OPTIMAL ODERING POLICIES AND CAPITAL INVESTMENT IN SETUP
REDUCTION FOR CONTINUOUS REVIEW INVENTORY SYSTEM WITH
RANDOM YIELDS
Kuo-Lung Hou1) and Li-Chiao Lin2)
1)
National Defense Management College ([email protected])
National Chinyi Institute of Technology ([email protected])
2)
Abstract
In existing production/inventory models with random yields, the setup cost is
given and fixed. However, in many practice situations, they can be reduced by
investment in modern production technology. In this study, we attempt to determine the
optimal capital investment in setup cost reduction for (Q,r) inventory system, jointly
with the optimal ordering policies. We assume that the setup cost is the function of
capital expenditure. Furthermore, we show that the expected total annual cost function
with capital investment is convex. With the convexity, an algorithm is developed to
determine the optimal order quantity and reorder point, and then to find the optimal
setup cost. Finally, numerical results are presented to illustrate the algorithm procedure
and compared to the results without considering capital investment. Our results
evidently show that substantial cost savings might be obtained through capital
investment in setup reduction.
Keywords: Inventory; Lot sizing; Random yields; Investment analysis
1
1. Introduction
The problem of random yields in production or procurement has become an
important research topic in production and inventory area. In particular, several papers
that have been reported lately examine the implications of yields randomness on lot
sizing decisions. For an extensive review on many other variations of lot sizing
problems with random yields, the interested readers are referred to the excellent paper
of Yano and Lee [18]. Silver [15] was one of the earliest authors who extend the classic
economic order quantity (EOQ) model by involving the effect of yield uncertainty. He
listed a number of sources for randomness causing the quantity received not matching
the quantity requisitioned. Following Silver [15], Gerchak [2] and Noori and Keller [9]
consider a variable-yield lot-sizing problem with stochastic demand. They show that
the solution for the backorders case is a simple extension of the continuous-review
reorder point models initiated by Hadley and Whitin [4]. Other related studies can be
found in Kalro and Gohil [8], Parlar and Wang [10], Shin [14], and Wang and Gerchak
[17].
On the other hand, the benefits of reduced setups are well documented. For
example, faster changeovers have been associated with lower inventories, faster
throughput, shorter lead times, improved quality, and lower unit costs. Quick setups are
also considered an important element for successfully implementing just-in-time
production or time-based competition. Much of the analytical work in setup reduction
examines the benefits of reduced setups in inventory and setup costs. In addition,
several relationships between the amount of capital investment and the setup cost level
have been reported by Billington [1], Hofmann [6], Hong [7], Porteus [11,12], and
Sarker and Coates [13].
Based on above arguments, we assume that the relationship between setup cost
reduction and capital investment can be described by the logarithmic investment
2
function. Hence, this article extends the work of Gerchak [2] and Noori and Keller [9]
to consider setup cost can be reduced through capital investment. Moreover, we show
that the expected total annual cost function with capital investment is convex. With the
convexity, an algorithm is developed to determine the optimal order quantity and
optimal reorder point. Therefore the optimal ordering policies and capital investments
in setup cost reduction are appropriately determined. Finally, numerical example is
provided to illustrate the algorithm procedure and to compare the optimal policies
without investment option in setup cost reduction.
2. NOTATIONS and ASSUMPTIONS
To establish the mathematical model, the following notations and assumptions are
used.
Notations.
D = average demand per year,
Q = order quantity, a decision variable,
r = reorder point, a decision variable,
Y Q = quantity received given that Q units are ordered, a random variable,
 = yield’s standard deviation,
K0 = original setup cost per setup,
K = nominal setup cost per setup, a decision variable,
 k = capital investment in setup cost reduction,
i = cost of capital/$/year,
E (YQ )   Q , expected value of YQ given that Q units are ordered,
 Y   Q , standard deviation of YQ given that Q units are ordered, is proportional
Q
to Q,
3
 = bias factor,  
E (YQ )
Q
,
h = inventory holding cost per item per year,
 = penalty cost per shortage,
L = replenishment lead time, in years,
z = safety factor ( z  0 ),
1
 (t ) 
2

exp(
t2
),
2

 ( z )   (t )dt ,
z


 ( z )  (t  z ) (t )dt   ( z )  z ( z ) ,
z
X = lead time demand,
x = mean of X,
s L = standard deviation of X,
f(x) = probability density function of X,
  (x  x)2 
exp 
,
2
2 s L
 2s L

