Optimal inventory policies for profit maximizing EOQ models under various cost functions

European Journal of Operational Research 174 (2006) 689–705

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Optimal inventory policies for profit maximizing EOQmodels under various cost functions

Hoon Jung a, Cerry M. Klein b,*

a Postal Technology Research Center, Electronics and Telecommunications Research Institute, Deajeon 305-350, Koreab Department of Industrial and Manufacturing Systems Engineering, The University of Missouri, Columbia, MO 65211, USA

Received 19 August 2002; accepted 16 June 2004Available online 13 May 2005

Abstract

In this paper, we establish and analyze three EOQ based inventory models under profit maximization via geometricprogramming (GP) techniques. Through GP, we find optimal order quantity and price for each of these models con-sidering production (lot sizing) as well as marketing (pricing) decisions. We also investigate the effects on the changes inthe optimal solutions when different parameters are changed. In addition, a comparative analysis between the profitmaximization models is conducted. By investigating the error in the optimal price, order quantity, and profit of thesemodels, several interesting economic implications and insights can be observed.� 2005 Elsevier B.V. All rights reserved.

Keywords: Inventory; Geometric programming; EOQ

1. Introduction

This paper establishes and analyzes three inventory models under profit maximization which extends theclassical economic order quantity (EOQ) model. The extensions to the EOQ model by these models are; (1)The cost per unit exhibits some type of economies of scale. In the EOQ model, the cost per unit is fixed. Inthe proposed models, the cost per unit is a power function of the demand per unit time (Model 1), a powerfunction of the order quantity (Model 2), or a power function of both of the demand per unit time and theorder quantity (Model 3). (2) For all models, we consider the demand per unit time as a power function ofthe price per unit whereas the demand per unit time is independent of the price per unit in the EOQ model.

0377-2217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2004.06.041

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

mailto:[email protected]

690 H. Jung, C.M. Klein / European Journal of Operational Research 174 (2006) 689–705

That is, we assume to have control over the demand through pricing. (3) In deriving and analyzing theoptimal solutions, geometric programming (GP) techniques as well as derivative based classical first orderconditions are used.

There are two main reasons for developing these models. The first is that geometric programming (GP)can be effectively applied to these models to derive the optimal solutions. Note that we cannot obtain opti-mal solutions of our models from the classical method of calculus. The second motivating factor is the com-parison of the models and the relationships between the optimal solutions that can be determined undercertain conditions. By comparing the models, we can derive managerial insights and based on these insightswe can select an optimal policy from different forms of the cost functions.

GP has been very popular in engineering design research since its inception in the early 1960s. Eventhough GP is an excellent method to solve nonlinear problems, the use of GP in inventory models has beenrelatively infrequent. Kochenberger [10] was the first to solve the basic EOQ model using GP. In Worralland Hall [16], GP techniques were utilized to solve an inventory model with multiple items subject to multi-ple constraints. Cheng [3,4] 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, sev-eral papers relaxed the assumption of the fixed cost per unit for the EOQ models. For example, Lee [12,13]assumed the cost per unit as a function of the order quantity. This assumption means that the productionexhibits economies of scale when the order quantity increases. In Cheng [4], Jung and Klein [8], and Lee andKim [14], the cost per unit was assumed to be a function of the demand per unit time which means that thedecision maker employs better equipment and more resources for the production of this product when thedemand per unit time increases. Also, a multiplicative term was frequently observed in the literature. Cheng[5] investigated a multiplicative term where the cost per unit is a function of the demand per unit time andthe process reliability. This indicates that the cost per unit time is affected by both of the demand per unittime and the process reliability. Several publications have relaxed the assumption of the fixed demand perunit time for the EOQ model. For example, Kim and Lee [9], Lee [12,13], and Lee et al. [15] relaxed thefixed demand per unit time and assumed the demand per unit time is a function of the price per unit. Thatis, the demand per unit time is assumed to be dependent upon the price per unit.

In Jung and Klein [8], the total cost minimization model and the profit maximization model were com-pared and the differences in the optimal order quantity of these two models were investigated. In this paper,we compare the three different profit maximization models (Models 1 and 2, Models 1 and 3, and Models 2and 3). By investigating the difference in the optimal order quantities, prices, and profits, we can deriveinteresting managerial implications. For example, the difference in the optimal order quantities of the dif-ferent models indicates the amount by which one can over-order/under-order due to an error in estimatingthe cost function. From the comparison, we provide relationships between the optimal solutions by com-paring our cost functions without computing the optimal solutions. This means that we can determine theoptimal inventory policy by estimating the cost functions and not solving for optimality. This is advanta-geous since the data used is often approximate and the correct policy is more important.

The remainder of this paper is organized as follows. First, we present assumptions and the three modelsfor profit maximization. We optimally determine the price and the order quantity for the each model. In thenext section, we present computational results investigating the changes in the optimal order quantity,price, and profit according to varied parameters. We then compare and contrast the three profit maximi-zation models to gain managerial insights. Finally, we make concluding remarks and comment on futureresearch areas.

2. Assumptions

We define the following variables and parameters for our models.

H. Jung, C.M. Klein / European Journal of Operational Research 174 (2006) 689–705 691

P price per unit (dollar/unit, decision variable)Q order quantity (units, decision variable)D demand per unit time (units/unit time)C cost per unit (dollar/unit)A ordering cost (dollar/batch)i inventory carrying cost rate (%/unit time)a scaling constant for D

a price elasticity of demandb scaling constant for C for Model 1b degree of economies of scale for Model 1d scaling constant for C for Model 2d quantity discount factor for Model 2f scaling constant for C for Model 3c 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.,[7]), 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.

