American Journal of Oper ations Research, 2011, 1, 46-50
doi:10.4236/ajor.2011.12007 Published Online June 2011 (http://www.SciRP.org/journal/ajor)
Optimal Policy and Simple Algorithm for a Deteriorated
Multi-Item EOQ Problem
Bin Zhang, Xiayang Wang
Lingnan College, Sun Yat-Sen University, Guangzhou, China
Received March 29, 201 1; revised April 19, 2011; accepted May 12, 2011
Abstract
This paper considers a deteriorated multi-item economic order quantity (EOQ) problem, which has been stu-
died in literature, but the algorithms used in the literature are limited. In this paper, we explore the optimal
policy of this inventory problem by analyzing the structural properties of the model, and introduce a simple
algorithm for solving the optimal solution to this problem. Numerical results are reported to show the effi-
ciency of the proposed method.
Keywords: Inventory, EOQ, Deterioration, Multi-Item
1. Introduction
Multi-item inventory problem with resource constraints
is an important topic of inventory management [1]. These
constrained inventory models are still hot topics in aca-
demic and practice fields, for example, see [2-6]. The re-
source constraints typically arise from shipment capacity,
warehouse capacity or budgetary limitation. Since some
items such as fruits, vegetables, food items, drugs and
fashion goods will deteriorate in the shipment or storage
process, many works have been done for investigating
inventory problems for deterioration items [7-11]. Since
items’ deterioration often takes place during the storage
period, some researchers have considered economic order
quantity (EOQ) models for deteriorating ite ms, f or ex am-
ples see [12,13].
Recently, Mandal et al. [14] present a constrained
multi-item EOQ model with deteriorated items. In [14],
the model is firstly formulated as the transcendental form
and the polynomial form, i.e., without and with trunca-
tion on the deterioration terms. These two versions of the
model are both solved by applying non-linear program-
ming (NLP) method (Lagrangian multiplier method). As
[14] points out, the polynomial form is an approximation
of the transcendental form. Secondly, the transcendental
form is converted to the minimization of a signomial
expression with a posynomial constraint, which is so lved
by applying a modified geometric programming (MGP)
method. However, we argue that the studied problem can
be solved using a simple algorithm without any model
approximation or conversion.
In this paper, we prove that the deteriorated multi-item
EOQ model is a special convex separable nonlinear
knapsack problem studied in [15], which is characterized
by positive marginal cost (PMC) and increasing marginal
loss-cost ratio (IMLCR). PMC requires p ositive marginal
cost of decision variable, and IMLCR means that the
marginal loss-cost ratio is increasing in decision v ariable.
Following [15], we explore the optimal policy for the
problem, and develop a simple algorithm for solving it.
The main purpose of this paper is twofold: 1) to explore
the optimal policy of this inventory problem by analyz-
ing the structural properties of the model; 2) to introduce
a simple algorithm for solving the optimal solution to
this problem.
The reminder of this paper is organized as follows. We
formulate the problem in the next section. In Section 3,
we explore the structural properties of the problem, and
provide the optimal policy and algorithm. Numerical
results are reported in Section 4, and Section 5 briefly
concludes this paper.
2. Problem Formulation
Consider a multi-item EOQ problem with a storage sp ace
constraint, in which all items (1, ,in) deteriorate
after certain periods.
Before presenting the model, we list all notation as
follows. Notice that the same notation used in [14] is
presented.

B. ZHANG ET AL.
47
n = Total number of items
Qi = Order quantity
c0i = Purchasing cost
c1i = Holding cost per unit quantity per unit time
c3i = Set-up cost
i

= Constant rate of deterioration (01
i

)
wi = Required storage quantity per unit time
Di = Demand rate per unit time
Ti = Time period of each cycle
1
(,, )
n
TC TT= Total average cost function
W = Available storage space
Following [14], we set 2
01 3iiiiiii i
acD cDc   
,
2
010
iiiiiii
bcD cD   
, and 1iiii
ccD

. Then the
deteriorated multi-item EOQ model can be expressed as
follows (denoted as problem P). We refer the reader to
[14] for the details of this model.
1
11
min (,,)
() ii
n
T
nn
i
iii i
ii
ii
TC TT
ae
f
Tbc
TT








