Optimal inventory policies for an economic order quantity model with decreasing cost functions

European Journal of Operational Research 165 (2005) 108–126

www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

Optimal inventory policies for an economic orderquantity model with decreasing cost functions

Hoon Jung a, Cerry M. Klein b,*

a Postal Technology Research Center, Electronics and Telecommunications Research Institute, Deajeon 305-350, South Koreab Department of Industrial and Manufacturing Systems Engineering, The University of Missouri, E3437 Engineering Building East,

Columbia, MO 65211, USA

Received 5 June 2001; accepted 3 September 2003

Available online 5 March 2004

Abstract

In this paper, three total cost minimization EOQ based inventory problems are modeled and analyzed using geo-

metric programming (GP) techniques. Through GP, optimal solutions for these models are found and sensitivity

analysis is performed to investigate the effects of percentage changes in the primal objective function coefficients. The

effects on the changes in the optimal order quantity and total cost when different parameters of the problems are

changed is also investigated. In addition, a comparative analysis between the total cost minimization models and the

basic EOQ model is conducted. By investigating the error in the optimal order quantity and total cost of these models,

several interesting economic implications and managerial insights can be observed.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Inventory; Geometric programming; EOQ

1. Introduction

This purpose of this work is to extend the classical economic order quantity (EOQ) model to more

realistic scenarios. The classic EOQ model assumes constant demand and a fixed purchasing cost. These

assumptions do not accurately reflect the time-based competition of today. To this end, three inventory

models considering total cost minimization are established and analyzed. The key feature differentiatingthese models from the basic EOQ model is that the cost per unit exhibits some type of economies of scale. In

the EOQ model, the cost per unit is fixed. In the three proposed models, the cost per unit is a power

function of the demand per unit time (Model 1), a power function of the order quantity (Model 2), or a

power function of both of the demand per unit time and the order quantity (Model 3). In deriving and

* Corresponding author. Address: Department of Industrial and Manufacturing Systems Engineering, The University of Missouri-

Columbia, E3437 Engineering Building East, Columbia, MO 65211, USA.

E-mail address: [email protected] (C.M. Klein).

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2002.01.001

mail to: [email protected]

H. Jung, C.M. Klein / European Journal of Operational Research 165 (2005) 108–126 109

analyzing the optimal solutions, geometric programming (GP) techniques as well as derivative basedclassical first and second order conditions are used.

GP can be effectively applied to these models to derive the optimal solutions and it gives several

advantages over the classical method of calculus. The advantages will be illustrated in finding the optimal

solutions for these minimization models and in the managerial implications that can be derived for the

optimal policy through bounding and sensitivity analysis.

GP has been very popular in engineering design research since its inception in the early 1960s. Even

though GP is an excellent method to solve nonlinear problems, the use of GP in inventory models has been

relatively infrequent. Kochenberger [12] was the first to solve the basic EOQ model using GP. In Worralland Hall [18], GP techniques were utilized to solve an inventory model with multiple items subject to

multiple constraints. Cheng [4,5] applied GP to solve modified EOQ models and to perform sensitivity

analysis.

There have been numerous publications on EOQ models with fixed cost per unit. Recently, however,

several papers relaxed the assumption of the fixed cost per unit for the EOQ models. For example, Lee

[14,15] assumed the cost per unit as a function of the order quantity. This assumption means that the

production exhibits economies of scale when the order quantity increases. In Cheng [5], Jung and Klein [11],

Lee and Kim [16], and Lee et al. [17], the cost per unit was assumed to be a function of the demand per unittime which means that the decision maker employs better equipment and more resources for the production

of the product when the demand per unit time increases. Cheng [6] investigated a multiplicative term where

the cost per unit is a function of the demand per unit time and the process reliability. This indicates that the

cost per unit time is affected by both of the demand per unit time and the process reliability.

In Jung and Klein [11], the total cost minimization model and the profit maximization model were

compared and the differences in the optimal order quantity of these two models were investigated, analyzed

and discussed. In this paper, we compare three different cost minimization models and investigate the

differences in the optimal order quantities and in the optimal total costs. The difference in the optimal orderquantity (total cost) of these models indicates the quantity (total cost) that is over-ordered/under-ordered

(over-cost/under-cost) due to the error in estimating the cost function of the models. From the comparison,

we derive relationships between the optimal solutions by comparing our cost functions without computing

the optimal solutions. This means that we can determine optimal inventory policy by estimating the cost

functions. We also compare the EOQ model to the minimization models. Since the EOQ model has a fixed

unit cost and our models extend the EOQ model by making the unit cost dynamic, we can observe how the

EOQ model can be improved.

The remainder of this paper is organized as follows. First, we present assumptions and the three modelsfor total cost minimization. We optimally determine the order quantity for the each model and perform a

sensitivity analysis. In the next section, we obtain the optimality results using the first and second order

conditions. That is, the changes in the optimal order quantity and total cost according to varied parameters

are analyzed to see the effect on inventory policy. Then, we compare and contrast the three cost minimi-

zation models as well as the EOQ model to the total cost minimization models to gain managerial insights.

Finally, we make concluding remarks and comment on future research areas.

2. Assumptions

We define the following variables and parameters for our models.

D demand per unit time (units/unit time)

Q order quantity (units, decision variable)

C cost per unit (dollar/unit)

110 H. Jung, C.M. Klein / European Journal of Operational Research 165 (2005) 108–126

A setup cost (dollar/batch)

i inventory carrying cost rate (%/unit time)

b scaling constant for C for Model 1

b degree of economies of scale for Model 1

d scaling constant for C for Model 2

d quantity discount factor for Model 2

f scaling constant for C for Model 3

c degree of economies of scale for Model 3l quantity discount factor for Model 3

In this paper, the following three assumptions, which are frequently found in the EOQ literature (see e.g.,

Hillier and Lieberman [9]), are used: (1) replenishment is instantaneous; (2) no shortage is allowed; (3) the

order quantity is ordered in batch.