(4) For all Models, D(P) = aP�a where a > 0 and a > 1.(5) For Model 1, C(D) = bD�b where b > 0 and 0 < b < 1.(6) For Model 2, C(Q) = dQ�d where d > 0, a > d, and 0 < d < 1.(7) For Model 3, C(D, Q) = fD�cQ�l where f > 0 and 0 < c + l < 1.

Assumption (4) indicates that the demand per unit time is a decreasing power function of the price perunit. That is, when the price per unit increases, the demand per unit time decreases. Assumption (5)–(7)imply that the cost per unit is a power function of the demand per unit time displaying economies of scale,a power function of the order quantity displaying quantity discounts, and a power function of the orderquantity and demand per unit time, respectively. That is, the cost per unit decreases as the demand per unittime, the order quantity, or both of the demand per unit time and the order quantity increases.

The price elasticity of the demand per unit time, a, represents the relative change in the demand with

respect to the corresponding relative change in the price i:e:; a ¼ oD=oPD=P

�� . The degree of economies of

scale, b (or c), which is the cost elasticity with respect to the demand per unit time, represents the relative

change in the cost with respect to the corresponding relative change in the demand i:e:; bðor cÞ ¼ oC=oDC=D

�� .

The cost elasticity with respect to the order quantity, d (or l), represents the relative change in the cost with

respect to the corresponding relative change in the order quantity i:e:; dðor lÞ ¼ oC=oQC=Q

�� . Requiring a, b, d,

and f > 0 is an obvious condition since D, P, C, and Q must be nonnegative. The condition a > 1 ofassumption (4) will be discussed later with the GP approach to find the dual feasibility condition. This con-dition is a mild one since a < 1 implies that the price per unit has little impact on the demand per unit time.The condition 0 < b < 1 of assumption (5) will be assumed since the variable cost per unit time


(= C(D)D = bD1�b) will approach zero as we produce more if b P 1. This condition shows that the costper unit decreases at a diminishing rate when the demand per unit time increases. The condition 0 < d < 1of assumption (6) will be assumed since any d > 1 represents too much discounting and would be unrealisticas has been demonstrated in the literature [1,2,12,13]. The condition a > d will be proved through the dualfeasibility condition. This indicates that the price elasticity with respect to the demand is greater than thecost elasticity with respect to the order quantity. The conditions of assumption (7), 0 < c + l < 1, will beproved through the dual feasibility condition as well. Note that the conditions, 0 < c < 1 and 0 < l < 1, areshown for Models 1 and 2, respectively.

Under the above definitions and assumptions, profit per cycle is the revenue per cycle minus the sum ofthe ordering cost per cycle, the variable cost per cycle, and the inventory holding cost per cycle. The profitper cycle is mathematically given by PQ � [A + C(D)Q + iC(D)Q2/[2D(P)]]. By dividing the profit percycle by the cycle length (Q/D), we can obtain the profit per unit time. Thus, we have the following formu-lation for the profit per unit time (=p(Q, P)).

Max pðP ;QÞ ¼ revenue per unit time� ðordering cost per unit timeþ variable cost per unit time

þ inventory holding cost per unit timeÞ¼ PDðP Þ � ½ADðP Þ=Qþ CðDÞDðP Þ þ iCðDÞQ=2�.

We denote Ci, Di, Qi and pi, as the cost per unit, the demand per unit time, the order quantity, and the profitof Model i, i = 1, 2, 3, respectively. The asterisk sign means that the value is optimal. From the GP perspec-tive, the primal problems of the models are given as

Max p1ðP 1;Q1Þ ¼ aP 1�a1 � aAP�a

1 Q�11 � a1�bbP ab�a

1 � 0.5ia�bbP ab1 Q1; ð1Þ

where D1ðP 1Þ ¼ aP�a1 and C1ðD1Þ ¼ bD�b

1 .

Max p2ðP 2;Q2Þ ¼ aP 1�a2 � AaP�a

2 Q�12 � adP�a

2 Q�d2 � 0.5idQ1�d

2 ; ð2Þ
where D2ðP 2Þ ¼ aP�a
2 and C2ðQ2Þ ¼ dQ�d2 .

Max p3ðP 3;Q3Þ ¼ aP 1�a3 � AaP�a

3 Q�13 � a1�cfP ac�a

3 Q�l3 � 0.5ia�cfP ac

3 Q1�l3 ; ð3Þ

where D3ðP 3Þ ¼ aP�a3 and C3ðD3;Q3Þ ¼ fD�c

3 Q�l3 .

The objective of our models is to maximize the profit per unit time with decision variables P and Q whenthe demand per unit time is a decreasing power function of the price per unit with constant elasticity for allmodels, and the cost per unit is a decreasing power function of the demand per unit time, the order quan-tity, and both of the demand per unit time and the order quantity for Models 1–3, respectively.

3. Optimal solutions

The development of the solution procedure for our models is similar to the work by Lee [12], Lee andKim [14], or Duffm et al. [6]. The above objective function is an unconstrained signomial problem with onedegree of difficulty. Although global optimality is not guaranteed for a signomial problem, the profit func-tion can be transformed into a posynomial problem with one additional variable and constraint [6]. Detailsfor the transformation and solutions of unconstrained signomial problems are provided in the Appendix A.

In the constrained posynomial GP problem, the dual feasible solutions, xi and k, provide the weights ofthe terms in the constraints of transformed primal problem by the following equation.

V i ¼ xi=k; i ¼ 1; 2; 3; 4 whereX4

i¼1

V i ¼ 1 and k ¼X4

i¼1

xi. ð4Þ


These weights represent proportions of the profit (V1), the ordering cost (V2), the variable cost (V3), andthe inventory holding cost (V4) to the total revenue. We then have the following relations for our models.

