252 European Journal of Operational Research 40 (1989) 252-256 North-Holland

Theory and Methodology

An economic order quantity model with demand-dependent unit cost

T.C.E. C H E N G Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Abstract: For the classical EOQ problem a relationship between demand and unit cost may exist under certain circumstances. We propose an EOQ model with demand-dependent unit cost and formulate the optimization problem as a geometric program (GP). We solve the GP analytically to obtain a closed-form optimal solution. An illustrative example is provided to show the working procedures of applying GP to solve a given problem. We also touch on some aspects of sensitivity analysis based on the GP approach. We demonstrate in this paper the point that GP has potential as a viable mathematical tool for the analysis of a certain class of inventory control problems.

Keywords: Inventory, geometric programming, EOQ

Introduction

Generally, inventory planning and control is concerned with the acquisition of all materials necessary for supporting various business oper- ations. More specifically, inventory management is primarily dominated by two vital decisions, namely determination of: (1) replenishment quan- tity and (2) timing of placing replenishment orders, as discussed in Hadley and Whitin (1963).

The problem of determining the optimal order quantity under rather stable conditions is com- monly known as the classical economic order quantity (EOQ) inventory problem. Over the years a great deal of research has been done on this topic and many interesting results have been re- ported in the literature, see, for example, the survey papers by Clark (1972), Urgeletti Tinarelli (1983), Veinott (1966) and Whitin (1954).

Received February 1988; revised May 1988

Two major assumptions in the classical EOQ model are that the demand rate is constant and deterministic and that the unit cost is constant and independent of demand (Hadley and Whitin (1963) and Silver and Peterson (1985)). However, in a realistic situation, the demand rate and unit cost of production may be related to each other. Such a relationship exists typically in two situa- tions. First, in a manufacturing environment, when demand is high a company can justify the use of more efficient production processes to produce items at lower unit production cost. Second, in a distribution environment, if sufficient demand ex- ists a firm can arrange long-term supply contracts with the suppliers to take advantage of quantity discount on the unit price. Thus it is quite legiti- mate to assume that demand rate and unit cost are inversely related to each other.

In this paper we propose a general equation to model the relationship between demand rate and unit production cost. The objective is to minimize

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T.C,E. Cheng / Economic order quantity model 253

a total cost function based on the values of de- mand rate, unit cost, inventory carrying cost and order quantity over a long period of time. Gener- ally, this can be shown to be mathematically equivalent to minimizing the average annual total cost, as has been pointed out by Hadley and Whitin (1963).

A geometric programming (GP) approach to solving the EOQ problem is presented in this paper. We show that GP is able to yield a closed-form optimal solution. This is due to the fact that the objective function of the EOQ prob- lem to be minimized is a posynomial which is most suitable for GP treatment. An additional advantage of the GP approach is that it allows a simple and easy sensitivity analysis to be per- formed on the optimal solution, as will be dis- cussed in the sequel.

This paper is organized as follows. The next section presents the model with demand-depen- dent unit cost and the underlying assumptions. The following sections derives the optimal solu- tion to the EOQ problem using the geometric programming solution technique. A numerical ex- ample is then solved to illustrate how a closed-form optimal solution of a given problem is derived, followed by a section on sensitivity analysis of the optimal solution. In the final section some con- cluding remarks are presented.

Model and assumptions

Consider the case where a company manufac- tures a product for which demand is in excess of supply. The market will take up virtually any quantity of the product rolled out from the pro- duction line. This situation is typical of a success- ful technologically-advanced product which is in the growth phase of its life cycle. Since demand for the product is high, the manufacturer may find it economical to adopt a more efficient method of production. This reduces the unit production cost, which in turn generates more revenue to the com- pany. To construct a model for this problem, we define the following variables:

S = setup cost per batch, H = inventory carrying cost per item per

unit time, V = unit production (item) cost, D = demand rate (a decision variable),

q = production (order) quantity per batch (a decision variable),

C(D, q) = average annual total cost. The following basic assumptions about the

model are made. (1) Production is instantaneous. (2) No back-order is allowed. (3) Demand for the product exceeds supply. (4) The unit production cost is inversely re-

lated to the demand rate according to the follow- ing equation:

(1) where a > 0 , /3> 1 are constant real numbers selected to provide the best fit of the estimated cost function. While a > 0 is an obvious condition since both V and D must be non-negative, the reason for /3 > 1 will be become clear when the details of the GP approach are discussed in the next section.

