Computers ind. Engng Vol. 18, No. 4, pp. 529-534, 1990 0360-8352/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ~) 1990 Pergamon Prt~s pie

A N E O Q M O D E L W I T H P R I C I N G C O N S I D E R A T I O N

T. C. E. CHENG Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba,

Canada R3T 2N2

(Received for publication 29 March 1990)

Ala~rat't--This paper presents an economic order quantity (EOQ) model that integrates the product pricing and order sizing decisions to maximize profit, The Kuhn-Tueker conditions are used to determine the optimal solution under conditions of storage space and inventory investment limitations. The aim of this paper is to explore the effect of relating the pricing and order sizing decisions on the optimal solution. Such a relationship, although it exists in real situations, has long been ignored in the classical EOQ model.

INTRODUCTION

The classical economic order quantity (EOQ) model is a well-known mathematical model developed to assist in inventory management decision making. Simply put, EOQ is concerned with determining the most desirable order quantities for the products offered by a firm that will result in the lowest average annual total material cost, which is the sum of the annual order, inventory carrying and material acquisition costs.

In the classical EOQ model it is assumed that the rates of demand for the products of a firm are fixed and that the demand rates are independent of the product prices [1-3]. This assumption is hardly valid in many realistic situations because a firm, especially a monpolistic price setter, is generally able to influence the demand for its products by manipulating their prices [4]. Under such circumstances the demand rate should be treated as a decision variable in the EOQ model, since it can be varied and indirectly controlled by the firm.

Although over the years much has appeared about the determination of the optimal order quantity [5,6], it seems that very little attention has been paid to the problem of integrating the order sizing decision with the product pricing decision. A notable exception is the paper of Whitin [7], who formulated an EOQ problem in which the demand is a negatively-sloped linear function of the price; but he did not solve the problem expliciily. In a more recent paper, Porteus [8] presented a sophisticated EOQ model to address the issue of setup reduction in which the demand rate is considered as a decision variable affected by the pricing decision.

In this paper we propose a profit maximization decision model that explicitly integrates the pricing and sizing decisions to detehnine the optimal ordering policy. We also include in the model two practical constraints, namely, storage space and inventory investment limitations, which are encountered in almost all realistic inventory management situations. We formulate this EOQ problem as a constrained non-linear optimization problem and apply the Kuhn-Tucker optimality conditions to determine the optimal solution.

MODEL FORMULATION AND ASSUMPTIONS

The following definitions and assumptions are made before we discuss the model in detail:

n = total number of products produced by the firm; Qi = demand rate for product i; Q = (Ql, Q2.- . Qn), the demand rate vector; q~ = order size of product i; q = (q~, q2.--qn), the order size vector; c~ = price-demand elasticity of product i, defined as

\ / dp-S, '

529

530 T.C.E. C~No

SI ri

J T f, P, v~

n(Q, q) F .4 I

= order cost per batch of product i; = unit cost of production of product i; ffi fractional inventory carrying cost rate; = length of a replenishment cycle; = storage space requirement per unit of product i; = unit selling price of product i; = unit batch production cost of product i; = total profit derived from the sale of the products; = total fixed cost of production and administration; = total storage space available; = maximum inventory investment allowable.

The following basic assumptions about the model are made:

(A1) All products have equal replenishment cycle length

T = q-J- 1 < ~ i ~ n . Qi'

(A2) Replenishments of the products are instantaneous. (A3) No backorder is permitted. (A4) The demand rates are uniform and continuous. (AS) The demand functions of the products are given as follows:

P i f ht(Q~), l <~ i <~ n (1)

where h;(.) is a function of Q~ which, in general, is monotonically decreasing.