1
=
F ( x) 

 f ( x)dx ,
r
B(r) = expected shortage at the end of a cycle,
=


r
( x  r ) f ( x ) dx
 s L (z) ,
K * = optimal setup cost per setup,
Q* = optimal order quantity,
r* = optimal reorder point.
Assumptions.
1. A single item is considered.
2. The inventory position of an item is continuously reviewed, and the policy is
4
to order a lot Q when the inventory position drops to a reorder point r.
3. The reorder point r = expected demand during lead time + safety stock (SS)
and SS = r  x = z  (standard deviation of lead time demand), i.e.,
r  x  z s L where z is the safety factor and considered as a non-negative
value, that is r  x , as assumed in Hariga and Ben-Daya [5] and Silver, Pyke,
and Peterson [16].
4. The lead time demand X, which is assumed to be independent of YQ, follows
the normal distribution with mean x and standard deviation s L .
5. The quantity received is a random variable depending upon the quantity
ordered.
6. The bias factor  is necessarily greater than zero, it need not be less than one,
as assumed in Gerchak and Palar [3], Kalro and Gohil [8], and Silver [15].
7. The relationship between setup cost reduction and capital investment can be
described by the logarithmic investment cost function. That is, setup cost, K,
and the capital investment in setup cost reduction,  k , can be stated as
 K0 

 K 
 k ( K )  a ln 
f o r0  K  K 0 .
(1)
where a is positive constants.
3. THE MODEL
The expected total annual cost, which is composed of setup, holding and
backorder costs, can be determined in a manner similar to that used by Gerchak [2] or
Noori and Keller [9] and is given by
TC(Q,r) =

KD hQ 2
D

   2  h( r  x ) 
B(r )
Q 2
Q


(2)

where B (r )  ( x  r ) f ( x)dx .
r
As it takes investment to reduce setup cost, we should include an amortized
investment cost in our proposal model. Therefore, the expected total annual cost of the
5
system, ETC (Q, r , K ) , is composed of Eq. (2) and the amortized total capital cost,
i k (K ) which shows the economic consequences of the investment per unit time, as
follows:
E T C(Q, r , K )

KD hQ
D

( 2   2  h(r  x ) 
B(r )  i k ( K )
Q 2
Q


(3)
where  k (k ) is based on Eqs. (1). Hence, Eq. (3) can be stated as
E T C(Q, r , K )

2
KD
K 
 Q
 h Q D
 h
 r  x 

B(r )  ia ln  0 
Q
2
Q
 2

 K 
(4)
It is easy to show that ETC in Eq. (4) is convex on K. In order to find the minimum cost,
the partial derivatives of the ETC (Q, r , K ) can be evaluated as
ETC (Q, r , K )
D ia

 0
K
Q K
(5)
From Eqs. (5), the optimal setup cost can be solved as
K * (Q)  ia
Q
(6)
D
Substituting Eq. (6) into Eq. (4) yields the following expression of the corresponding
expected total annual cost EAC:
EAC(Q, r )  ia  h(
Q
2
 r  x) 
K D
h 2 Q D

B(r )  ia ln  0 
2
Q
 ia Q 
(7)
Hence, the following results will be shown.
Theorem 1. EAC (Q, r ) is convex with respect to Q  0 and r  x .
The proof of Theorem 1 is found in the appendix.
Based on the convex nature of the function, EAC(Q,r). The first-order conditions
for a minimum are given by


