t  1,2,..., T
h
yt
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t
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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Outline
•
•
•
•
•
•
•
Introduction
Related Literature
Model
Analysis with No Set-up Costs
Analysis with Set-up Costs
Value of Flexible Capacity
Conclusions and Future Work
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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
 ytc

 xt  U
*
yt  xt    p
 yt
x
 t
 b 
y G 

hb
p
T
1
T
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if xt  ytc  U
if ytc  U  xt  ytp  U
if ytp  U  xt  ytp
if ytp  xt
 b  cc 
y G 

 hb 
c
T
1
T
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Optimal Control Parameters in Time
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Outline
•
•
•
•
•
•
•
Introduction
Related Literature
Model
Analysis with No Set-up Costs
Analysis with Set-up Costs
Value of Flexible Capacity
Conclusions and Future Work
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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
S  x 
y x   
x
*
T


max s u x , s c x 
 u
s  x   s  x 
 p
s  x 
if x  sx 
otherwise
if x  yTc  U
if yTc  U  x  yTp  U
if yTp  U  x
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 yTc
if sx   s c x 

S x    x  U if sx   s u x 
y p
p


x 
if
s
x

s
 T
Optimal order quantities for a single-period problem with set-up costs
200
150
s(x)
100
y*
Q*
50
0
0
20
40
60
80
100
x
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120
140
160
180
200
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
100
80
60
y*
40
Q*
20
0
-80
-60
-40
-20
0
-20
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Initial Inventory
20
40
60
80
Outline
•
•
•
•
•
•
•
Introduction
Related Literature
Model
Analysis with No Set-up Costs
Analysis with Set-up Costs
Value of Flexible Capacity
Conclusions and Future Work
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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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Outline
•
•
•
•
•
•
•
Introduction
Related Literature
Model
Analysis with No Set-up Costs
Analysis with Set-up Costs
Value of Flexible Capacity
Conclusions and Future Work
/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
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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 multiperiod problem with set-up costs
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