A NEW ANALYSIS OF A LOT-SIZE MODEL WITH PARTIAL BACKLOGGING

David Rosenberg

School of Business Administration Old Dominion Universiry

~ o ~ ~ l k , Virginia

ABSTRACT

We reformulate the cost equation for the lot-size model with partial back- logging. The formulation is in terms o f "fictitious demand rate," a new invento- ry decision variable that simplifies the analysis. Using decomposition by projec- tion, we obtain an optimal solution in a straightforward manner. The form o f the solution sheds addifional light on the behavior of the model. Some o f these insights are elucidated by numerical examples.

INTRODUCTION

Most works in inventory theory on infinite-time-horizon lot-size models have been con- cerned with the extreme cases, wherein all demand occurring during stockout is backlogged or not backlogged (lost sales). The abundance of such models contrasts sharply with the scarcity of inventory models that consider the hybrid situation of partial backlogging. A realistic appli- cation of the partial-backlogging concept is in the demand for spare parts. Intuitively, it seems reasonable to assume that, during stockout, critical needs will be satisfied from other sources and less urgent needs will be met by backordered items.

The scant inventory literature on infinite-tim~-horizon lot-size models with partial back- logging contains only one such model for which an optimal solution has been obtained ([ l ] , pp. 256-259). The purpose of this paper is to show that the analysis of this solved model and the mathematical form of the resuiting optimal solution can be greatly s impli~ed. This simplification results from reformulating the model in terms of a new inventory decisian vari- able, which we term the "fictitious demand rate" (FDR). The decision variable FDR is a crucial modeling concept in the analytical study of monopoly price - inventory models [21. An important byproduct of the simplified analysis of partial backlogging is a new economic interpre- tation of the circumstances under which this operating doctrine is optimal.

The remaining sections of the paper review the optimization procedure of Montgomery, Bazaraa, and Keswani (MBK) i l l , develop the revised version of the infinite-time-horizon lot- size model with partial backlogging, establish an optimal policy, discuss and interpret the policy, and give numerical illustrations.

The following variable-cost-rate model for a lot-size inventory system with partial backlog- ging is given in (111 p. 256):

349

350 D. ROSENBERG

( la)

where

D = Q = c = I = A = s - 7 F = r = I

To = b =

demand rate ( D R 1 , D > 0, order quantity, Q > 0, unit cost, C > 0, carrying rate as a per cent of unit cost, I > 0, cost per order, A > 0, stockout demand per inventory review cycle, S > 0, unit-shortage penalty, 7r 2 0, backorder unit-shortage cost rate, if 2 0, unit profit, ro 2 0, fraction of stockou~-demand backordered, 0 < b < 1.

The nonconvexity of ( la) motivated MBK to create the ad hor optimization procedure which we proceed to outline. By use of the nonsingular transforma~ion

U = Q + S ( l - 6 )

V = Q - b S

model (1 a) became AD +- (lb) K ( U , V ) = - U

ICY' + r D ( U - Y ) + Z ( U - V)' + r o D ( 1 b ) ( U - V ) U 211 U 2u

Observing the nonconvexity of ( lb ) , MBK applied the projection concept. To facilitate the use of projection, the transformation

was applied to ( lb) , which became

Then decomposition by projection was applied sequentially in the following manner:

The outer minimization, unfortunately, was not straightforward. This optimization difficulty is circumvented by our proposed reformulation which yields a sequence of strictly convex sub- probtems.

REFORMULATION

Model (.la) is based on the assumption of a uniform demand rate, which implies that the following formula for inventory review cycle length T holds:

T [ Q + (1 - b ) S ] / D .

For exuositorv convenience we rewrite model ( la) in terms of 7:

LOT-SIZE MODEL WITH PARTIAL BACKLOGGING 351

The analysis will be greatly simplified if we introduce the new inventory decision variable, fictitious demand rate X, which we define by means of the following transformation:

X = (Q - bS)/T.

Substituting X in the appropriate piaces in model ( lc) , we obtain the following reformulation:

In Figure 1 we give the geometry of partial backlogging and indicate the key decision vari- ables the reader has encountered up to this point. From a geometric standpoint we are model- ing the stockout level as the product of the inventory review cycle length T and the term (DR - FDR), which is the difference between the actual demand rate and an artificial demand rate.

INVENTORY LEVEL

T- I (DR - FDR)T

\ \ \

\ \ L \

Ir T 4-- FIGURE 1. The inventory ~ u ~ ~ u ~ ! ~ o n s in a lo!-size model with Partial Bdcklogging

Preparatory to seeking an optimal solution, we write model (2a) in the following equivalent form,

2 (2b) C ( X , T ) = x + [T + ~ o ( 1 - 6)ID.