EAC (Q, r )  0 . Then, we
EAC(Q, r )  0 and
r
Q
obtain:
6
Q  Q(r ) 
F (r ) 
ia 
ia 2  2h 2   2 DB (r )

h  2  2

(8)
h Q
D
(9)
h Q 1
 . In such a situation, the
D
2
2ia
model says that the lower r is, the lower the cost is. Let QP =
, it represents
h  2  2
Since r  x , Eq. (9) will have no solution if


that the optimal order quantity when shortages are not allowed. From Eq. (8) we know
that
Q  QP
(10)
Then, we have the following result.
Theorem 2.
If
h  QP 1
 , then
D
2
ia 
r *  x and Q * 
ia 2  2h

h 
2
2
2


  2 Ds L
2
.
(11)
The proof of Theorem 2 is included in the appendix.
Algorithm
The optimal solution (Q * , r * ) can be obtained by solving the above Eqs. (8) and (9)
iteratively until convergence. We can easily prove the convergence of the algorithm by
adopting a similar graphical technique used in Hadley & Whitin [4]. Therefore, we
obtain the optimal setup cost from Eq. (6). Thus, we can use the following iterative
algorithm to find the optimal solution (Q * , r * , K * ) . This algorithm is to initially set Q
=
2ia
, then solve for a corresponding r value in Eq. (9), and then use this
h  2  2


value in Eq. (8) to find a new Q value, and so forth. Let   Q(0)  QP . Eq. (8) implies
  0 . Then the detail about the algorithm can be described as follows:
7
Step 1.
Let     0 , Qopt = QP =
Step 2. If
h  Qopt
D