In addition, the following power function relations are assumed for our models. For Model 1, the cost

per unit is a power function of the demand per unit time displaying economies of scale. That is,

CðDÞ ¼ bD�b where b > 0 and b > 0. This indicates that when the demand per unit time increases, the costper unit decreases. Requiring b > 0 is an obvious condition since C and D must be nonnegative. Likewise,

b > 0 is required since the cost per unit is a decreasing power function of the demand per unit time.

For Model 2, the cost per unit is a power function of the order quantity displaying quantity discounts.

That is, CðQÞ ¼ dQ�d where d > 0 and 0 < d < 1. This implies that the cost per unit is a decreasing function

of the order quantity. d > 0 is an obvious condition since C and Q must be nonnegative. The condition

0 < d < 1 will be discussed with the details of the GP approach to find the dual feasibility condition. This

condition has been found in the literature [1,2,14,15]. We will focus on the cases where the discount factor is

relatively small.For Model 3, the cost per unit is a power function of the order quantity and the demand per unit time.

That is, CðD;QÞ ¼ fD�cQ�l where f > 0, c > 0, and 0 < l < 1. This means that we are considering

economies of scale and cost reduction due to better allocation of resources at the same time. As before,

f > 0 is an obvious condition since C, D, and Q must be nonnegative. We have c > 0 since the cost per unit

is a decreasing function of the demand per unit time. The condition 0 < l < 1 will be discussed with the

optimal solution procedure. Note that we also use a multiplicative term, which is frequently used in the

literature (see e.g., Cheng [4] and [6], and Lee and Kim [16]).

Given the above definitions and assumptions, we have the following mathematical formulation for thetotal cost per unit time ð¼ TCÞ:

MinTCðQÞ ¼ setup cost per unit time

þ variable cost per unit timeþ inventory holding cost per unit time

¼ AD=Qþ CðDÞDþ iCðDÞQ=2:

We denote Ci, Qi, and TCi as the cost per unit, the order quantity, and the total cost per unit time of Model

i, i ¼ 1; 2; 3, respectively. The asterisk sign means that the value is optimal. From the GP perspective, the

primal problems of the models are given as

MinTC1ðQ1Þ ¼ ADQ�11 þ bD1�b þ 0:5ibD�bQ1; where C1ðDÞ ¼ bD�b; ð1Þ

MinTC2ðQ2Þ ¼ ADQ�12 þ dQ�d

2 Dþ 0:5idQ1�d2 ; where C2ðQ2Þ ¼ dQ�d

2 ; ð2Þ

MinTC3ðQ3Þ ¼ ADQ�13 þ fD1�cQ�l

3 þ 0:5 if D�cQ1�l3 ; where C3ðD;Q3Þ ¼ fD�cQ�l

3 : ð3Þ


The objective of our models is to minimize the total cost per unit time with decision variable Q when the

cost per unit is a decreasing power function of the demand per unit time, the order quantity, and both of the

demand per unit time and the order quantity for Models 1, 2, and 3, respectively.

3. Optimal solutions

The objective function in Model 1 is an unconstrained posynomial with zero degree of difficulty. Hence,we can easily obtain the closed-form optimal solution by using the technique developed by Beightler and

Phillips [3] or Duffin et al. [8]. We note that the second term (i.e., bD1�b) of the objective function is a

constant as far as the decision variable Q is concerned. Hence, we will omit this term when we calculate the

optimal order quantity Q�. However, when the total cost is calculated, it is added back in. For Model 2 and

Model 3, the objective function is an unconstrained posynomial problem with one degree of difficulty. The

development of the solution procedure for this model is similar to the work by Cheng [4–6].

In the unconstrained posynomial GP problem, the dual variable, wi, provides the weight of ith term of

the primal problem over Q by the following equation:

U �i ¼ w�

i dðw�Þ: ð4Þ

dðw�Þ is the optimal dual objective function and the optimal weights, w�1, w

�2, and w�

3, represent proportions

of the setup cost (U �1 ), the variable cost (U

�2 ), and the inventory holding cost (U �

3 ) to the total cost per unit

time, respectively. We then have the following relations for our models.

Model 1: U �1 ¼ AD½Q�

1��1; U �

2 ¼ 0:5ibD�bQ�1; ð5Þ


2��1; U �

2 ¼ dD½Q�2��d; U �

3 ¼ 0:5id½Q�2�1�d

; ð6Þ


3��1; U �

2 ¼ fD1�c½Q�3��l; U �

3 ¼ 0:5ifD�c½Q�3�1�l

: ð7Þ
From the above equations, the corresponding primal solutions can be obtained (see e.g., Appendix A).
Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2AD1þbÞ=ðibÞ

p; ð8Þ

Q�2 ¼ AD=ðw�

1dðw�ÞÞ ¼ ½dD=ðw�2dðw�ÞÞ�1=d ¼ ½ðw�

3dðw�Þ=ð0:5idÞ�1=ð1�dÞ; ð9Þ

Q�3 ¼ AD=ðw�

1dðw�ÞÞ ¼ ½fD1�c=ðw�2dðw�ÞÞ�1=l ¼ ½ðw�

3dðw�Þ=ð0:5ifD�cÞ�1=ð1�lÞ: ð10Þ

4. Bounding and sensitivity analysis

4.1. Sensitivity analysis for Model 1

The sensitivity analysis employed here is used to evaluate the effects of percentage changes in input data

to the variations in the optimal solution. We will find the optimal solution using sensitivity analysis without

re-solving the problem given the change of data. We know that any change in the coefficients (A, D, i, b) andexponent (b) of the total cost function with zero degree of difficulty will not change the fraction which is

accounted for by each term of this function (i.e., ADQ�11 and 0:5ibD�bQ1). This is because the normality and

orthogonality equations are independent of these coefficients. Hence, the dual variables are not changed

even though such parameters are changed. This property makes it easy to find a new optimal solution for a


change in A, D, i, b, and b. The sensitivity analysis in this model is based on the work of Cheng [4–6].Therefore, if the change in a parameter is known (let Dx represent the change in x), then the new optimal

values can be computed directly from the previous optimal solution and the change in the given variable.

