Inventory of damagable items with variable replenishment rate, stock-dependent demand and some units in hand

Applied Mathematical Modelling 23 (1999) 799±807

www.elsevier.nl/locate/apm

Inventory of damagable items with variable replenishment rate, stock-dependent demand and some units in hand M. Mandal a, M. Maiti a

b,*

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, West Bengal, India b Department of Mathematics, Midnapore College, Midnapore 721 101, West Bengal, India Received 26 May 1998; received in revised form 30 March 1999; accepted 16 April 1999

Abstract Items made of glass, ceramics, etc., break/get damaged during the storage due to the accumulated stress of heaped stock. For the ®rst time, a deterministic inventory model of such a damagable item is developed with variable replenishment when both demand and damage rates are stock-dependent in polynomial form. Here replenishment rate for the ®rst cycle is partly instantaneous and partly varies with demand. For the next cycle, the variable replenishment is augmented when the inventory level falls to Q0 , the instantaneous replenishment amount for the ®rst cycle. After this, the cycle repeats itself. The amount, Q0 is also here varied and the optimum Q0 and Q (inventory level) are evaluated following the pro®t maximization principle in integral form. The model is illustrated numerically and sensitivity analyses are presented. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Inventory; Polynomial demand; Variable replenishment; Stock-dependent demand; Damagable items

1. Introduction For classical inventory models, normally the rate of replenishment/production is assumed to be constant. In practice, it is observed that the replenishment/production is in¯uenced by the demand. This situation generally arises in the case of inventories of highly demandable products. If the demand goes up, consumption obviously will be more and to meet the extra requirement, a retailer increases the replenishment/production rate of the item. On the other hand, if the demand of a commodity goes down, the retailer takes steps to avoid unnecessary inventory and cuts down the replenishment/production rate. Although the inventory policy with in®nite and ®nite replenishment/production rate is available in the text books of Hadley and Whitin [1], Naddor [2], Arrow and Kerlin [3] and others, a very few O.R. scientists (Goswami and Chaudhuri [4], Bhunia and Maiti [5], Balkhi and Benkherouf [6], etc.) attempted the inventory models with variable replenishment/production rate. Goswami and Chaudhuri [4] considered inventory of deteriorating items with ®nite production rate proportional to the time dependent demand rate as K…t† ˆ bD…t†.

*

Corresponding author. Fax: +91 032 266 2329

0307-904X/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 9 9 ) 0 0 0 1 8 - 9

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Bhunia and Maiti [5] developed two inventory models ± one with replenishment rate as on-hand inventory dependent i.e., K…t† ˆ a ÿ bq…t† and other with demand-dependent replenishment rate i.e., K…t† ˆ a ‡ bD…t†, where K…t† is the replenishment rate, D…t† the demand rate, q…t† stock level at time t and a; b being constants. Balkhi and Benkherouf [6] allowed the production to vary with time in their inventory model of deteriorating items. Till now, none has attempted the inventory problem of variable replenishment with stock-dependent demand. In this paper, an attempt has been made to consider the inventory model with both variable replenishment and stock-dependent demand. During inventory, deterioration of goods in the form of direct spoilage or gradually physical decay in course of time is a realistic phenomenon and hence, it can not be ignored in inventory modelling. Normally, rate of deterioration is taken as either constant or a function of time, t. Mandal and Phaujdar [7], Datta and Pal [8] and others have considered the deterioration of items along with the stock-dependent demand. But, in real life, there are some items, mainly made of glass, ceramics, china-clay, etc., which break/get damaged during the storage due to the accumulated stress of stocked items. Till now, none except Mandal and Maiti [9] has considered the inventory of items made of these materials. Damageability of this type of items has been taken into account in this paper. According to Whitin [10], ``for retail stores, the inventory control problem for stylish goods is further complicated by the fact that inventory and sales are not independent of one another. An increase in inventories may bring about increased sales of some items''. Wolfe [11] presented empirical evidence in favour of it. Levin et al [12] noted that ``it is common belief that large piles of goods displayed in a supermarket will lead the customers to buy more''. Now-a-days, marketing research also recognizes this relationship. Due to this fact, O.R. scientists recently have concentrated on the inventory problems taking the e€ect of displayed inventories on demand into account. Normally, there are two types of stock-dependent demand. These are (i) D…t† ˆ a ‡ bq…t† (linear form) and (ii) D…t† ˆ aq…t†b (polynomial form). Mandal and Phaujdar [7], Datta and Pal [8] and others have solved the inventory problems with linear form of stock-dependent demand whereas Urban [13], Pal et al. [14,15] and others attempted the problems with polynomial form of on-hand inventory dependent demand. Till now, none has formulated the inventory model with both ®nite replenishment and polynomial form of stock-dependent demand. Here, variable replenishment has been assumed for the formulation of the present inventory model with stock-dependent demand in polynomial form from which the ®nite replenishment case is derived as a particular case. In this paper, an inventory model with demand-dependent replenishment/production and stock-dependent consumption rate is formulated for breakable/damagable items which get damaged due to the accumulated stress of heaped stocks during storage. Although the accumulated stress is a function of on-hand inventory and time, for simplicity, it has been assumed that the number of damaged items depends only on the current stock level. This dependency may be both linear and non-linear. Here, at the beginning of the ®rst cycle, an amount of units, Q0 (say) is procured instantaneously and after that, replenishment/production rate is ®nite and varies linearly with demand. At the end of the ®rst cycle, reorder is placed i.e., varied replenishment/production starts when Q0 amount of units is in hand. Then, this process is continued for all cycles (cf. Fig. 1). In this formulation, demand is stock-dependent in polynomial form and the on-hand units, Q0 is considered to be a variable. We solve the model with the help of pro®t maximization principle in integral form and the optimum inventory and ordering point are determined for maximum average pro®t. Model is illustrated with numerical values.