Model 1: V 1 ¼ a�1P a�11 z1;

V 2 ¼ AP�11 Q�1

1 ;

V 3 ¼ a�bbP ab�11 ;

V 4 ¼ 0.5ia�1�bbP abþa�11 Q1.

ð5Þ

Model 2: V 1 ¼ a�1P a�12 z2;

V 2 ¼ AP�12 Q�1

2 ;

V 3 ¼ dP�12 Q�d

2 ;

V 4 ¼ 0.5a�1dP a�12 Q1�d

2 .

ð6Þ

Model 3: V 1 ¼ a�1P a�13 z3;

V 2 ¼ AP�13 Q�1

3 ;

V 3 ¼ a�1fP ac�13 Q�l

3 ;

V 4 ¼ 0.5ia�1�cfP acþa�13 Q1�l

3 .

ð7Þ

From the above equations, the corresponding primal solutions can be obtained:

P 1 ¼ abV 3=b� �½1=ðab�1Þ� ¼ 0.5iAb½abV 3=b�½ðabþa�1Þ=ab�1�

h i.ða1þbV 2V 4Þ

¼ a1þbV 2V 4½abV 3�½1=ðab�1Þ�h .

½0.5iAb½ab=ðab�1Þ��i½1=ðabþa�1Þ�

; ð8Þ

Q1 ¼ Ab½1=ðab�1Þ�� ½abV 3�½1=ðab�1Þ�V 2

h i.¼ ½a1þbV 4� 0.5ib½abb�1V 3�½ðabþa�1Þ=ðab�1Þ�

h i.; ð9Þ

P 2 ¼ A�d dV d2=V 3

� �½1=ð1�dÞ� ¼ adV 2V 4=ð0.5idAV 3Þ½ �½1=ða�1Þ�; ð10Þ

Q2 ¼ AV 3=ðdV 2Þ½ �½1=ð1�dÞ� ¼ aV 4V ða�1Þ=ð1�dÞ3 0.5id½A�d dV d

2�ða�1Þ=ð1�dÞ

h i.h i½1=ð1�dÞ�; ð11Þ

P 3 ¼ AlacV 3=ðfV l2Þ½ �½1=ðlþac�1Þ� ¼ 0.5iA1þ2a�lf ða1þcV 1þ2a�l

2 V 4Þ�� ½1=ð2þa�ac�lÞ�

; ð12Þ

Q3 ¼ Aac�1f ðacV ac�12 V 3Þ

�� ½1=ðlþac�1Þ� ¼ a1þcV acþa�12 V 4 ð0.5iAacþa�1f Þ

�� ½1=ð2þa�ac�lÞ�. ð13Þ

4. Computational results

In this section, we perform a computational analysis to determine the effects of changes in the param-eters. From this analysis, we observe whether the optimal solutions and the optimal profit are increasing(or decreasing) and convex (or concave) functions with respect to each parameter. That is, we can seehow the optimal solution would change as parameter values vary. We investigate the changes in Q*, P*

and p* according to the changes in the parameters, A, i, a, b, d, f, a, b, d, and c. For the experiment, weuse the basic parameter values: A = 50, i = 0.1, a = 500 000, b = d = f = 5, a = 2.5, and b = c = 0.2.


The values of the parameters to analyze with optimal solutions are allowed to vary ±10%, ±20%,±30%, . . ., ±90%, ±100% for A, i, a, b, d, f, ±1%, ±2%, ±3%, . . ., ±9%, ±10% for a, and±5%, ±10%, ±15%, . . ., ±45%, ±50% for b and c.

The results for Model 1 are as follows:From Fig. 1, it can be seen that any increase (decrease) in ordering cost, A, results in a larger (smaller)

optimal order quantity and a higher (lower) optimal price. However, optimal profit decreases (increases) asordering cost increases (decreases). These relationships indicate that increases in A lead to higher inventorycost and therefore, at the same time, to higher Q*. Also, higher inventory cost will in turn lead to a higherP* and lower p*. Fig. 2 indicates that any increase (decrease) in inventory holding cost, i, results in a smal-ler (larger) optimal order quantity, a higher (lower) optimal price, and lower (higher) optimal profit. If theinventory holding cost, i, which is the part of total cost, is increased, total cost will be increased and there-fore profit will be reduced. In this case, a decision maker will reduce Q* to save the expense of storing inven-tory. From Fig. 3, we can observe that any increase (decrease) in a scaling constant for demand, a, results ina larger (smaller) optimal order quantity, a lower (higher) optimal price, and higher (lower) optimal profit.Since a is a constant for demand and demand is a decreasing function of price, price will be reduced as a

increases. This will result in higher profit and order quantity from the increased demand. Fig. 4 indicatesthat any increase (decrease) in a scaling constant for unit cost, b, results in a smaller (larger) optimal orderquantity, a higher (lower) optimal price, and lower (higher) optimal profit. The fact that when we increase

Fig. 1. Changes in the order quantity (a), the price (b), and the profit (c) with respect to change in the ordering cost of Model 1.

Fig. 2. Changes in the order quantity (a), the price (b), and the profit (c) with respect to changes in the inventory holding cost ofModel 1.

Fig. 3. Changes in the order quantity (a), the price (b), and the profit (c) with respect to change in the scaling constant for demand ofModel 1.