2ia
.
h  2  2


1
, set ropt = x and go to Step 6. Otherwise, solve for a
2
corresponding r value in Eq. (9) and set ropt = the corresponding r value.
Step 3. Use Eq. (8) to set Qnew = Q (ropt ).
Step 4. If
h  Qnew 1
 , then set rnew = x . Otherwise, solve for a corresponding r
D
2
value in Eq. (9) and set rnew = the corresponding r value.
Step 5. If Qnew  Qopt   and rnew  ropt   , go to Step 6. Otherwise, set
Qopt = Qnew , ropt = rnew and go to Step 2.
Step 6. Get the optimal order quantity Q* = Q (ropt), the optimal reorder point
r *  ropt ,
and
(i.e.,  k ( K * )  a ln(
then
the
optimal
setup
cost
K *  ia
Q *
D
K0 D
) ).
iaQ *
When no capital investment in setup cost reduction is made, then K *  K 0 .
Hence, from Eq. (6), we have
ia 
K0D
Q
(12)
Thus, the corresponding expected total annual cost EAC in Eq. (7) can reduce to the
expression of Eq. (2) which is the result of Gerchak [2] and Noori and Keller [9].
Therefore, the lot size reorder point inventory model with random yield in this paper is
an extension of both Gerchak [2] and Noori and Keller [9]. Next, substituting Eq. (12)
into Eq. (8), the optimal order quantity will reduce to
Q
2K 0 D  2DB (r )
h( 2   2 )
(13)
Therefore, a similar algorithm as mentioned above can be used to solve Eqs. (13) and
(9). Then, the results for special case such as no capital investment in setup cost
8
reduction are derived.
4. ILLUSTRATIVE EXAMPLE
For comparison, two different models used here are specified as follows.
Model 1: (Q,r) model with random yields and with capital investment in setup cost
reduction [see Eq. (7)], and
Model 2:(Q,r) model with random yields and no capital investment [see Eq. (2)].
The percentage of savings of excepted total annual cost is defined by
 EAC(Q * , r * ) of model 1 
%EAC  1 
  100%
*
*
 EAC(Q , r ) of model 2 
Next, to illustrate the above algorithm, let us consider the following numerical
example:
Example: We use the following parameter values: D = 1000 units/year, K0 = $100 per
setup, h = $10 per unit per year,  = $40 per shortage, x  36 units, s L = 10 units,
 = 1.0,  = 1.5, a = 1600, i = 0.15, and   0.001. Then, we obtain the following
associated values that are required in the algorithm procedure.
1. Qp = 22.1538, Q(0) = 110.7906,  = 88.6368 >  , and
hQP
1
= 0.0083 <
2
D
for Model 1, and
2. Qp = 78.4465, Q(0) = 78.5572,  = 0.1107 >  , and
hQP
1
= 0.0294 <
2
D
for Model 2.
Applying the proposed algorithm, we provide the results of procedure solution for two
Models as in Table 1. For comparison, a summary of optimal solution for two Models
is presented in Table2. From the results shown in Table 2, we see that significant
savings of the expected total annual cost are achieved through capital investment. Here,
we get the optimal setup cost K* = $9.0948. Then, the investment in the setup cost
reduction,  k ( K * ) = 1600 ln 100 9.0948 = $3835.9478, namely, the optimal
9
investment require $3835.9478 when Q* = 25.2632 and r* = 59.4656. The
corresponding expected total annual cost EAC(Q * , r * ) = $1357.4260, which is less
than the cost of without capital investment in setup cost reduction. The percentage of
savings of expected total annual cost is given by 29.55%. This is, 29.55% expected
total annual cost savings are relative to that without capital investment in setup cost
reduction.
5. CONCLUSIONS
Considering that the reduction of setup cost is an important strategy in
manufacturing, the present investigations on setup cost in a manufacturing environment
are warranted. In this paper, we develop the ordering policies when setup cost can be
reduced through capital investment. To explore these policies, the expected total annual
cost function with capital investment is formulated. We show that the cost function is
convex and develop the algorithm to determine the optimal order quantity and reorder
point, then to locate optimal setup cost. Therefore the optimal capital investment is
appropriately determined. Finally, a numerical example is given to illustrate the
algorithm and evaluate the effects of utilizing capital investment. It should be
emphasized that these results evidently show that significant cost savings can be
achieved by adopting capital investment and it is consistent with the JIT manufacturing
philosophy which calls for reducing setup cost to achieve the inventory reductions.
10
Table 1
Results of procedure solution of two different models for example 1.
Model Iteration
type
Model 1
Model 2
1
2
3
4
5
6
Number
Q
22.1538 24.8848 25.2179 25.2583 25.2632 25.2632
r
59.9511 59.5218 59.4723 59.4664 59.4656 59.4656
Q
78.4465 80.2092 80.2539 80.2550 80.2550
r
54.8943 54.7964 54.7940 54.7939 54.7939
Table 2
Summary of the results of the optimal procedure solution for example1.
Model
Q*
r*
z*
Model 2
EAC(Q*, r*)
% EAC
1357.4260
29.55
1926.7980
---
(k (K * ) )
type
Model 1
K*
25.2632
59.4656
2.3466
9.0948
(3835.9478)
80.2550
54.7939
100
1.8794
(0)
11
REFERENCES
1)
P. J. Billington, The classical economic production quantity model with setup
cost as a function of capital expenditure, Decision Sciences 18 (1987) 25-42.
2)
Y. Gerchak, Order point/order quality models with random yield, International
Journal of Production Economics 26 (1992) 297-298.
3)
Y., Gerchak, M. Parlar, Yield variability, cost tradeoffs and diversification in
the EOQ, Model. Naval Research Logistics 37 (1990) 341-354.
4)
G. Hadley, T.M. Whitin, Analysis of inventory systems, Prentice-Hall,
Englewood Cliffs, NJ, 1963.
5)
H. Hariga, M. Ben-Daya, Some stochastic inventory models with deterministic
variable lead time, European Journal of Operational Research 113 (1999)
42-51.
6)
C. Hofmann, Investments in modern production technology and the cash
flow-oriented EPQ-model, International Journal of Production Economics 54
(1998) 193-206.
7)
J.D. Hong, Optimal production cycle, procurement schedules, and joint
investment in an imperfect production system, European Journal of
Operational Research 100 (1997) 413-428.
8)
A.H. Kalro, M.M. Gohil, A lot size model with backlogging when the amount
received is uncertain, International Journal of Production Research 20 (1982)
775-786.
9)
A.H.
Noori,
G.
Keller,
The
lot-size
reorder-point
model
with
upstream-downstream
10) uncertainty, Decision Sciences 17 (1986) 285-291.
11) M. Parlar, D. Wang, Diversification under yield randomness in inventory
models European Journal of Operational Research, 66 (1993) 52-64.
12
12) E.L. Porteus, Investing in reduced setups in the EOQ model, Management
Sciences, 31 (1985) 998-1010.
13) E.L. Porteus, Optimal lot sizing, process quality improvement and setup cost
reduction, Operations Research 34 (1986) 137-144.
14) B.R. Sarker, E. R. Coates, Manufacturing setup cost reduction under variable
lead time and finite opportunities for investment, International Journal of
Production Economics, 49 (1997) 237-247.
15) W. Shih, Optimal inventory policies when stockouts result from defective
products, International Journal of Production Research 18 (1980) 677-685.
16) E.A. Silver, Establishing the reorder quantity when the amount received is
uncertain, INFOR 14 (1976) 32-39.
17) E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production
Planning and Scheduling, New York: John Wiley, 1998.
18) Y, Wang, Y. Gerchak, Periodic review production models with variable
capacity, random yield, and uncertain demand, Management Science 42 (1996)
130-137.
19) C.A. Yano, H.L. Lee, Lot sizing with random yields: a review, Operations
Research 43 (1995) 311-334.
13
APPENDIX
Proof of Theorem 1
Eq. (7) yields

