Comment on Bose S, Goswami A and Chaudhuri KS (1995). An EOQ model for deteriorating items with linear time-dependent de
- PDF / 91,797 Bytes
- 4 Pages / 595 x 842 pts (A4) Page_size
- 25 Downloads / 183 Views
#2001 Operational Research Society Ltd. All rights reserved. 0160-5682/01 $15.00 www.palgrave-journals.com/jors
Viewpoint Comment on Bose S, Goswami A and Chaudhuri KS (1995). An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting In an article published in JORS, Bose et al1 considered the inventory replenishment problem for a deteriorating item with linear (positive) trend in demand, finite shortage cost and equal replenishment intervals over a fixed planning horizon under inflation and time discounting. They provided the procedure of finding the optimal solution of the problem and also studied the sensitivity of the decision variables to changes in the parameter values of the model. Unfortunately, their model contains mathematical errors in the formulation of the holding cost and the purchase cost which lead to incorrect total cost function over the fixed planning horizon. In this Viewpoint, we point out the errors and present the appropriate theory for the problem. Here we use the following assumptions and notation, most of which are similar to Bose et al:1 (a) H is taken to be the fixed time horizon. (b) n orders are placed during the time horizon H and the replenishment rate is infinite, ie replenishment is instantaneous. (c) The demand f (t) at time t is a continuous function of time. (d) A constant fraction y(0 < y < 1) of on-hand inventory deteriorates per unit of time. (e) r is the discount rate representing the time value of money. (f) At time t ¼ 0, c11 and c12 are internal and external holding costs per unit item per unit time; c21 and c22 are internal and external shortage costs per unit item per unit time. (g) At time t ¼ 0, A is the fixed internal ordering cost per order and p is the external purchase cost. (h) Internal and external inflation rates are denoted by i1 and i2 respectively. (i) Co, Ch, Cs and Cp are the present worth of total replenishment cost, total inventory holding cost, total shortage cost and total purchase cost during the fixed time horizon H, respectively. ( j) K is the fraction of the replenishment interval for which there is no shortage and K is same for each replenishment cycle. (k) tj ¼ ( j 7 1)H=n is the time of the jth replenishment, j ¼ 1, 2, . . . , n.
(l)
(m) (n) (o) (p)
sj ¼ tj þ KH=n ¼ (K þ j 7 1)H=n is the time at which the inventory level in the jth replenishment cycle drops to zero, j ¼ 1, 2, . . . , nÿ 1 and sn ¼ H. I1(t) is the inventory level at any time t in [tj, sj], j ¼ 1, 2, . . . , n. I2(t) is the shortage level at any time t in [sj, tj þ 1], j ¼ 1, 2, . . . , n 7 1. Shortages are allowed and are fully backlogged. Shortages are not allowed in the last replenishment cycle.
A typical variation of the inventory level with time for increasing demand pattern are shown in Figure 1. During the period of positive inventory, the differential equation describing the instantaneous state of I1(t) is given by dI1 ðtÞ ¼ ÿf ðtÞ ÿ yI1 ðtÞ dt
tj 4 t 4 sj ; j ¼ 1; 2; . . . ; n ð1Þ
with the boundary c
Data Loading...