Let K; be the total manufacturing cost to produce a batch of qt product i sufficient to meet the demand over a replenishment cycle of length T. The total manufacturing cost per cycle is equal to the sum of order cost, production cost and inventory carrying cost incurred in a cycle, so

jriq~ T Ki = s~ + rlq i + - - - ~ , (2)

o r

jr~q~ Kl = si + r~q, + 20"-"-7" (3)

Let V~ be the unit manufacturing cost. It follows that

which, on substituting (3), becomes

V~: K~, (4) qi

jr~ q~ Vi = ri + s_, + . (5) q~ 2Qi

Our objective is to maximize profit, which is the excess of revenue over cost, subject to both storage and inventory investment constraints. That is

Maximize

subject to

n(Q, q)= ~ {P,Q,- V,Q,}- F (6) i - - I

n

~, f~q, <<..4 (7) i - I

An EOQ model with pricing consideration 531

and

]r,q ~, ,~., ~ <~ i. (8)

This constrained maximization problem equations (6)-(8) can be readily solved by the Lagrangian methods of undetermined multipliers [4, 9]. To do this, we first write the Lagrangian function

I Jr~ q i L f f i , . , { P , Q , - V , Q , } - F + 2 , A - - +22 - = ~ / } (9)

where 2 = (2~, 22) I> 0 is a vector of La.,vangian multipliers which is yet to be determined. Next, we apply the Kuhn-Tucker optimality conditions to the Lagrangian function (9) to obtain

the necessary conditions for the optimal solution.

Condition 1

o r

OL Q,= ~-~= O, l<~i<<.n

_ Oil ~ O E . 2zjr, q l ) e, e , . = o.

Substituting equations (1) and (5) into equation (10) yields

Condition 2

or

Condition 3

o r

1 Q j { h j ( Q i ) I 1 - ~ ] - r l - ~ + ~ 2zjr~q~l

dL aQ<~O, l<<.i<~n

1 ] Sa 22jrfq~ h,(Q,) 1- -~ -- r, -- -- + ~ <~ O,

q/ 2Q/

OL qi~q -- O, l<~i<~n

q,{ Is' Q, ~'~12 2QdJ~ri l - - 2'f"-- 2~:q '} =0 '

1 ~<i ~<n. (10)

l <<.i <<.n. (11)

l <~ i <~ n.

l ~ i ~ n .

(12)

(13)

Condition 4

or

aL dq---~ < O, l<<.i<<.n

22jr ~q~ ~ ..

L~, 2Q,_l l < i < n . (14)

532 T.C.E. CHENG

Condition 5

o r

Condition 6

o r

dL

[ 2, - i =0. i I

t~L o

(15)

n

fq , ~< A. (16) i = l

Condition 7

o r

Condition 8

o r

aL 22 ~-~2 = 0

2211- ~ j r ' q~]=O ,~, 2Q, J "

dL

(17)

n • 2 V;" jr~q7 t-'Y', 2Q, ~< I. (18)

Condition 9

Qi, q~, 2,, 22 ~< 0, l~<i~<n. (19)

Expressions (10)--(19) are only necessary conditions for a relative maximum of H(Q, q). But if the objective function (6) and the constraints (7) and (8) can be shown to be convex, the Kuhn-Tucker conditions are both necessary and sufficient for a global maximum. However, it is often very difficult to ascertain whether the objective function and constraints involved in a problem are convex [10]. Thus, there are three possible scenarios when this method of solution is used:

(1) The objective function and feasible region are found to be convex. Therefore, the solution is optimal.

(2) The objective function and feasible region are found to be nonconvex, and thus, optimality cannot be guaranteed.

(3) It is mathematically infeasible or impractical to determine convexity. Consequently, no claims about the optimality of the solution can be made.

Under the last two scenarios, it is necessary to evaluate all relative maxima determined from expressions (10)-(19) to find the global maximal solution to FI(Q, q).

Finally, let Q* and q* denote the optimal solutions determined by solving the system of nonlinear equations and inequalities (10)-(19). It follows that the optimal product prices P* and the maximum profit H*(Q, q) can be easily calculated by substituting Q~ = Q* and qi = q*, i <<. i <<. n, into equations (1) and (6), respectively.

An EOQ model with pricing consideration 533

In practice, the solution of a large-scale system of nonlinear equations necessitates the use of a computer. Given the availability on the market of some powerful computer software for solving nonlinear equations such as GINO, Eureka and the like, the solution of a large-scale problem can be efficiently determined even on a microcomputer.