Fig. 4. Changes in the order quantity (a), the price (b), and the profit (c) with respect to change in the scaling constant for the unit costof Model 1.


the scaling constant for C, Q* decreases, represents that if the cost per unit is increased by the scaling con-stant, Q* will be decreased because of economies of scale. Increases in unit cost will lead to higher total costand price, and hence, lower profit. From Fig. 5, it can be seen that any increase (decrease) in the price elas-ticity of demand, a, results in a larger (smaller) optimal order quantity, a lower (higher) optimal price, andhigher (lower) optimal profit. Highly elastic demand conditions yield higher demand and thus larger orderquantity, which in turn lead to lower price and higher profit. Fig. 6 indicates that any increase (decrease) inthe degree of economies of scale, b, results in a larger (smaller) optimal order quantity, a lower (higher)optimal price, and higher (lower) optimal profit. If the cost per unit is decreased by b from the cost func-tion, C1 ¼ bD�b

1 , Q�1 will be increased because of economies of scale. Decreases in unit cost will lead to lowertotal cost and price, and hence, higher profit. Fig. 7 indicates that any decrease (increase) in d results in asmaller (larger) optimal order quantity, a higher (lower) optimal price, and a lower (higher) optimal profit.If the cost per unit is decreased by d from the cost function, C2 ¼ bQ�d

2 , Q�2 will be increased. Decreases inunit cost will lead to lower total cost and price, and hence, higher profit.

The above results for i, a, a of Model 1 are similar to the results for the same parameters of Models 2 and3. Also, the results for b are similar to the results for d and f, and the results for b are similar to the results

Fig. 6. Changes in the order quantity ((a), (d)), the price ((b), (e)), and the profit ((c), (f)) with respect to change in the degree ofeconomies of scale of Models 1 and 3.

Fig. 7. Changes in the order quantity (a), the price (b), and the profit (c) with respect to change in the quantity discount factor ofModel 2.

Fig. 5. Changes in the order quantity (a), the price (b), and the profit (c) with respect to change in the price elasticity of demand ofModel 1.


for c. Therefore, managerial insights of the results for inventory holding cost, scaling constants, price elas-ticity, and degree of economies of scale of Models 2 and 3 are similar to those used in Model 1.


4.1. Effects of changes in ordering costs of Models 2 and 3

For Models 2 and 3, we investigate what changes occur in Q* and P* when changes occur in the orderingcost, A by computational analysis. The purpose of this analysis is to determine possible relationships bycomparing the results of Models 2 and 3 with those of the EOQ model since we are not able to determineanything from the previous computational results for A. The basic parameter values are: A = 100, i = 0.1,a = 50 000 000, b = 5, a = 1.5, d = 0.01, c = 0.2, and l = 0.01. The value of the ordering cost, to analyzewith optimal solutions, is allowed to vary ±10%, ±20%, ±30%, . . ., ±90%, ±100%.

In the classical EOQ model, the order quantity always increases when the ordering cost goes up.Note that Q� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2AD=ðiCÞ

pin the EOQ model. In fact, the optimal order quantity and price typically

increase for a wide range of input parameter values as the ordering cost increases. However, Tables 1and 2 indicate that for Models 2 and 3, the order quantity sometimes decreases as the ordering cost in-creases, especially when a decrease in the demand due to a higher price is large enough to offset the positiveeffect of an increase in the ordering cost on the lot size. The reason for the different results between the EOQmodel and Models 2 and 3, is that our demand and unit cost functions are decreasing functions of the priceand the order quantity, respectively and these functions are affected by the scaling constant for demand, a.That is, the larger the number for the scaling constant such as a = 50 000 000 causes more effects on thedemand and therefore, increases the optimal order quantity although A decreases. That is, a decrease inthe ordering cost provides an economic incentive for reducing the order quantity on the one hand andcauses an increase in the demand, thereby increasing the order quantity, on the other hand. If the lattereffect dominates the former, the optimal order quantity will increase as the ordering cost becomes lesscostly.

The above changes indicate the sensitive interrelationships of the parameters of the new models and howwhen all aspects are considered results different than expected can occur.

Table 1Effects of changes in the ordering cost for Model 2

Ordering cost Unit cost Demand Price Order quantity

50 4.4228 1 034 455.67 13.2690 212 598.6655 4.4232 1 034 280.60 13.2705 210 338.2460 4.4236 1 034 134.52 13.2718 208 490.6665 4.4227 1 034 456.74 13.2690 212 950.3470 4.4231 1 034 314.01 13.2702 211 139.4275 4.4234 1 034 191.86 13.2713 209 594.4780 4.4227 1 034 448.17 13.2691 213 170.1085 4.4230 1 034 328.68 13.2701 211 660.4890 4.4232 1 043 222.10 13.2710 210 336.2795 4.4235 1 034 126.28 13.2718 209 165.28100 4.4229 1 034 330.89 13.2701 212 026.25105 4.4231 1 034 236.76 13.2709 210 868.86110 4.4234 1 034 150.75 13.2716 209 827.49115 4.4228 1 043 326.12 13.2701 212 296.97120 4.4230 1 034 241.65 13.2708 211 269.66125 4.4232 1 034 163.52 13.2715 210 333.18130 4.4228 1 034 316.76 13.2702 212 505.30135 4.4230 1 034 240.00 13.2709 211 582.07140 4.4232 1 034 168.32 13.2715 210 731.85145 4.4228 1 034 304.24 13.2703 212 670.47150 4.4229 1 034 233.77 13.2709 211 832.33