We shall assume that model (2) is twice continuously differentiable, since this validates the subsequent use of the projection technique for obtaining an optimal solution.

352 D. ROSENBERG

OPTIMAL SOLUTION

We obtain an optimal solution by applying decomposition by projection in the following stepwise manner:

C ( X " , T*) = min min C ( X , T ) = min C ( X , T I , T X X , T

For fixed positive T, (2b) is strictly convex in A'. Hence, the optimal fictitious demand rate X*( T ) is easily obtained from model (2b),

( 3 a ) X * ( T ) = 1562' + T + ~ o ( l - b)ID/[(IC + 7Tb) TI,

(3b)

We now substitute X*( T) into model (2b), which yields the following expression in T :

if T > IT + vo(f - b)J/ZC.

(4) C[X*(T). TI = ( A - D [ T + ~ o ( 1 - b)12/2('irb + IC))T-' + ?FbD[ZC/(?b + IC)IT/2 + [T -I- ~ o ( 1 - b)I[ZC/(?Fb + ZC)lD.

Model (4) is strictly convex in .T. Thus the optimal inventory-review-cycle length T" is easily obtained from the above expression and is given below:

( 5 ) I f 2

T" = ( { 2 , 4 ( Z b + ZC) - [a + ao(l - b)12D}/F~ZCDl .

We now obtain the minimum cost rate solution C(X*, T") by substituting T" for T i n model (4), ( 6 ) C(X*, T*) = {5bZCD[2 ,4 (~b + IC) - [T + ~ o ( 1 - b)12DI}1f2 / (Fb + ZC)

+ [T + a00 - b)][ZC/(?Fb + ZC)lD.

If we now substitute T* for T i n inequality (3b), we get, after simplifying, the following inequality:

( 7 )

The above inequality is equivalent to its counterpart (ill, p. 259), but here has been put into the more meaningful form.

The optimal policy can now be stated with the above inequality serving as a test criterion. Satisfaction of ( 7 ) indicates that a partial backlogging solution is optimal; otherwise, the solu- tion of the classical lot-size model is optimal.

In words, the left side of inequality ( 7 ) is the classical infinite-time-horizon formula for the optimal interval between replenishments. Therefore, we can describe the optimal policy in a more informative way. Whenever the optimal inventory-review-cycle length computed by the formula derived for the classical lot size model exceeds the test criterion on the right side of (71, partial backlogging is optimal. Otherwise, the classical lot-size operating doctrine should be employed.

DISCUSSION

By examining in detail the test criterion on the right side of in~quality ( 7 ) we will gain insight concerning the circumstances under which the optimal policy specifies partial backlog- ging. We begin by observing that the numerator of the test criterion ratio contains parameters

LOT-SIZE M O D E L WITH P A R T I A L B A C K L O G G I N G 353

that appear nowhere else in the inequality. This implies that, for any given value of the left side, under suitable conditions, namely, the value of the numerator of the right side, the right side can exceed the left side.

The terms comprising the numera~or have an economic ~hara~terization, They represent penalty costs incurred when a demanded item is not inventoried. Thus the economic content of inequality (7) is that, whenever the penalty for not inventorying demanded items is rela- tively small, then partial backlogging is a viable operating doctrine. In the numerical examples given below, this economic result is illustrated. We will demonstrate the fact that the value of the right side of (7) can be varied to the extent that the optimal operating doctrine will be changed by variation in the unit profit.

NUMERICAL EXAMPLES

We now introduce numerical cases to elucidate the concepts. Both cases have the follow- ing data in common:

D = 250, C = 10, I = 0.2, A = 1 0 , r r=O.2 , ?f-0.1, b = O . 5 ,

The above omp put at ion implies that the partial backlogging operating doctrine is optimal. The optimal replenishment interval is

1/2 6) T+ = [{2(10)[0.1(0.5) + 0.2(10>1 - 10.2 + 0.2(1 - 0.5~12~250~}/0.1~0.5~~0.2~~10~~2SO~] = 0.86.

The above computation implies that the classical lot-size operating doctrine is optimal. The optimal replenishment interval is obviously 0.2.

REFERENCES

I . Montgomery, D.C., M.S. Bazaraa, and A.K. Keswani, "Inventory Models with a Mixture of

2. Rosenberg, D. " ~ o n o p o l y Inventory Models," Doctoral Dissertation, New York University, Backorders and Lost Sales," Naval Research Logistics Quarterly, 20, 255-63 (1973).

New York, N.Y. (1977).