The first two assumptions are the basic as- sumptions used in the classical EOQ model. Since demand is in excess of supply the manufacturer is free to choose a preferred level of production, for it is certain that any quantity of the product produced will be absorbed by the market. Thus the third assumption implies that demand rate is equal to production rate. The fourth assumption follows the third assumption that high demand justifies the use of a more efficient production method which will result in a lower unit produc- tion cost. Clearly, (1) is an appropriate expression to model the inverse relationship between demand and unit cost, since

dV/dD = - a/3D -~+ 1) < 0

and V ~ 0 as D ~ ~ and vice versa as required. Our objective is to minimize the average total

annual cost (i.e. the sum of setup, production and inventory carrying costs) which, according to the basic assumptions of the EOQ model, is

C(D, q) = (Total cost per cycle)

× (Average number of cycles per year).

that is

( C(D, q )= S+ V q + ~ q . (2)

Substituting (1) into (2) yields

C ( D , q ) = S D q - I + a D 1 ~+ Hq -2 -" (3)

254 T.C.E. Cheng / Economic order quantity model

Geometric programming solution

Applying GP to solve the minimization prob- lem expressed in (3), we let

U 1 = SDq- 1, (4)

u2 = aD l-is, (5)

Hq (6) u3 = 2 "

Using the fact that the arithmetic mean of a set of non-negative real numbers is greater than or equal to its geometric mean, we write

C( D, q) = gl 1 -I- u 2 .-{- u 3

>/ t a 2 j a3/ = f ( a ) ' (7)

subject to

a I + a 2 + a 3 = 1, a l , a 2 , a 3 > 0 . ( 8 )

In GP terminology C(D, q) and f (a) are known as the primal and pre-dual functions, re- spectively; and (8) is called the normality condi- tion, for example, see Duffin et al. (1967).

Substituting (4), (5) and (6) into (7), we get

f ( a ) = 7 , a2 2a3 J D°'

+ (1 - / 3 ) a 2 q -'~ +"'. (9)

Since the dual variable vector a = (a 1, a2, a 3 )

is arbitrary and can be chosen according to con- venience subject to (8), we chose a such that the exponents of D and q are zero thus making the right hand side of (9) independent of the decision variables. To do this we require

a 1 + (1 - f l ) a 2 = 0 (10)

and

- - a 1 + a 3 = 0 . (11)

These are called the orthogonality conditions (Duffin et al., 1967), which together with (8) are sufficient to determine the values of a~. Solving (8), (10) and (11) for a* yields

a~* = a~' = (fl - 1) / (2f l - 1), (12)

a~' = 1 / ( 2 / 3 - 1). (13)

In view of the positivity constraint (8), a/* > 0 (i = 1, 2, 3) entails/3 > 1, which explains why such

a condition was specified in the previous section. With a* values determined, (7) is reduced to

( S )a~( ot la~[ H I a~ C ( D , q ) > ~ f ( a * ) = a---~ ~,a~] ~ a ~ 1 "

(14)

Since equality is possible for (14),

min C(D, q) = max f ( a * ) = f ( a * ). (15)

This is because f (a * ) is a constant in the present case. Thus the original minimization problem (3) has been reduced to a mere evaluation of the quantity f ( a* ) as expressed in (14), where all a* are determined from (12) and (13).

To find the optimal values of D and q, we proceed as follows. For equality in (7) all ui/a i (i = 1, 2, 3) must be equal. Hence, from (15), for rain C(D, q), all ui/a* (i --- 1, 2, 3) are equal. Let

ui a---~- = k for i = 1, 2, 3. (16)

Then

E u i = k E a * = k , (17) 1~<i~<3 1~<i~<3

and so

rain C( D, q) = f (a* ) = k. (18)

Hence

u i = a * f ( a * ) f o r i = l , 2 , 3 . (19)

Substituting (6) and (12) into (19) and solving for q, we obtain

q* = 2(/3 - 1 ) f ( a * ) / ( H ( 2 f l - 1)). (20)

Similarly, substituting (4), (12) and (20) into (19) and solving for D yields

D* = 2 ( ( f l - 1 ) f ( a * ) / ( 2 f l - 1)}2/HS. (21)

An illustrative example

For a particular EOQ problem, it is estimated that f l - - 2 in (1). Thus the unit production cost assumes the following form,

V= aD -2. (22)

The two decision variables are D and q whose values are to be determined to minimize the aver-

T.C.E. Cheng / Economic order quantity model 255

age annual total cost, that is

min C ( D , q) = SDq -1 + aD -1 + Hq (23) 2

To find the optimal demand rate D* and pro- duction batch quantity q*, we use (12) and (13) to determine a* (i = 1, 2, 3) as follows:

a~* = a~" = a~" = ½. (24)

Thus the corresponding f ( a * ) as determined from (14) has the following value,

It follows from (20) and (21) that the optimal values of production batch quantity and demand rate are respectively

q . = [ 4aS 11/3, (26) -ffT/

D* = ( 2ct2]a/3 (27)

and the minimum average annual total cost is

rain C ( D , q) = C * ( D * , q* ) = f ( a * )

= 3 ( - ~ ) 1/3. (28)

It is noted that the above optimal result can be obtained using calculus by first differentiating (23)

with respect to D and q once, letting the results to equal zero and then solving for D and q. For this simple problem, calculus is able to obtain a closed-form optimal solution. However, for a more involved problem, for example fl = v/3, applying the GP approach not only simplifies the computa- tion but also yields a closed-form optimal solu- tion.