Table 2Effects of changes in the ordering cost for Model 3

Ordering cost Unit cost Demand Price Order quantity

50 0.069921 730 110 897.14 0.167387 147 459 010.9955 0.069933 729 919 061.35 0.167416 145 669 735.5360 0.069919 730 131 825.30 0.167383 147 670 591.8765 0.069930 729 967 275.09 0.167409 146 133 492.6970 0.069918 730 146 360.83 0.167381 147 822 061.6175 0.069927 730 002 329.14 0.167403 146 475 422.3680 0.069918 730 156 894.47 0.167380 147 935 841.2085 0.069926 730 028 836.79 0.167399 146 737 948.3890 0.069917 730 164 756.98 0.167378 148 024 434.1895 0.069924 730 049 480.58 0.167396 146 945 845.61100 0.069917 730 170 747.95 0.167377 148 095 364.15105 0.069923 730 065 926.72 0.167394 147 114 553.14110 0.069916 730 175 376.79 0.167377 148 153 429.50115 0.069922 730 079 264.77 0.167391 147 254 193.14120 0.069916 730 178 983.42 0.167376 148 201 834.88125 0.069921 730 090 237.22 0.167390 147 371 678.20130 0.069916 730 181 803.35 0.167376 148 242 802.16135 0.069921 730 099 367.53 0.167388 147 471 890.06140 0.069915 730 184 004.88 0.167375 148 277 920.33145 0.069920 730 107 035.30 0.167387 147 558 374.35150 0.069925 730 185 711.52 0.167375 148 308 355.86


5. Comparative analysis

The comparative analysis here is to study the relationship between the optimal solutions of the threemaximization models. We focus on the comparative analysis of the optimal solutions of models wherecertain conditions in the cost per unit and the demand per unit time are given. This analysis is reasonablebecause we consider the different models according to the different shapes of the cost function (see [11]). We

investigate the error, j Q�i � Q�j j j P �i � P �j j or j p�i � p�j j� �

, for all i and j by comparing Q*s (P*s or p*s)

of Models 1 and 2, Models 1 and 3, and Models 2 and 3. That is, the error in Q*(P* or p*) shows the situa-tion where Model i is used when Model j should have been used for all i and j.

To compare the models, we need to know the form of each cost function and demand function of themaximization models. To estimate the cost function for the models, we first assume that we have previousdata for C, D, and Q so that we can draw the cost function forms of Models 1–3 if b, d, l, and c are ob-tained 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 canestimate b by taking the logarithm to transform the cost function into a linear model and then apply simplelinear regression. Similar procedures can be performed to estimate d, l, and c for Models 2 and 3, and toestimate a of the demand function for all models.

Now, suppose for a problem we have determined all cost function forms. Then, based on the analysis, wecan 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 decisionmaker 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 decreasingpower function of the demand per unit time, then select Model 2 which means that production exhibitseconomies 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 reductiondue to better allocation of resources. Also, we note that the multiplicative effect for Model 3 is assumed.

Furthermore, we assume the following parameters are identical for these models: A, i, and a. Under theseassumptions, we can select appropriate cost functions under certain conditions after investigating the errorin Q*(P* or p*).

The analysis in the previous section was done with GP because of the advantages GP gives us in deter-mining optimal solutions for real problems and for when parameters change. Here however, it is easier tosee the relationships between optimal solutions and their relationships by using first order conditions, eventhough they cannot be directly solved. By studying these relationships, several interesting properties can bederived.

From the first derivative of p1, p2, and p3 with the demand and cost functions, we have

op1

oQ1

¼ Aa½P �1��a½Q�1�

�2 � 0.5ia�bb½P �1�ab ¼ AD1½Q�1�

�2 � 0.5iC1 ¼ 0; ð14Þ

op2

oQ2

¼ Aa½P �2��a½Q�2�

�2 þ dad½P �2��a½Q�2�

�d�1 � 0.5ð1� dÞid½Q�2��d

¼ AD2½Q�2��2 þ dC2D2½Q�2�

�1 � 0.5ð1� dÞiC2 ¼ 0; ð15Þ

op3

oQ2

¼ Aa½P �3��a½Q�3�

�2 þ la1�cf ½P �3�ac�a½Q�3�

�l�1 � 0.5ð1� lÞia�cf ½P �3�ac½Q�3�

�l

¼ AD3½Q�3��2 þ lC3D3½Q�3�

�1 � 0.5ð1� lÞiC3 ¼ 0. ð16Þ

First, we compare the optimal order quantities of Models 1 and 2. By rewriting (14) and (15), we have

0.5i ¼ AC�11 D1½Q�1�

�2 ¼ AC�12 D2½Q�2�

�2 þ dD2½Q�2��1

1� d. ð17Þ

By manipulating (17), we have the following relationship.

C�11 D1½Q�1�

�2 � C�12 D2½Q�2�

�2 ¼ dC�11 D1½Q�1�

�1 þ dA�1D2½Q�2��1. ð18Þ

Eq. (18) gives us the following inequality relationship.

C2D1

C1D2

>Q�1Q�2

2

. ð19Þ

These relationships indicate that if C1 P C2 and D1 6 D2, then Q�1 < Q�2 and P �1 P P �2 since Q�1=Q�2 mustbe less than 1 and the demand is a decreasing function of the price. If C1 P C2 and D1 > D2, or C1 < C2,then all the cases Q�1 < Q�2, Q�1 > Q�2, and Q�1 ¼ Q�2 can result.

We can summarize the relationship between optimal solutions of Models 1 and 2 as follows:

Property 1a. If C1 P C2 and D1 6 D2, then P �1 P P �2 and Q�1 < Q�2.Property 1b. Otherwise, all cases for the optimal solutions result.

The difference in the optimal order quantity (the optimal price) implies the quantity (price) is over-ordered/under-ordered (over-priced/under-priced). This means that we used Model 1 (Model 2) when Mod-el 2 (Model 1) should have been used. Therefore, Property 1a shows that we can determine an optimalinventory policy for both models by estimating C1, C2, D1, and D2 (we assume that we have previous data


of cost and demand functions) without computing optimal order quantities and prices of both models ifC1 P C2 and D1 6 D2.