h h 2 D
ia
EAC(Q, r ) 


B(r ) 
2
Q
2
2 Q
Q
(A1)
2
2DB (r ) ia
EAC (Q, r ) 
 2 0
2
Q
Q3
Q
(A2)

D
E A C(Q, r )  h 
F (r )
r
Q
(A3)
2
D
EAC(Q, r ) 
f (r )  0
2
Q
r
(A4)
2
D
E A C(Q, r ) 
F (r )  0 .
Qr
Q 2
(A5)
and
Let H(Q,r) denote the determinant of the Hessian of EAC (Q, r ) . Then
H (Q, r )
 2 EAC (Q, r )  2 EAC (Q, r )   2 EAC (Q, r ) 


 
Qr
Q 2
r 2


2
2
2
 D
 
1   D 
QD i a
 4 2  B(r ) f (r ) 
f (r )   F (r )  
Q   


 

Q4 is always positive. With respect to r, we notice that:
lim H (Q, r )  0
r 
(A6)
  D  2
H (Q, r )  (r  x )
Q D i a



f
(
r
)
2
B
(
r
)


0
2
 
r
 
Q 4 sL
   
(A7)
and
for all r  x
Eq. (A7) means that, with increasing r ( r  x ), the determinant of Hessian tends to
decrease monotonically to zero. Hence,
H(Q,r)  0 for all r  x .
(A8)
14
Thus, we conclude that EAC(Q,r) is convex on Q > 0 and r  x .
Proof of Theorem 2
Eq. (A3) can be rewritten as


D  h  Q *

E A C(Q * , r ) 
 F (r ) 
* 
r
 Q  D

Eq. (13) implies that if
(A9)
h  QP 1
 , then
D
2
h Q* 1
  F (r ) f o r r  x.
D
2
(A10)
Eq. (A10) implies

EAC (Q * , r )  0
r
for all r  x .
(A11)

EAC (Q * , r ) is non-decreasing with respect to
r
Therefore, Eq. (A11) reveals that
r  x and EAC(Q*,r) has a minimum at r  x . So r *  x . On the other hand, Eq.
(A2) shows that EAC(Q, x ) is convex with respect to Q > 0. Q* can be determined by
solving the following equation

E A C(Q, x )  0
Q
(A12)
hence, we can obtain
ia 
Q* 
ia 2  2h

h 
2
2
2


  2 Ds L
2
.
15
(A13)