This is a very important concept when dealing with real systems in which parameters change quite often.

These new optimal values can be determined for changes in A, D, i, b, and b are as follows:

new dðw�Þ ¼ dðw�Þ½1þ DA=A�w�1 ;

new dðw�Þ ¼ dðw�Þ½1þ DD=D�w�1�bw�

2 ;

new dðw�Þ ¼ dðw�Þ½1þ Di=i�w�2 ;

new dðw�Þ ¼ dðw�Þ½1þ Db=b�w�2 ;

new dðw�Þ ¼ dðw�Þ½D�Db�w�2 ;

new Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Að1þ DA=AÞD1þb=ðibÞ

p;

new Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2A½Dð1þ DD=DÞ�1þb

=ðibÞq

;

new Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD1þb=ðið1þ Di=iÞbÞ

p;

new Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD1þb=ðibð1þ Db=bÞÞ

p;

new Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD1þbð1þDb=bÞ=ðibÞ

q;

ð11Þ

where w�1 ¼ w�

2 ¼ 0:5 and Dh=h is the percentage change in a primal function parameter h.

4.2. Bounding and sensitivity analysis for Model 2 and Model 3

According to the duality theorem of GP (see e.g., Beightler and Phillips [3] or Duffin et al. [8]) derived

from the arithmetic–geometric mean inequality, we have the following relationship:

TCðQÞP TC� ¼ dðw�ÞP dðwÞ: ð12Þ
This relationship indicates that any dual feasible solution provides a lower bound and its corresponding
primal solution gives us an upper bound. Also, this shows that the maximized dual objective function,dðw�Þ, is equal to the minimized primal objective function, TC�. This is very useful because the dual problem

is usually easier to solve. One advantage of (12) is that this can be used for obtaining quick estimates of the

optimal value of the objective function (see e.g., Beightler and Phillips [3]).

The sensitivity analysis applied here is based on the work of Dinkel and Kochenberger [7]. They

developed a formulation that gives the change in the optimal values of the dual variables resulting from the

changes in the primal objective function coefficients. Hence, we can investigate the effects of percentage

changes in the primal objective coefficients, A, i, D, d, and f with their method. The effects on the dual

variables are as follows.

Model 2:

dw1 ¼ ½1=ð1� dÞ�J�1R;

dw2 ¼ ½�ð2� dÞ=ð1� dÞ�J�1R;

dw3 ¼ J�1R;

ð13Þ


where

J�1 ¼ ½½1=ð1� dÞ�2=w�1 þ ½�ð2� dÞ=ð1� dÞ�2=w�

2 þ 1=w�3��1;

R ¼ ½1=ð1� dÞ�DðADÞ=ðADÞ þ ½�ð2� dÞ=ð1� dÞ�DðdDÞ=ðdDÞ þ Dð0:5idÞ=ð0:5idÞ:

Model 3:

dw1 ¼ ½1=ð1� lÞ�J�1R;

dw2 ¼ ½�ð2� dÞ=ð1� dÞ�J�1R;

dw3 ¼ J�1R;

ð14Þ

where

J�1 ¼ ½½1=ð1� lÞ�2=w�1 þ ½�ð2� lÞ=ð1� lÞ�2=w�

2 þ 1=w�3��1;

R ¼ ½1=ð1� lÞ�DðADÞ=ðADÞ þ ½�ð2� lÞ=ð1� lÞ�DðdfD1�cÞ=ðdfD1�cÞ þ Dð0:5ifD�cÞ=ð0:5ifD�cÞ:

Dh=h is the percentage change in the primal objective function coefficient h and J�1 is the inverse of the

Jacobian associated with the substituted dual problem and the Hessian of � log dðw3Þ. Once the effects are

computed, we readily have the new dual variable, wi ¼ w�i þ dwi, i ¼ 1, 2, 3. These new dual variables are

used to calculate the associated primal variable.

5. Model analysis

The previous analysis was done with GP because of the advantages GP gives us in determining new

optimal solutions when parameters change. New solutions are computed directly and no re-optimization is

necessary as would be the case if derivative based methods were used. Here however, the first and second

derivatives of the optimal order quantity, Q�, and the optimal total cost, TC� are used to investigate thechanges in Q� and TC� as different parameters are varied. By studying these changes, we can derive several

interesting economic implications and managerial insights.

5.1. Model 1

We analyze this model directly from the closed-form optimal solution obtained from the first and second

derivatives. We investigate the changes in Q�1 and TC�

1 according to the changes in the parameters A, i, b, D,and b.

First, Q�1 and TC�

1 are obtained from the derivative of Eq. (1).

Q�1 ¼


p; ð15Þ

TC�1 ¼ C1Dþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AiDC1

p¼ bD1�b þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AibD1�b

p: ð16Þ

From the first and second derivatives of (15) and (16) with respect to the parameters, A, i, b, D, and b, thefollowing results are obtained:

oQ�1

oA> 0;

o2Q�1

oA2< 0;

oTC�1

oA> 0;

o2TC�1

oA2< 0; ð17Þ

oQ�1

oi< 0;

o2Q�1

oi2> 0;

oTC�1

oi> 0;

o2TC�1

oi2< 0; ð18Þ


oQ�1

ob< 0;

o2Q�1

ob2> 0;

oTC�1

ob> 0;

o2TC�1

ob2< 0; ð19Þ

oQ�1

oD> 0;

o2Q�1

oD2< 0 if b < 1;

oTC�1

oD> 0 if b < 1;

o2TC�1

oD2< 0 if b < 1;

o2Q�1

oD2¼ 0 if b ¼ 1;

oTC�1

oD¼ 0 if b ¼ 1;

o2TC�1

oD2¼ 0 if b ¼ 1;