A graph of D* versus q* over a range of fl values is shown in Figure 1. It is interesting to note that both D* and q* decrease with increase in fl, as they should, and D* appears to be gradually approaching a limit for ever increasing values of ft. This observation is intuitively in agreement with the economic law of diminishing returns. Therefore there exists a demand level beyond which further reduction in unit production cost is impossible because productivity cannot be improved further unless more efficient production processes are made available. This is probably the demand level implicitly assumed by the classical EOQ model at which the effect of demand on unit production cost need not be considered.

Sensitivity analysis

In GP theories, the dual variables a* represent the fractions of the minimum cost C* (D*, q* ) accounted for by the respective terms in the objec-

0

3 0 . 0 0

2 8 . 0 0 -

2 8 . 0 0 -

2 4 . 0 0 -

2 2 . 0 0 -

2 0 . 0 0 .

1 8 . 0 0 -

1 6 . 0 0 -

1 4 , 0 0 -

1 2 . 0 0 -

1 0 . 0 0

8 . 0 0

e , o o

4 . 0 0

2 . 0 0

0 , 0 0

0 , 1 0

2 , 0 0

2 .

I l I I I l | l I l I 1 I t l l

0 , 3 0 0 . 5 0 0 . 7 0 0 , 9 0 1 . 1 0 1 . 3 0 1 . 5 0 1 . 7 0

O~o.,,~-) O~m~ EOQ

Figure 1. Demand vs. EOQ

256 T.C.E. Cheng / Economic order quantity model

tive function of the minimization problem (3). Derivation of the a* requires simultaneous solu- tion of the linear equations (8), (12) and (13). Since these equations are independent of the coef- ficients of the total cost function a, H and S, any change in these coefficients will not change the fractions. In fact, a* only changes with the expo- nent /3 in (1). This property makes it very easy to find a new optimal solution for a change in a, H or S. For example, suppose the setup cost S is changed by a per cent. Then the new minimum cost C~(D*, q*)=fa(a*) can be calculated as follows:

f ~ ( a * ) [ ( S ( l + A ) / a ~ ) ( - ~ ) ~ ( ~-~-f )a~] -1

= f ( a * ) a---( t a X I ~2a~

or

fa(a* ) = f ( a * )(1 + A) af . (29)

It is then an easy matter to determine the new optimal production quantity q* and demand D* using (20) and (21), with f (a * ) replaced by fa(a * ) and S by S(1 + a ) .

Occasionally, there will be change in the expo- nent fl of (1). Suppose fl is changed to /3 ' , giving rise to a new set of orthogonality conditions (12) and (13) which, taken with the normality condi- tion (8), are sufficient to determine a new set of dual variables a~,. Substituting this new a~, into the modified pre-dual function fa,(a) yields the new minimum total cost fB,(a~). It follows that the new optimal quantity q~*, and optimal demand D~ can simply be determined with the aid of (20) and (21), which should also have been modified to account for the new exponent factor fl ' . I t is noted that the above sensitivity analysis is similar to the one presented in a recent paper by Cheng (1989) to deal with another related inventory control problem.

Conclusions

In this paper we first propose a general equa- tion to model the relationship between unit pro- duction cost and demand rate. We then formulate the EOQ problem as a GP and apply the theories of GP to help derive a closed-form optimal solu- tion. We demonstrate through a numerical exam- ple the power and efficiency of GP as an analyti- cal tool for the analysis of certain types of inven- tory control problems.

Acknowledgements

The author wishes to thank two anonymous referees for their many constructive comments. This research was partially supported by a grant from the University of Manitoba Faculty of Management Associate Funds.

References

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Cheng, T.C.E. (1989), "An economic production quantity model with flexibility and reliability considerations", European Journal of Operational Research 39, 174-179.

Duffin, R.J., Peterson, E.L., and Zener, C. (1967), Geometric Programming, Wiley, New York.

Hadley, G., and Whitin, T.M. (1963), Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ.

Silver, E.A., and Peterson, R. (1985), Decision Systems for Inventory Management and Production Planning, Wiley, New York.

Urgeletti Tinarelli, G. (1983), "Inventory control: models and problems", European Journal of Operational Research 14, 1-12.

Veinott, Jr., A.F. (1966), "The status of mathematical inven- tory theory", Management Science 12, 745-777.

Whitin, T.M. (1954), "Inventory control research: a survey", Management Science 1, 32-40.