Inventory decisions in product-mix models

03,[EGA, The Int. J1 of M g m t Sci., Vol. 4, No. 3, 1976. P e r g a m o n Press. Printed in Great Britain

Memoranda Inventory Decisions in Product-Mix Models A a R o D U CT-mix model is one in which scarce resource inputs are transformed into an

where Kj and hj are the order and holding costs, respectively; the economic order quantity which minimizes I C j is

optimal product-mix of outputs. A common

/(2.~1.~, 3 hj /"

vehicle for solving such models is linear prograrnming. In many instances, various combinations of resource inputs (right hand side coefficients) and/or product outputs (optimal values of the decision variables) are subject to inventory decisions. However, economic order quantities for the inputs and economic production quantities for the outputs are typically determined outside the context of the product-mix model. Separate consideration of interrelated decisions can lead to suboptimal results. The purpose of this note is to formulate several models which integrate product-mix and inventory decisions and to apply an existing algorithm to their solution. A retail sales-mix model might be stated

QJ = d \

I C j = Oj.xl * u 2 [where 01 = a/(2.Kj.hj)]. (5)

Incorporating the nonlinear inventory costs (5) into the objective function (2) results in: Maximizef(x) = Z' (cjxj

as:

Maximizer(x) = .~' c j x j

(1)

subject to,E, a u x J = b~ (i = 1 ..... m)

(2)

where Z' stands for z~ and where the decision 1-1

variables xj >_ 0 (_/= 1..... n) represent the number of units of product ] to sell over a given time period. The model coefficients, the cj's, bt's and au's, represent the per unit profit, the initial amount of resource i available, and the amount of resource i consumed per unit of product j, respectively. Once the optimal values of the decision variables in the linear programming model, the xj*'s, have been determined, the retailer is faced with the decision of how often and in what quantities to order each of the products from the manufacturer. Using standard economic order quantity analysis, the inventory cost for ordering Q~ is given by: IG = Kjxj* OMEGA

4/' ~-"-'G

+

hj Q j

--5-"

(4)

so that

Ix = ( i l i ~ Iz=

r,i=

-

1 ..... r}

{i[i> r , i = r + l

0#..rjt/2) (6)

]

..... m} jk

(7)

and replace bt with a new decision variable x.+~ for i E I1. The corresponding integrated model would be: Maximizef(x) = ,Sc.txl -- Z O,,+lx.+t ttz (8) i~:I1

subject to Z a u x j -- x , + t = 0 2~ a u x j

(3)

-

subject to (2). This model is very similar to one proposed by Raiborn and Harris [2]. Next, consider the problem faced by a manufacturer, where some combination of resource inputs, b~'s, might be subject to inventory decisions, as were the outputs of the retail model. It is possible that by treating these inputs as decision variables and integrating their nonlinear inventory costs into the product-mix objective function, he may be able to achieve a greater total profit than by considering product-mix and inventory decisions separately. Without loss of generality, let us assume that we wish to integrate inventory decisions concerning the first r resource inputs, 0 < r < m, into the product-mix model and adopt the notation

and again .rj ___0.

339

= bt

i~11

is 12

} (9)

Memoranda When a combination of both inputs and outputs is subject to inventory decisions, the follov,ing integrated model might be appropriate : Maximize f(x) = Z [cjxj

-

-

Ojxj t/2 ]

- X O,.,++x,,+,t/+' (10) i+It subject to (9). Since the objective functions of (6), (8) and (10) are convex, global maximum solutions occur at extreme points of their respective solution spaces. However, there may be local maxima different from the global maximum. Inventory setup and order costs can be thought of as fixed charges. If every decision variable is subject to an inventory setup or order cost, then every extreme point of the solution space is a local maximum. As would be expected, this fact greatly complicates the problem of determining a global solution. Soland [3] has proposed a finite branch and bound algorithm for the global maximization of a separable convex function over linear polyhedra. It is a simplified version of an earlier algorithm by Falk and Soland [1] and can be applied to problems described here.

340

An advantage of an integrated model is that it represents a closer, more realistic approximation to the actual problem situation. A potential disadvantage is that the effort required for its solution tends to be greater than the sum of the efforts required to solve separate product-mix and inventory models. REFERENCES 1. FALK JE and SOLAND RM (1969) An algorithm for separable nonconvex programming problems. Mgmt Sci. 15 (9). 2. RAtBORN MH and HAP,aXS WT (1974) Integration of inventory and product sales-mix models. Decis. Sci. 5 (4). 3. SOLAND R M (1974) Optimal facility location with concave costs. Ops. Res. 22 (2). GR Reeves

(February 1976) Assistant Professor School of Business Administration :VIiami University Oxford, Ohio 45056 USA