ð20Þ

o2Q�1

oD2> 0 if b > 1;

oTC�1

oD< 0 if b > 1;

o2TC�1

oD2> 0 if b > 1;

oQ�1

ob< 0 if D < 1;

o2Q�1

ob2¼ 0 if D ¼ 1;

oTC�1

ob> 0 if D < 1;

o2TC�1

ob2¼ 0 if D ¼ 1;

oQ�1

ob¼ 0 if D ¼ 1;

o2Q�1

ob2> 0 otherwise;

oTC�1

ob¼ 0 if D ¼ 1;

o2TC�1

ob2> 0 otherwise;

oQ�1

ob> 0 if D > 1;

oTC�1

ob< 0 if D > 1:

ð21Þ

From the above analysis, it can be seen that any increase (decrease) in A or D, or decrease (increase) in i or bresults in a larger (smaller) optimal order quantity. These results are consistent with the fact that

Q�1 ¼


pincreases as A or D increases, and Q�

1 decreases as i or b increases. Since Q�1 is a

concave and increasing function with respect to the setup cost A, the relationship between the setup cost and

the optimal order quantity indicates that increases in A lead to higher inventory cost and therefore, at the

same time, to higher Q�1. If the inventory holding cost, i, which is part of total cost, increases, total cost

increases. In this case, a decision maker will reduce Q�1 to save the expense of storing inventory. The fact

that when we increase the scaling constant for C1, Q�1 decreases represents that if the cost per unit is in-

creased by a scaling constant, Q�1 will be decreased because of economies of scale. We can also observe that

any increase (decrease) in A, i, or b results in a larger (smaller) optimal total cost. These results are con-

sistent with the fact that TC� ¼ AD½Q�1��1 þ bD1�b þ 0:5ibD�bQ�

1 increases when A, i, or b, which is the partof total cost, increases. The change in the optimal order quantity against the change in b depends on the

demand per unit time. Also, the change in the optimal total cost against the change in bðDÞ depends onDðbÞ.

5.2. Model 2

In this model, we use the implicit function theorem (see e.g., Hildebrand [10]) with the first and second

order conditions since the closed-form solutions cannot be obtained. When the parameters A, i, d, D, and dvary, the changes in Q�

2 and TC�2 are investigated. From the total cost function (2), the first and second order

conditions for the global optimality are

oTC2

oQ2

¼ �ADQ�22 � ddDQ�1�d

2 þ 1

2ð1� dÞidQ�d

2 ¼ 0; ð22Þ

o2TC2

oQ22

¼ 2ADQ�32 þ dð1þ dÞdDQ�2�d

2 � 1

2dð1� dÞidQ�1�d

2 > 0: ð23Þ


First, we investigate the change of the optimal order quantity as the setup cost A varies. From the first order

condition, (22), using the implicit function theorem, we have

�D½Q�2��2 þ 2AD½Q�

2��3 oQ�

2

oAþ dð1þ dÞdD½Q�

2��2�d oQ�

2

oA� 1

2dð1� dÞid½Q�

2��1�d oQ�

2

oA¼ 0: ð24Þ

Eq. (24) can be expressed by

oQ�2

oA¼ D½Q�

2��2

2AD½Q�2��3 þ dð1þ dÞdD½Q�

2��2�d � 1


2��1�d : ð25Þ

By applying (23) to (25), we obtain the following result:

oQ�2

oA> 0: ð26Þ

Hence, the optimal order quantity is an increasing function with respect to the setup cost A. The secondderivative of (25) is

o2Q�2

oA2¼ X

½2AD½Q�2��3 þ dð1þ dÞdD½Q�

2��2�d � 1


2��1�d�2

; ð27Þ

where

X ¼ �2D½Q�2��3 oQ�

2

oA2AD½Q�

2��3

�þ dð1þ dÞdD½Q�

2��2�d � 1


2��1�d

�� D½Q�

2��2

2D½Q�2��3

�

� 6AD½Q�2��4

�þ dð1þ dÞð2þ dÞdD½Q�

2��3�d � 1

2dð1� dÞð1þ dÞid½Q�

2��2�d

�oQ�

2

oA

�

¼ �2D½Q�2��3 oQ�

2

oAddD½Q�

2��2�d dð1� dÞ

2

�þ 1

2ð1� dÞid½Q�

2��1�d ðd� 2Þðd� 1Þ

2

�:

From the fact that o2TC2

oQ22

> 0 andoQ�

2

oA > 0, we can obtain

o2Q�2

oA2< 0: ð28Þ

Therefore, the optimal order quantity is a concave function with respect to the setup cost A. For the totalcost, we can apply the above procedure and obtain

oTC�2

oA> 0;

o2TC�2

oA2< 0: ð29Þ

The following results for i, d, D, and d are obtained by similar procedures shown for A:

oQ�2

oi< 0;

o2Q�2

oi2> 0;

oTC�2

oi> 0;

o2TC�2

oi2< 0; ð30Þ

oQ�2

od< 0;

o2Q�2

od2> 0;

oTC�2

od> 0;

o2TC�2

od2< 0; ð31Þ

oQ�2

oD> 0; ð32Þ

oQ�2

od> 0: ð33Þ


The above results indicate that any increase (decrease) in A or decrease (increase) in i or d results in a larger

(smaller) optimal order quantity, and any increase (decrease) in A, i, or d results in a larger (smaller)

optimal total cost. These results of A, i, and d are identical to the results of A, i, and b shown in Model 1,

respectively. Therefore, the managerial insights for A, i, and the scaling constant are similar to those given

for Model 1. The results of D and d show that the optimal order quantity is an increasing function with

respect to D and d.