, (1)
subject to
11
() ((1))
ii
nn T
i
ii i
ii
i
D
g
Twe W




 , (2)
0
i
T, 1, ,in. (3)
The order quantity is given by (1)
ii
T
i
ii
D
Qe



,
1, ,in. The first and second order derivatives of
()
ii
f
T,1, ,in, and the first order derivative of
()
ii
g
T,1, ,in, are calculated as follows:
2
()(1 )
ii
T
iii iii
ii
df TabeT
dT T

 
, (4)
222
23
() 2(22)
ii
T
iii iiiii
ii
dfTa beTT
dT T

  
, (5)
() ii
T
ii ii
i
dg T
D
we
dT

. (6)
3. Structural Properties and Algorithm
In this section, we first establish structural properties of
problem P, and then we present an efficient procedure for
solving the optim al solution to problem P.
3.1. Structural Properties
Before presenting the structural properties of problem P,
we give two basic equations, which will be used in our
proofs. Since Taylor expansion of exponential function is
1
1!
ii
kk
Tii
k
T
ek




, then we have
22
11 2
ii
T
iiiii i
TTTe

  , (7)
for 0
i
T, 1, ,in

.
By comparing the definitions of ai and bi, we have
ii
ab

, 1, ,in

. (8)
Considering the objective function of problem P, we
have the following proposition.
Proposition 1. The cost 1 (,, )
n
TC TT is strictly
convex.
Proof. Since the function 1 (,, )
n
TC TT is separable,
we only need to prove that ()
ii
f
T is strictly convex in
i
T, 1, ,in

. According to Equations (7) and (8), we
have.
22
2222 44
2(22 )
221 1
222
ii
T
ii iiii
iiii iii
ii iiii
abeT T
TTbT
bb TT




 

 


.
Substituting the above equation into Equation (5), we
have.


244 4
23 0
2 2
ii ii iii i
ii
dfT bT bT
dT T


for 0
i
T.
Thus, ()
ii
f
T and 1
(,, )
n
TC TT are strictly convex.
QED.
The strictly convexity of 1
(,, )
n
TC TT ensures that
the optimal solution to problem P is unique. This prop-
erty has also been indicated in [14] by using a more
complicated proof procedure.
Denote ProductLog( )z as the principal solution for x
in
x
zxe, which has the same function name in Ma-
thematica to stand for the Lambert W function. Let ˆi
T,
1, ,in

, be a value such that () 0
iii
dfTdT . Denote
problem CP as problem P without the constraint in Equa-
tion (2), then the following proposition characterizes the
optimal solution to problem CP.
Proposition 2. The optimal solution to problem CP is
1 ProductLog
ˆ
i
ii
ii
a
be
T





, 1, ,in. (9)
Proof. Since ()0
ii i
dfTdT  at the point ˆi
T, we have
ˆˆ
(1) 0
ii
T
ii ii
abe T



 . This equation can be rewritten
as ˆ1ˆ
(1)
ii
T
iii
i
aeT
be




, and hence we have
1 ProductLog
ˆ
i
ii
ii
a
be
T





.
From Equation (8), we know 1i
iii
a
ebe
 . According

B. ZHANG ET AL.
48
to [16], we know that ProductLog( )z is an increasing
function for 1
i
ze  , then we have ProductLog( )
i
ii
a
be

1
ProductLog( ) 1
i
e
 . Substituting this equation into
ˆi
T, we have hence ˆ0
i
T, which satisfies the positive
constraint in Equation (3). Thus, ˆi
T, 1,,in, is an
optimal solution to problem CP. Since the optimal solu-
tion is unique, we know that the optimal solution to
problem CP is ˆi
T, 1,,in. QED.
Following [15], we define the marginal loss-cost ratio
of item i (1,,in) as.
2
()
() ()
ii
ii T
ii iiii
ii ii iii
i
df T
dTa ebTb
rT dg TDwT
dT
 