A numerical analysis is performed to observe the relationships between the optimal profits of Models 1and 2 since the comparative analysis of the optimal profits by using the first order conditions is intractable.The basic parameter values which are randomly chosen are: A: 0–100, i: 0–0.2, a: 500 000–1 000 000, b: 0–10,d: 0–10, a: 2.25–2.75, b: 0.1–0.3, and d: 0.005–0.015. Note that we can use any values if all values are po-sitive and satisfied with the dual feasibility conditions and our assumptions. Fig. 8 shows the results forrelationships between the profits of Models 1 and 2 after 100 iterations with the conditions on the costper unit given. We obtain 17 cases for C1 > C2 and 83 cases for C1 < C2, among 100 iterations. These re-sults indicate that p�1 < p�2 if C1 > C2, and p�1 > p�2 if C1 < C2. These results are consistent with the fact thatp = PD � [AD/Q + CD + iCQ/2]. This means that we can directly have information about the relation-ship between profits of both models by estimating C1 and C2 without computing p�1 and p�2, and the optimalpolicy can be determined from the information.

Similar to the above analysis, we can obtain the following relationships for Models 1 and 3 by manipu-lating (14) and (16) with 0 < l < 1, and for Models 2 and 3 by manipulating (15) and (16) under threeassumptions, d = l, d > l, and d < l.

Property 2a. If C1 P C3 and D1 6 D3, then P �1 P P �3 and Q�1 < Q�3.Property 2b. Otherwise, all cases for the optimal solutions result.

Property 3a. If d = l, C2 > C3, and D2 < D3, then P �2 > P �3 and Q�2 < Q�3.Property 3b. If d = l, C2 < C3, and D2 > D3, then P �2 < P �3 and Q�2 > Q�3.Property 3c. If d = l, C2 = C3, and D2 = D3, then P �2 ¼ P �3 and Q�2 ¼ Q�3.Property 3d. Otherwise, all cases for the optimal solutions result where d ¼ l.

Property 4a. If d > l, C2 6 C3, and D2 P D3, then P �2 6 P �3 and Q�2 > Q�3.Property 4b. Otherwise, all cases for optimal solutions result where d > l.

Property 5a. If d < l, C2 P C3 and D2 6 D3, then P �2 P P �3 and Q�2 < Q�3.Property 5b. Otherwise, all cases for optimal solutions result where d < l.

Property 2a indicates that the optimal inventory policy for Models 1 and 3 can be determined by esti-mating C1, C3, D1, and D3 without computing optimal order quantities and prices of both models ifC1 P C3 and D1 6 D3.

Fig. 8. Comparative analysis of profits of Models 1 and 2: (a) C1 > C2, (b) C1 < C2.


Assumption d = l means that the elasticity with respect to the order quantity of Model 2 is identical tothat of Model 3. From C2 ¼ d½Q�2�

�d and C3 ¼ fD�c3 ½Q�3�

�l, we can see that the above property is consistentwith the same elasticity for both models. We note that d and fD�c are constant, and d and l are betweenzero and one. Assumption d > l(d < l) means that the elasticity with respect to the order quantity of Model 2is greater (less) than that of Model 3. Property 4a (Property 5a) is consistent with C2 ¼ d½Q�2�

�d andC3 ¼ fD�c½Q�3�

�l where d > l(d < l).In Properties 3–5, the relationship between the optimal order quantities (or the optimal price) of Models

2 and 3 depends on the cost and demand functions as well as the elasticity with respect to the order quan-tity. Since we can estimate the cost and demand functions and the elasticity of Models 2 and 3 by usinglinear regression, we can determine an optimal inventory policy from the cost and demand functionsand the elasticity without computing the optimal order quantities.

Similar to the above numerical analysis, the results for Models 1 and 3 show that p�1 < p�3 if C1 > C3, andp�1 > p�3 if C1 < C3. This means that we can determine an optimal inventory policy by estimating C1 and C3

without computing p�1 and p�3. The results are consistent with the fact that p = PD � [AD/Q + CD + iCQ/2]. For the numerical analysis between p�2 and p�3, the results show that p�2 < p�3 if C2 > C3, and p�2 > p�3 ifC2 < C3, and managerial implications are similar to those shown in Models 1 and 3.

The above properties and figure indicate that if the demand and cost functions can be estimated, the rela-tionship between the optimal order quantities as well as the optimal profits can be analyzed. Since we canestimate the demand and cost functions from the previous data using linear regression, we can determine anoptimal inventory policy by estimating the demand and cost functions without computing optimal solu-tions if the difference in the optimal solutions is found. For example, if we know Q�i > Q�j (i.e., Q�i isover-ordered or Q�j is under-ordered) from Ci < Cj, we can increase Ci, (or decrease Cj) to reduce the errorin the optimal order quantity. This adjustment will give us an optimal policy for our models.

6. Conclusion

In this paper, we have developed and analyzed three EOQ based inventory models under profit maximi-zation via GP techniques. We demonstrated that GP is an excellent method to solve our nonlinear problemsconsisting of power functions. The theories of GP were applied to help derive closed-form optimal solu-tions, which is not possible by classical methods. The change in the optimal order quantity, price, and profitaccording to varied parameters was analyzed to see the effect on inventory policy from the computationalanalysis. Comparisons between these models showed the relationship between the optimal solutions of dif-ferent forms for cost functions where certain conditions in the demand per unit time and the cost per unitare given.