5.3. Model 3

As in Model 2, we use the implicit function theorem with the first and second order conditions for this

model. When parameters A, i, f , D, l, and c vary, the changes in the optimal order quantity are investi-

gated. By using the first and second derivatives of (3), we obtain the following results. The procedure is

similar to those used in Model 2.

oQ�3

oA> 0;

o2Q�3

oA2< 0;

oTC�3

oA> 0;

o2TC�3

oA2< 0; ð34Þ

oQ�3

oi< 0;

o2Q�3

oi2> 0;

oTC�3

oi> 0;

o2TC�3

oi2< 0; ð35Þ

oQ�3

of< 0;

o2Q�3

of 2> 0;

oTC�3

of> 0;

o2TC�3

of 2< 0; ð36Þ

oQ�3

oD> 0; ð37Þ

oQ�3

ol> 0; ð38Þ

oQ�3

oc> 0: ð39Þ

We can see that the results of this model are identical to the results of Model 1 and Model 2 for A, i, and the

scaling constant. Therefore, the managerial insights of A, i, and the scaling constant are the same to those

shown in Model 1.

5.4. Effects of changes in demands of Model 1 and Model 3

For Model 1 and Model 3, we investigate the effects of the changes in Q� according to the changes in the

demand, D, with the different values of degree of economies of scale, b and c. That is, we develop man-

agerial insights by comparing the results of Model 1 and 3 with those of the EOQ model. In the classical

EOQ model, the optimal order quantity is always an increasing and concave function with respect to the

demand since oQ�

oD ¼ 12D�1=2

ffiffiffiffi2AiC

q> 0 and o2Q�

oD2 ¼ � 14D�3=2

ffiffiffiffi2AiC

q< 0 where Q� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD=ðiCÞ

p. However, we

know that for Model 1, the optimal order quantity can be a concave or convex function with respect to the

demand which depends on the degree of economies of scale, b, from Eq. (20). That is, Q�1 is a concave

function with respect to D if b < 1 and is a convex function with respect to D if b > 1. Note that the degree

of economies of scale, b, which is the cost elasticity with respect to the demand per unit time, represents the

relative change in the unit cost with respect to the corresponding relative change in the demand (i.e.,

Fig. 1. Effects of changes in the demand for Model 3: (a) c ¼ 0:2 < 1, (b) c ¼ 2 > 1.


b ¼ oC1=oDC1=D

�� ). In other words, Q�1 is a concave (convex) function with respect to D if the percent changes in

the unit cost are less (greater) than the percent changes in the demand for Model 1.

If b < 1, the unit cost decreases at a diminishing rate when the demand increases and therefore, the

optimal order quantity will be increased at a decreasing rate which indicates a concave function with respect

to the demand. If b > 1, the unit cost decreases at an increasing rate and the variable cost

(¼ C1ðDÞD ¼ bD1�b) will approach zero when the demand increases. Therefore, the optimal order quantitywill approach infinity since we produce more due to the lower variable cost. This indicates that the optimal

order quantity is a convex function with respect to the demand.

For Model 3, we perform a computational analysis to observe the effects of the changes in Q�3 with

respect to the changes in D with different values of degree of economies of scale, c. The basic parameter

values are A ¼ 50, i ¼ 0:1, b ¼ 5, and l ¼ 0:01. The value of the demand to analyze with optimal solutions

is allowed to vary from 10 to 10,000 in steps of 500. The value of c is 0.2 for c < 1 and 2 for c > 1. Note that

we can also obtain the same results for the other values of c (< 1 or > 1).

Fig. 1 indicates that for Model 3, Q�3 is a concave function with respect to D if c < 1 and is a convex

function with respect to D if c > 1. This result is identical to those shown for Model 1 and therefore, the

analysis is similar to the above analysis. The reason for the different results between the EOQ model and

our models, Model 1 and Model 3, is that the unit costs for our models are decreasing functions of the

demand and these cost functions are affected by degree of economies of scale, b and c whereas the EOQ

model has a constant unit cost.

6. Comparative analysis

The comparative analysis here is to study the relationship between the optimal solutions of the three

minimization models. We focus on the comparative analysis of the optimal order quantities and total costs

of models where certain conditions in the cost per unit are given. This analysis is reasonable because we

consider the different models according to the different shape of the cost function (see Ladany and Sternlieb

[13]). Also, the EOQ model, which has a fixed unit cost, is compared with the minimization models to

observe how the EOQ model can be improved and used to more realistically model real scenarios in terms


of the unit cost which generally depends on the demand, the order quantity, and the demand and orderquantity. We denote Model E as the EOQ model.

For this analysis we investigate the error that can occur in quantity and total cost when the wrong model

is used to determine these values. That is we compute jQ�i � Q�

j j (jTC�i � TC�

j j), for all i and j by comparing

Q��s (TC��s) of Models E and 1, Models E and 2, Models E and 3, Models 1 and 2, Models 1 and 3, and

Models 2 and 3. The error in Q� and TC� shows the situation where Model i is used when Model j shouldhave been used for all i and j.

To compare the models, we need to know the form of each cost function of the minimization models. To

estimate the cost function for the models, we first assume that we have previous data for C, D, and Q so thatwe can draw the cost function forms of Model 1, 2, and 3 if b, d, l, and c are obtained from C, D, and Q.For the cost function of Model 1, C1 ¼ bD�b

1 , C2 ¼ bD�b2 ; . . . ;Cn ¼ bD�b

n , we can estimate b by taking the

logarithm to transform the cost function into a linear model and then apply simple linear regression.

Similar procedures can be performed to estimate d, l, and c for Models 2 and 3.

Now, suppose for a problem we have determined all cost function forms. Then, based on the analysis

previously derived, we can make the following three assumptions.

Assumption 1. If the cost per unit is a decreasing power function of the demand per unit time, but not adecreasing power function of the order quantity, then we should use Model 1 which means that the decision

maker employs better equipment and more resources for the production of this product.