 . (10)
Then we have the following proposition.
Proposition 3. ()
ii
rT is strictly increasing in
ˆ
(0, ]
ii
TT, 1, ,in.
Proof. From Equation (7), we know 1ii
T
ii
Te

  for
0.
i
TThe convexity of ()
ii
fT guarantees ii
T
ii
abe


(1)0
ii
T

 for ˆ (0, ]
ii
TT. Using ii ab in Equa-
tion (8) and the above two equations, we have
(2) (2)
[(1)][(1)]
[(1)][(1)]0
ii
ii ii
ii ii
T
iiiiii
TT
iiiiiiii
TT
iiii iii
beT aT
abeTaT be
abeTbT e







 
 
,
for ˆ (0, ]
ii
TT. Hence we have
3
()[(2 )(2 )]0
ii ii
TT
iiiii iii
ii
dr TebeTaT
dT DT w




 .
Thus, ()
ii
rT is strictly increasing in ˆ (0, ]
ii
TT,
1, ,in. QED
Since this proposition illustrates that ()
ii
rT is strictly
increasing in ˆ
(0, ]
ii
TT, 1, ,in

, we let ()
ii
Tr ,
(,0]
i
r, be the inverse function of ()
ii
rT . Although
it is difficult to write ()
ii
Tr in a closed form, the strict
monotony of ()
ii
Tr ensures that ()
ii
Tr can be easily
determined by applying a bi-section search procedure
over ˆ
(0, ]
ii
TT, for any given (,0]
i
r .
3.2. Optimal Policy and Algorithm
We now demonstrate that the deteriorated multi-item
EOQ model is a special convex separable nonlinear
knapsack problem studied in [15]. Firstly, Proposition 1
illustrates that P is a convex problem; Secondly, from
Equation (6), we know () 0
ii
i
dg T
dT , for 0
i
T,1, ,in

,
which means there are positive marginal costs in problem P;
Finally, Proposition 3 ensures that the marginal loss-cost
ratio ()
ii
rT is increasing in ˆ
(0, ]
ii
TT, 1,,in

.
Therefore, the theoretical results and solution procedure
proposed in [15] are both applicable for problem P.
Denote by *
i
T, 1, ,in

, the optimal solution to
problem P. By directly applying the theoretical results in
[15], we can summarize the optimal policy for the dete-
riorated multi-item EOQ problem in the following pro-
position.
Proposition 4. The optimal policy of problem P is (a)
*ˆ
ii
TT

, 1, ,in

, if 1
ˆ
()
n
ii
i
g
TW


; (b) *
1()
n
ii
i
g
TW



and **
() ()
ii kk
rT rT, , 1,,ik n

, specify the optimal
solution *
i
T, 1, ,in

, if 1
ˆ
()
n
ii
i
g
TW


.
This proposition is obtained by directly applying the
theoretical results in [15] to problem P, since problem P
has PMC and IMLCR. Based on Propositions 2-4, the
idea of the algorithm proposed in [15] can be used for
solving problem P. The basic idea of the algorithm is as
follows: If the constr ain t in Equ a tion (2) is inactive, i.e. ,
1
ˆ
()
n
ii
i
g
TW


, then the optimal solution to problem P
equals to the optimal solution to the unconstrained prob-
lem, i.e., *ˆ
ii
TT

, 1, ,in

; Otherwise, Proposition 4(b)
means that obtaining the exact value of ** *
() ()
ii kk
rrT rT,
, 1,,ik n

, is the key to solving the optimal solution to
problem P. The optimal value *
r can be searched by
applying a binary search method over the interval
(,0]rM

, where
M
is a sufficient large value such
that ()
1(( 1))
ii
nTM
i
i
ii
D
we W





.
Main steps of the above solution procedure for solving
the optimal solution to problem P are summarized in the
following algorithm.
The Algorithm
1
*
ˆ
1: Solve , 1,,, from Equation (9);
ˆ
2: If (), t h e n
ˆ
let , 1,,, go to 8;
3: Let ,0;
4: Le t ()2;
5: Calculate
i
n
ii
i
ii
LU
LU
ii
StepT in
Stepg TW
TTin Step
SteprMr
Steprr r
StepT T




 





1
1
(), 1,,;
6: If (), then let ;
If (), then let ;
n
ii L
i
n
ii U
i
ri n
Stepg TWrr
gT Wr r