Our three models can be applied in many real industrial situations. Applications for our models are: (1)For Model 1, the unit cost is related to the demand. This implies that decision maker employs better equip-ment and more resources for the production of this product. That is, decision maker will focus on produc-tion management as the demand increases. This cost function can be typically found in the industries whichproduce a successful technologically-advanced product with the growth phase of its life cycle such as com-puter industry since the decision maker will justify more efficient production processes as the demand ishigh. It can be also applied to help model supply chain in which the major unit of the chain pushes costsdownstream but wants to maintain price. Hence, its costs become a function of upstream demand. (2) ForModel 2, the unit cost is related to the order quantity. This means that the production exhibits economies ofscale. This function can be applicable in the most industries where decision maker focuses on managementoperations with quantity discount offer from the arrangement of long-term supply contracts as the orderquantity is large. (3) For Model 3, the unit cost is a function of both the demand and the order quantity.This means that we consider economies of scale and cost reduction due to better allocation of resources at


the same time. This can be found in the situation that decision maker focuses on both production manage-ment and management operations.

The newly proposed models extend the models in the literature by using a mutiplicative term in Model 3(i.e., the demand and the order quantity in the cost function). By using this multiplicative term, we are ableto consider economies of scale of production as well as cost reduction due to more efficient production pro-cesses at the same time. In addition, we have extended current models by using the cost function and thedemand function as a function of the demand per unit time and a function of the price per unit, respectivelyin Model 1. This assumption means that the unit cost is a function of the price per unit. That is, when theprice is increased (decreased), then unit cost will be increased (decreased). If the price is decreased, decisionmaker will reduce the unit cost to increase the profit. Also, if the price is increased, people involved in theproduction processes tend not to be as concerned about cost control, but just about production to meetdemand. This can result in a higher unit cost. These phenomena with the above cost function can be foundin large industries such as automobile and munitions industries, and tend to adequately reflect what hap-pens in many supply chains as costs are pushed through the chain.

The three models we have investigated may provide the basis for numerous further research areas. Ourmodels could be a basis for inventory models integrated with quality, ordering cost, and process improve-ment issues (see e.g., [3,5]). Due to the suitability of GP for dealing with exponential functions, we can ap-ply these to more comprehensive models, where the effects of marketing mix variables, such as advertisingand promotion activities on demand, are represented by an exponential function. We can also extend ourmodels to the multi-product case where the nonlinear interactions among related products with respect totheir demands are taken into account.

Appendix A

For Model 1, We rewrite the unconstrained signomial problem, Eq. (1), to demonstrate the transforma-tion as follows:

Max z1;

s.t. aP 1�a1 � AaP�a

1 Q�11 � a1�bbP ab�a

1 � 0.5ia�bbP ab1 Q1 P z1; ðA:1Þ

which is equivalent to the following transformed primal problem:

Min z�11 ;

s.t. a�1P a�11 z1 þ AP�1

1 Q�11 þ a�bbP ab�1

1 þ 0.5ia�1�bbP abþa�11 Q1 6 1. ðA:2Þ

This transformed primal function is a constrained posynomial problem with one degree of difficulty whichis guaranteed to have a global optimal.

Instead of dealing with (A.2) to obtain the solution, we use the following dual problem which is usuallyeasier to solve.

Max dðxÞ ¼ ½1=x0�x0 ½a�1k=x1�x1 ½Ak=x2�x2 ½a�bbk=x3�x3 ½0.5ia�1�bbk=x4�x4 ; ðA:3Þ

s.t. x0 ¼ 1;

� x0 þ x1 ¼ 0;

ða� 1Þx1 � x2 þ ðab� 1Þx3 þ ðabþ a� 1Þx4 ¼ 0; ðA:4Þ

� x2 þ x4 ¼ 0; xi > 0 for i ¼ 0; 1; 2; 3; 4;

where k = x1 + x2 + x3 + x4.


The dual variables, xi, i = 0, 1, 2, 3, 4, are often referred to as the weight that is not normalized. The firstconstraint is the normality condition and the second, third, and fourth are the orthogonality conditions.From these constraints, we can express x0, x1, x2, x3, and k in terms of x4.

x0 ¼ x1 ¼ 1;

x2 ¼ x4;

x3 ¼ ½1� a� ðabþ a� 2Þx4�=ðab� 1Þ;k ¼ ½ab� aþ ðab� aÞx4�=ðab� 1Þ.

ðA:5Þ

In order to guarantee that all dual variables are positive, the expressions for x2( = x4) and x3 should bepositive. Hence, the dual feasibility condition from x3 of (A.5) is essentially x3 = [1 � a �(ab + a � 2)x4]/(ab � 1) > 0. From our assumption 0 < b < 1 and k = [a(b � 1) + a(b � 1)x4]/(ab �1) > 0, the dual problem is infeasible if ab > 1. That is, if ab > 1, a becomes negative to satisfy k > 0. Itis clear that x3 > 0 and ab < 1 indicate a > 1. Therefore, the following dual feasibility condition is true.

Lemma 1 (Dual feasibility condition for Model 1). If 0 < b < 1, ab < 1, and a < 1, then the dual problem is

feasible.

Since the model has one degree of difficulty, we can solve this with the following substituted dual func-tion, d(x4) to find an optimal solution. By substituting (A.5) into the dual objective function (A.3), wherex4 is the only variable, the substituted dual problem is formed.

Max dðx4Þ¼ a�1½A=x4�x4 ½ðab�1Þa�bb=½1�a�ðabþa�2Þx4��ð1�a�ðabþa�2Þx4Þ=ðab�1Þ�½0.5ia�1�bb=x4�x4

�½ðab�aþðab�aÞx4Þ=ðab�1Þ�½ðab�aþðab�aÞx4Þ=ðab�1Þ�

¼ a�1½ðab�1Þa�bb=½ð1�a�ðabþa�2Þx4Þ��½ð1�a�ðabþa�2Þx4Þ=ðab�1Þ�½0.5iAa�1�bb=x24�

x4

�½ðab�aþðab�aÞx4Þ=ðab�1Þ�ðab�aþðab�aÞx4Þ=ðab�1Þ�. ðA:6Þ

By taking the logarithm of the objective function of the substituted dual problem, we obtain the follow-ing concave function in one variable (see e.g., Duffin et al. [6], pp. 121–122 for the proof of concavity oflog dðx4Þ).