Assumption 2. If the cost per unit is a decreasing power function of the order quantity, but not a decreasing

power function of the demand per unit time, then select Model 2 which means that production exhibits

economies of scale.

Assumption 3. If the cost per unit is a decreasing power function of the demand per unit time and the orderquantity, then choose Model 3 which means that we assume both of economies of scale and cost reduction

due to better allocation of resources. Also, we note that the multiplicative effect for Model 3 is assumed.

Furthermore, we assume that the parameters A, i, and D are identical for these models. Under these

assumptions, we can select appropriate cost functions under certain conditions after investigating the error

in Q� and TC�.

From the first derivative of TC1 and TC2 with bD�b ¼ C1 and d½Q�2��d ¼ C2, we have

i2D

¼ Ab�1Db½Q�1��2 ¼ AC�1

1 ½Q�1��2 ð40Þ

and

i2D

¼ Ad�1½Q�2�d�2 þ d½Q�

2��1

1� d¼ AC�1

2 ½Q�2��2 þ d½Q�

2��1

1� d: ð41Þ

By manipulating (40) and (41), we have the following relationship:

ð1� dÞAC�11 ½Q�

1��2 ¼ AC�1

2 ½Q�2��2 þ d½Q�

2��1: ð42Þ

Eq. (42) with 0 < d < 1 gives us the following inequality relationship:

AC�11 ½Q�

1��2

> AC�12 ½Q�

2��2: ð43Þ

By rewriting (43), we can obtain the error in the optimal order quantity under certain conditions.

C1=C2 < ½Q�2=Q

�1�2: ð44Þ


This relationship indicates that if C1 PC2, then Q�1 < Q�

2 since Q�2=Q

�1 must be greater than 1. If C1 < C2,

then all cases Q�1 < Q�

2, Q�1 > Q�

2, and Q�1 ¼ Q�

2 can result.

Similar to the above analysis, we can obtain the relationships for Model 1 and Model 3, and for Model 2

and Model 3 under three assumptions, d ¼ l, d > l, and d < l. These results are summarized in the fol-

lowing properties.

Property 1a. If C1 PC2, then Q�1 < Q�

2.

Property 1b. If C1 < C2, then Q�1 < Q�

2, Q�1 > Q�

2, or Q�1 ¼ Q�

2.

Property 2a. If C1 PC3, then Q�1 < Q�

3.

Property 2b. If C1 < C3, then Q�1 < Q�

3, Q�1 > Q�

3, or Q�1 ¼ Q�

3.

Property 3a. If d ¼ l and C2 > C3, then Q�2 < Q�

3.

Property 3b. If d ¼ l and C2 < C3, then Q�2 > Q�

3.

Property 3c. If d ¼ l and C2 ¼ C3, then Q�2 ¼ Q�

3.

Property 4a. If d > l and C2 6C3, then Q�2 > Q�

3.

Property 4b. If d > l and C2 > C3, then Q�2 > Q�

3, Q�2 < Q�

3, or Q�2 ¼ Q�

3.

Property 5a. If d < l and C2 PC3, then Q�2 < Q�

3.

Property 5b. If d < l and C2 < C3, then Q�2 > Q�

3, Q�2 < Q�

3, or Q�2 ¼ Q�

3.

Properties 1a and 2a are consistent with the fact that Q�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD=ðiC1Þ

pdecreases as C1 increases. The

converse of the fact does not apply to Properties 1b and 2b in general because of the complex interaction

among the terms involving the setup cost, the variable cost, and the inventory holding cost.

The difference in the optimal order quantity implies the quantity that is over-ordered/under-ordered.This means that we used Model 1 (Model 2) when Model 2 (Model 1) should have been used for Model 1

and Model 2. Therefore, Property 1a shows that we can determine optimal inventory policy for both

models by estimating C1 and C2 without computing Q�1 and Q�

2 if C1 PC2. Note that we can estimate the

cost functions for both models by estimating the parameters with the linear regression since we assume that

we have previous data of the unit costs, the order quantity, and the demand for both models.

Assumption d ¼ l means that the elasticity with respect to the order quantity of Model 2 is identical to

that of Model 3. From C2 ¼ d½Q�2��d

and C3 ¼ fD�c½Q�3��l, we can see that Property 3 is consistent with the

same elasticity for both models. We note that d and fD�c are constant, d and l are between zero and one.Assumption d > l (d < l) means that the elasticity with respect to the order quantity of Model 2 is greater

(less) than that of Model 3. Property 4a (Property 5a) is consistent with C2 ¼ d½Q�2��d

and C3 ¼ fD�c½Q�3��l

where d > l (d < l).The comparative analysis of the optimal total costs of the models by using the first and second order

conditions is intractable since the cost per unit as well as the entire objective function are complex func-

tional forms. Hence, we perform numerical analysis to observe the relationships between the optimal total

costs of the models. The basic parameter values which are randomly chosen are D: 0–100, A: 0–100, i: 0–0.5,b: 0–10, d: 0–10, f : 0–10, b: 0–0.5, d: 0–0.1, l: 0–0.1, and c : 0–0.5. Figs. 2–4 show the results for

Fig. 2. Comparative analysis of total costs of Model 1 and Model 2: (a) C1 > C2, (b) C1 < C2.



relationships between the total costs of Models 1 and 2, Models 1 and 3, and Models 2 and 3 after 100iterations with the conditions on the cost per unit given. These results indicate that TC�

1 > TC�2 if C1 > C2,

and TC�1 < TC�

2 if C1 < C2 for Model 1 and Model 2, TC�1 > TC�

3 if C1 > C3, and TC�1 < TC�

3 if C1 < C3 for

Model 1 and Model 3, and TC�2 > TC�

3 if C2 > C3, and TC�2 < TC�

3 if C2 < C3 for Model 2 and Model 3. This

means that we can directly have information about the relationship between total costs of both models by

estimating the unit costs without computing the optimal total costs, and the optimal policy can be made

from the information.