B. ZHANG ET AL.
49
*
**
Go to 4;
7: Let , 1,,;
8: Calculate (,,), and output.
ii
in
Step
StepTT in
StepTC TT


In comparison with the two methods in [14], there are
two main advantages of our algorithm: 1) It is a polyno-
mial algorithm of O(n) order, which ensures that the al-
gorithm is applicable for large-scale problems; 2) It does
not need any approximation or conversion of the original
model, thus it always solves the optimal solution to prob-
lem P.
4. Numerical Results
In this section, numerical experiments are provided to
show the efficiency of the proposed algorithm for solv-
ing problem P. The instances of problem P are all ran-
domly generated. We use the notation x ~ (,)U

 to
denote that x is uniformly generated over [,]

 . The
parameters of test instances are generated as follows:
i
w~ (1,10)U, 0i
c~ (1,10)U, 1i c~ (0.5,1.0)U, 3i c~
(40,100)U, i

~ (0.01,0.10)U, i
D~ (200,500)U,
1, ,in, and 100Wn.
In this numerical study, we set n=100 and 1000, re-
spectively. For each problem size n, 100 test instances
are randomly generated. The statistical results on number
of iterations of the binary search and computation time
(in milliseconds) are reported in Table 1, where 95% C.I.
stands for 95% confidence interval.
From Table 1, we can conclude that the proposed al-
gorithm can solve large-scale deteriorated multi-item
EOQ models very quickly in few iteration times. Since
the ranges of parameters are large, the standard devia-
tions of number of iterations and computation time are
quite low, reflecting the fact that the algorithm is quite
effective and robust.
We also use our algorithm to solve the illustrative exam-
ple studied in [14], which outputs the optimal solution:
*
10.2899T, *
20.2176T, *
1102.6397Q, * 298.6801Q,
*2587.1382TC  with *= 0.5925r. Unfortunately, this
result cannot be directly compared with that solved by [14],
because there is something wrong with the values of *
i
T
and *
i
Q, 1,2i, shown in Tables 2 and 3 of [14], since
they violate the basic equation
Table 1. Performance of our algorithm for solving the ran-
domly generated instances.
Number of iterations Computation time
Problem size n 100 1000 100 1000
Mean 27.9 31.8 217.0 2619.3
Std. Dev. 1.7 1.7 8.9 38.8
95% Lower27.6 31.4 215.2 2611.6
C.I. Upper28.3 32.1 218.8 2627.0
(1)
ii
T i
i
i
D
Qe



, 1, 2i

. For example, in the Table 2 of [14],
when *
10.2414712T and *
20.2419020T, (1)
ii
T
i
ii
D
Qe




gives 185.3365Q

, 2109.7828Q

. In addition, their
mistake can also be verified by our proved optimal pol-
icy ** **
11 22
() ()rT rT. For example, the values of ** ()
ii
rT ,
1, 2i

for the MGP solution presented in Table 3 of
[14] are **
11
( )0.6017rT  , and **
21
( )0.5857rT  , which
does not satisfy ** **
11 22
() ()rT rT.
From the above analysis, we illustrate that the solution
provided in [14] does not satisfy the optimal policy
proved in this paper, which is easily used to verify the
optimality of a solution to problem P. Thus, the numeri-
cal results in [14] are incorrect. Since some comparison
of NLP and MGP given by [14] were established based
on the numerical results, especially the results in Tables
2 and 3 of [14], we argue that the comparison of NLP
and MCP in [14] are questionable.
5. Conclusions
In this paper, we explore the structural properties of de-
teriorated multi-item EOQ model and propose a simple
algorithm for solving the optimal solution by proving
that the studied problem is a special convex separable
nonlinear knapsack problem. In addition, it is obvious
that the basic idea and obtained results in this paper can
be simply modified for solving the classical constrained
multi-item EOQ problem.
6. Acknowledgements
The authors would like to thank the two reviewers for
their insightful comments, which helped to improve the
manuscript. This work is supported by national Natural
Science Foundation of China (No. 70801065), the Fun-
damental Research Funds for the Central Universities of
China and Natural Science Foundation of Guangdong
Province, China (No. 10451027501005059).
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