Max log dðx4Þ ¼ � log a� ½ð1� a� ðabþ a� 2Þx4Þ=ðab� 1Þ�� logbð1� a� ðabþ a� 2Þx4Þ=ððab� 1Þa�bbÞc � x4 logbx2

4=ð0.5iAa�1�bbÞcþ ½ðab� aþ ðab� aÞx4Þ=ðab� 1Þ� log½ðab� aþ ðab� aÞx4Þ=ðab� 1Þ�. ðA:7Þ

Setting the first derivative to zero yields

o log dðx4Þox4

¼ ½ðabþ a� 2Þ=ðab� 1Þ� log½ð1� a� ðabþ a� 2Þx4Þ=ððab� 1Þa�bbÞ

� log½x24=ð0.5iAa�1�bbÞ þ ½ðab� aÞ=ðab� 1Þ� log½ab� aþ ðab� aÞx4Þ=ðab� 1Þ�

¼ log½1� a� ðabþ a� 2Þx4Þ�ðabþa�2Þ=ðab�1Þ½ab� aþ ðab� aÞx4�ðab�aÞ=ðab�1Þ

x24

" #

� log½ðab� 1Þa�bb�ðabþa�2Þ=ðab�1Þ½ab� 1�ðab�aÞ=ðab�1Þ

0.5iAa�1�bb

" #¼ 0. ðA:8Þ

The unique solution can be easily found by any line search technique. After x�4 is obtained from (A.8), x�3,and k can be calculated from (A.5). The optimal weights are then calculated from (4). Hence, the dualobjective function, dðx�4Þ, can be obtained from (A.6). The corresponding P* and Q* are calculated from


(8) and (9). According to the duality theorem of GP, we can obtain p* from the relationship (1/z*) = d(x*)where p* = z* = Max z. Therefore, p* = 1/d(x*). Also, p* can be obtained from the profit function (1)after P* and Q* are substituted.

Since the procedures for Models 2 and 3 are identical to Model 1, we only show the dual problems anddual feasibility conditions for Models 2 and 3.

For Model 2, the dual problem from the transformed primal function is

Max dðxÞ ¼ ½1=x0�x0 ½a�1k=x1�x1 ½Ak=x2�x2 ½dk=x3�x3 ½0.5ia�1dk=x4�x4 ; ðA:9Þ

s:t: x0 ¼ 1; �x0 þ x1 ¼ 0; ða� 1Þx1 � x2 � x3 þ ða� 1Þx4 ¼ 0; ðA:10Þ

� x2 � dx3 þ ð1� dÞx4 ¼ 0; xi > 0 for i ¼ 0; 1; 2; 3; 4;

where k = x1 + x2 + x3 + x4.From (A.10), we can express x0, x1, x2, x3 and k in terms of x4.

x0 ¼ x1 ¼ 1;

x2 ¼ ½dð1� aÞ þ ð1� adÞx4�=ð1� dÞ;

x3 ¼ ½a� 1þ ðaþ d� 2Þx4�=ð1� dÞ;

k ¼ aþ ax4.

ðA:11Þ

From our assumption 0 < d < 1 and the positivity condition (xi > 0), we know that the dual problem isinfeasible if a P 1 and ad P 1 (in this case, x2 can not be positive) or if a 6 1 and a + d 6 2 (in this case,x3 can not be positive). Hence, the following dual feasibility condition is true.

Lemma 2 (Dual feasibility condition for Model 2). If a > 1, ad < 1, 0 < d < 1, and 0 < d < a < 1 then the

dual problem is feasible.

For Model 3, the dual problem from the transformed primal function is

Max dðxÞ ¼ ½1=x0�x0 ½a�1k=x1�x1 ½Ak=x2�x2 ½a�cf k=x3�x3 ½0.5ia�1�cf k=x4�x4 ; ðA:12Þ

s:t: x0 ¼ 1; �x0 þ x1 ¼ 0;

ða� 1Þx1 � x2 � ðac� 1Þx3 þ ðacþ a� 1Þx4 ¼ 0; ðA:13Þ

� x2 � lx3 þ ð1� lÞx4 ¼ 0; xi > 0 for i ¼ 0; 1; 2; 3; 4;

where k = x1 + x2 + x3 + x4.From (A.13), we can express x0, x1, x2, x3 and k in terms of x4.

x0 ¼ x1 ¼ 1;

x2 ¼ ½lð1� aÞ þ ð1� ac� alÞx4�=ð1� ac� lÞ;

x3 ¼ ½a� 1þ ð2� ac� a� lÞx4�=ð1� ac� lÞ;

k ¼ ½a� ac� alþ ða� ac� alÞx4�=ð1� ac� lÞ.

ðA:14Þ

From our assumption a > 1 and the positivity condition, x3 > 0, of (A.14), we know that the dual problemis infeasible if 1 � ac � l 6 0 (in this case, x3 cannot be positive). Then, from x2 > 0 and k > 0, we obtain0 < c + l < 1. Hence, the following dual feasibility condition is true.

Lemma 3 (Dual feasibility condition for Model 3). If a > 1 and 0 < c + l < 1, then the dual problem is

feasible.


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Optimal inventory policies for profit maximizing EOQ models under various cost functions