Now, we compare our models with the classic EOQ model to observe the relationship between our

models and the EOQ model under certain conditions. The EOQ model is the basic inventory model and hasconstant unit cost. By comparing the EOQ model to our models, we can observe how the EOQ model can



be improved by using unit cost which depends on the demand, the order quantity, and the demand and

order quantity.

By manipulating the first derivative of TCE and TC1 with bD�b ¼ C1, we can obtain relationships between

Q�E and Q�

1. This relationship can then be used to obtain relationships between TC�E and TC�

1 . We also obtain

the relationships for the Model Es and 2, and the Models E and 3 with similar procedures as before. The

results are summarized in the following property.

Property 6a. If CE > C1, then Q�E < Q�

1 and TC�E > TC�

1 .

Property 6b. If CE < C1, then Q�E > Q�

1 and TC�E < TC�

1 .

Property 6c. If CE ¼ C1, then Q�E ¼ Q�

1 and TC�E ¼ TC�

1 .

Property 7a. If CE PC2, then Q�E < Q�

2.

Property 7b. If CE < C2, then Q�E < Q�

2, Q�E > Q�

2, or Q�E ¼ Q�

2.

Property 8a. If CE PC3, then Q�E < Q�

3.

Property 8b. If CE < C3, then Q�E < Q�

3, Q�E > Q�

3, or Q�E ¼ Q�

3.

Property 6 is consistent with the fact that Q�E ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD=ðiCEÞ

pdecreases (increases) as CE increases (de-

creases). Property 6, 7.a, and 8.a indicate that optimal inventory policies for EOQmodel and our models canbe determined by estimating the cost functions without computing the optimal order quantities (or TC�

E and

TC�1). Fig. 5 shows that TC

�E > TC�

2 if CE > C2, and TC�E < TC�

2 if CE < C2. We also observe that TC�E > TC�

3 if

CE > C3, and TC�E < TC�

3 if CE < C3 from Fig. 6. As the parameter values, we randomly choose CE: 0–1 (other

parameter values are the same as in the previous analysis). These figures show that we can determine optimal

inventory policy by estimating the cost functions without computing the optimal total costs.

The above properties and figures indicate that if the cost functions can be estimated, the relationship

between the optimal order quantities as well as the optimal total costs can be analyzed. Since we can

Fig. 5. Comparative analysis of total costs of EOQ model and Model 2: (a) CE > C2, (b) CE < C2.

Fig. 6. Comparative analysis of total costs of EOQ model and Model 3: (a) CE > C3, (b) CE < C3.


estimate the cost function from the previous data using linear regression, we can determine an optimalinventory policy by estimating the cost function without computing optimal solutions if the difference in the

optimal solutions is found. For example, if we know Q�i > Q�

j (i.e., Q�i is over-ordered or Q�

j is under-or-

dered) from Ci < Cj, we can increase Ci (or decrease Cj) to reduce the error in the optimal order quantity.

This adjustment will give us an optimal policy for our models.

7. Conclusion

In this paper, we have developed and analyzed three EOQ based inventory models under total cost

minimization via geometric programming (GP) techniques. By using GP, we determined the order quantity


for these models with the cost per unit as a function of the demand per unit time, the order quantity, andboth the demand per unit time and the order quantity. For each of these models, the inventory problem was

formulated as a posynomial problem. Because a posynomial problem guarantees global optimality, we

obtain the optimal solution. Also, we derived the optimal solutions from the dual GP problem. The sen-

sitivity analysis for these three models allows us to determine a new optimal solution, without resolving the

problem, for a given change in the data.

The change in the optimal order quantity and total cost according to varied parameters was analyzed to

see the effect on inventory policy. Comparisons between these models showed the relationship between the

optimal solutions of different forms for cost functions where certain conditions in the cost per unit aregiven. We also compared our minimization models to the basic EOQ model and examined how the EOQ

model can be improved by using these more realistic cost functions.

Even though our extended EOQ models are easy to solve using the corresponding GP approaches, our

models can give the basis for complicated functions such as the problem with several terms in the primal

objective function and constraints (it will be difficult to solve the problem if the degree of difficulty is greater

than 1).

These three models we have investigated may provide the basis for numerous further research areas.

For example, profit maximization models with price function related to the demand per unit time, theorder quantity, and both the demand per unit time and the order quantity can be combined with our

three cost functions. Then, we can compare our minimization models with the maximization models to

observe managerial insights. That is, the minimization model with cost function related to demand per

unit time (order quantity, or both of order quantity and demand per unit time) can be compared with the

profit maximization model with cost function related to demand per unit time (order quantity, or both of

order quantity and demand per unit time), respectively. In addition, these models could be a basis for

inventory models integrated with quality, setup cost, and process improvement issues (see e.g., Cheng

[4,6]).

Appendix A

For Model 1, the dual problem from the primal problem without the second term (i.e., bD1�b) is

max dðwÞ ¼ ½AD=w1�w1 ½0:5ibD�b=w2�w2 ðA:1Þs:t: w1 þ w2 ¼ 1;

� w1 þ w2 ¼ 0; ðA:2Þw1;w2 > 0:

In (A.2), the first constraint is the normality condition and the second is the orthogonality condition. We

note that all weights are positive in GP. The degree of difficulty is zero so we can easily obtain the optimal

weight directly, i.e., w�1 ¼ w�

2 ¼ 0:5. By using Eq. (4)

U �1 ¼ AD½Q�

1��1; U �

2 ¼ 0:5ibD�bQ�1: ðA:3Þ

From the fact that the primal objective function is equal to the dual objective function at optimality, weobtain

U �1 =w

�1 ¼ U �

2 =w�2 ¼ AD½Q�

1��1=0:5 ¼ 0:5ibD�bQ�

1=0:5 ¼ dðw�Þ: ðA:4Þ

The optimal order quantity, Eq. (8), can be calculated from (A.4).