LINEAR DEMAND FUNCTIONS

Following Whitin's paper [7], we now consider a special case of the EOQ model in which the relationship between the unit selling price and demand rate for the products is linear. Under the assumption of linear demand functions, we can develop closed-form optimal solutions to the EOQ problem. If the price P~ is a negatively-sloped linear function of the demand rate Q;, we have

P, = h,(Q,) = po _ m , Q . 1 ~ i ~ n (20)

where p0, mi > 0 are arbitrary constants. The problem formulation is identical to that expressed in equations (6)-(8). Assuming for the

time being that the two constraint sets (7) and (8) are inactive, we obtain the following necessary conditions for the optimal solution.

~ n w _ _ . 0 0q~

and

s1 jri ~--~0 q2 2Q,

q * = ]----~-'. l~<i~<n (21) ~1 j r ~ ,

OH = 0 dQ,

Q, eP'- v, _ ~v, ::~ Pi "~ OQ i i -- Qi - ~ i "~ 0

=} P, - r , - 2m, Q, = q 2Q*

Substituting x/Q* = y~ > 0 into (22) yields

1 ~< i ~< n. (22)

where

For solution to (23), let

y~ + a ~ y i + b ~ = O , l <~i < n

r , - p0 a;-- 2m~ l~<i~<n

l <~ i <~ n.

l <<.i <~n

l <~ i <~ n

(23)

(24)

(25)

(26)

(27)

534 T .C .E . Cr~NG

then the values of Yi will be given by (see Ref. [11])

yi = Ai + Bi

or

- (A, + &) ~--3(A~ - B,) 1 ~< i ~< n. (28) Y~ = 2 t- 2 '

It follows that Q* = y~ and, from equation (21), q* = Y i ~ . Finally, we have to determine whether the solutions Q* and q* satisfy the constraints. If they do, then they are optimal. When this is not the case, then the constraints are active and we have to use the Lagrangian method and the Kuhn-Tucker conditions, as discussed in the preceding section, to find the optimal solutions.

CONCLUSIONS

In this paper we present an integrated EOQ model that combines the decisions of setting product prices and determining order quantities for maximizing profit. The Kuhn-Tucker conditions are applied to determine the optimal solution with storage space and inventory investment constraints. It is hoped that this paper will provide some insights into the relationship between the pricing and order sizing decisions that has long been ignored in the classical EOQ model.

Acknowledgements--This research was supported in part by a grant from the Associates Fund of the Faculty of Managraent, University of Manitoba. The author is thankful to the two anonymous referees for their many helpful comments.

REFERENCES

I. G. Hadley and T. M. Whiten. Analysis of Inventory Systems, Prentice-Hall, New Jersey (1963). 2. L. A. Johnson and D. C. Montgomery. Operations Research in Production Planning, Scheduling and Inventory Control,

Wiley, New York (1974). 3. E. A. Silver and R. Peterson. Decision Systems for lnventorv Management and Production Planning, Wiley, New York

(1985). 4. W. J. Baumol. Economic Theory and Operations Analysis, Prentice-Hall, New Jersey (1977). 5. A. J. Clark. An informal survey of multi-echelon inventory theory. Nay. Res. Logis. Q. 19, 621-650 (1972). 6. G. Urgeletti Tinarelli. Inventory control: models and problems. Fur. J. Opl Res. 14, 1-12 (1983). 7. T. M. Whitin. Inventory control and price theory. Mgmt Sci. 2, 61-68 (1955). 8. E. L. Porteus. Investing in reduced setups in the EOQ model. Mgmt Sci. 31, 998-1010 (1985). 9. D. G. Luenberger. Linear and Non-Linear Programming, Addison-Wesley, Reading (1984).

10. S. S. Rao. Optimization: Theory and Applications, Wiley, New York (1978). 11. W. H. Beyer. Standard Mathematical Tables, CRC Press, Florida (1981).