For Model 2, we use the following dual problem to solve our problem.

max dðwÞ ¼ ½AD=w1�w1 ½dD=w2�w2 ½0:5id=w3�w3 ðA:5Þs:t: w1 þ w2 þ w3 ¼ 1;

� w1 � dw2 þ ð1� dÞw3 ¼ 0; ðA:6Þw1;w2;w3 > 0:

There are not enough equations to determine the optimal weights since we have two linear equations

and three variables (i.e., under-determined). However, we can express the weights, w1 and w2, in terms of

w3.

w1 ¼ ð�dþ w3Þ=ð1� dÞ;w2 ¼ ð1� ð2� dÞw3Þ=ð1� dÞ: ðA:7Þ

The normality condition in conjunction with the dual variables being positive yields 0 < w1, w2, w3 < 1. By

substituting (A.7) into 0 < w1, w2, w3 < 1, we have the following conditions:

d < w3 < 1;

d=ð2� dÞ < w3 < 1=ð2� dÞ:ðA:8Þ

From (A.7), if dP 1, then w3 6 d and w3 P 1=ð2� dÞ to satisfy the positivity condition of the dual variables.

But, this does not coincide with (A.8). Hence, we know that the dual problem is infeasible if dP 1, w3 6 d,and w3 P 1=ð2� dÞ. Therefore, the positivity condition is

0 < d < 1; w3 > d and w3 < 1=ð2� dÞ: ðA:9Þ

After combining (A.8) and (A.9), we can obtain the following dual feasibility condition.

Lemma 1 (Dual feasibility condition). If 0 < d < 1, 0 < w1 < ð1� dÞ=ð2� dÞ, 0 < w2 < 1, andd < w3 < 1=ð2� dÞ, then the dual problem is feasible.

Since the model has one degree of difficulty, we can solve this with the following substituted dualfunction, dðw3Þ, to find an optimal solution. By substituting (A.7) into the dual objective function (A.5),

where w3 is the only variable, the substituted dual problem is formed.

max dðw3Þ ¼ ½AD=½ð�dþ w3Þ=ð1� dÞ��½ð�dþw3Þ=ð1�dÞ�

� ½dD=½ð1� ð2� dÞw3Þ=ð1� dÞ��½ð1�ð2�dÞw3Þ=ð1�dÞ�½0:5id=w3�w3 : ðA:10Þ

By taking the logarithm of the objective function of the substituted dual problem, we obtain the followingconcave function in one variable. Because constraints of the dual problem are linear, these constraints form

a convex region. Therefore, the dual problem is to find a stationary point for the concave objective function

subject to the set of convex constraints. This function has a guaranteed global optimal solution. The proof

of the concavity of log dðw3Þ is shown by Duffin et al. [8].

max logdðw3Þ ¼ �½ð�dþ w3Þ=ð1� dÞ� log½ð�dþ w3Þ=½ADð1� dÞ�� ½½1� ð2� dÞw3�=ð1� dÞ� log½½1� ð2� dÞw3�=½dDð1� dÞ�� w3 log½w3=ð0:5idÞ�:

ðA:11Þ


Setting the first derivative to zero

o logdðw3Þow3

¼ �½ð�dþ w3Þ=ð1� dÞ�½ADð1� dÞ=ð�dþ w3Þ�½1=½ADð1� dÞ��

� ½1=ð1� dÞ� log½ð�dþ w3Þ=½ADð1� dÞ�� ½½1� ð2� dÞw3�=ð1� dÞ�� ½dDð1� dÞ=½1� ð2� dÞw3��½�ð2� dÞ=½dDð1� dÞ�� ½�ð2� dÞ=ð1� dÞ�� log½½1� ð2� dÞw3�=½dDð1� dÞ�� w3½0:5id=w3�½1=ð0:5idÞ� � log½w3=ð0:5idÞ�

¼ log½1� ð2� dÞw3�ð2�dÞ=ð1�dÞ

w3½�dþ w3�1=ð1�dÞ

" #þ log

½ADð1� dÞ�1=ð1�dÞ½0:5id�½dDð1� dÞ�ð2�dÞ=ð1�dÞ

" #¼ 0: ðA:12Þ

Eq. (A.12) can be easily maximized by any line search technique. After the optimal weight w�3 is obtained

from (A.12), w�1 and w�

2 can be calculated from (A.7). After applying GP to solve the minimization primal

problem expressed in (2), we can obtain the optimal order quantity from the following relationship:

U �1 =w

�1 ¼ U �

2 =w�2 ¼ U �

3 =w�3 ¼ AD½Q�

2��1=w�

1 ¼ dD½Q�2��d=w�

2 ¼ 0:5id½Q�2�1�d

=w�3: ðA:13Þ

The dual problem for Model 3 is

max dðwÞ ¼ ½AD=w1�w1 ½fD1�c=w2 �w2 ½0:5ifD�c=w3�w3 ðA:14Þs:t: w1 þ w2 þ w3 ¼ 1;

� w1 � lw2 þ ð1� lÞw3 ¼ 0; ðA:15Þw1;w2;w3 > 0:

By solving for the first and second weights of the dual problem in terms of the third, we obtain

w1 ¼ ð�lþ w3Þ=ð1� lÞ;w2 ¼ ð1� ð2� lÞw3Þ=ð1� lÞ:

ðA:16Þ

By using the normality condition and (A.16) with the method shown in Model 2, we obtain the following

lemma.

Lemma 2 (Dual feasibility condition). If 0 < l < 1, 0 < w1 < ð1� lÞ=ð2� lÞ, 0 < w2 < 1, andl < w3 < 1=ð2� lÞ, then the dual problem is feasible.

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Optimal inventory policies for an economic order quantity model with decreasing cost functions

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