M. Mandal, M. Maiti / Appl. Math. Modelling 23 (1999) 799±807

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Fig. 1.

2. Model formulation The following notations and assumptions are used in developing the model. 2.1. Notations Q C1 C3 Q0 q…t† p m T s B…q† D…q† K Z…Q; Q0 †

optimal inventory level holding cost per unit quantity per unit time set-up cost per cycle ordering point inventory level at time, t purchasing price per unit item mark-up price duration of each cycle length ˆ t1 ‡ t2 (refer Fig. 1) selling price ˆ mp (m > 1, m being the mark-up price) number of damaged units per unit time at time t and is a function of current stock level, q ( P Q0 ) consumption rate per unit time per unit quantity ˆ aqb ; a > 0; 0 6 b 6 1 replenishment=production rate ˆ b ‡ cD…q† pro®t per unit time

2.2. Assumptions For the present model, it is assumed that (i) replenishment of stock is demand dependent, (ii) the time horizon of the inventory system is in®nite,

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(iii) demand D…q† is dependent on the current stock level and is of polynomial form i.e., D…q† ˆ aqb ; a > 0; 0 6 b 6 1; (iv) at the beginning of the ®rst cycle, a replenishment of Q0 units is made instantaneously. After that, further replenishment is made at variable rate, K…D† where K ˆ b ‡ cD…q†; b; c P 0, (v) towards the end of each cycle, order is placed when the stock of Q0 units remains at the storage, (vi) breakable units B…q† is a known function of current stock level and B…q† ˆ aqc ; 0 < a < 1; 0 6 c 6 1 and (vii) lead time is zero. 3. Mathematical formulation In this model the inventory level gradually decreases mainly to meet demand and partly for damage. Here the inventory level, q…t† at the beginning …t ˆ 0† and end …t ˆ T † of the cycle is Q0 which is also a variable. Therefore, if q…t† be the inventory level at time, t, we have  dq K ÿ D…q† ÿ B…q†; 0 6 t < 6 t1 ; …1† ˆ ÿD…q† ÿ B…q†; t1 6 t 6 t1 ‡ t2 ; dt where at t ˆ 0, and t ˆ T ; q…t† ˆ Q0 . The time length of each cycle is ZQ T ˆ Q0

dq ‡ k ÿ D…q† ÿ B…q†

ZQ Q0

dq : D…q† ‡ B…q†

…2†

The holding cost for each cycle is given by C1 G…Q; Q0 † ‡ C1 Q0 T , where ZQ G…Q; Q0 † ˆ Q0

q dq ‡ K ÿ D…q† ÿ B…q†

ZQ Q0

q dq: D…q† ‡ B…q†

…3†

B…q† dq: D…q† ‡ B…q†

…4†

The total damaged units are ZQ h…Q; Q0 † ˆ Q0

B…q† dq ‡ K ÿ D…q† ÿ B…q†

ZQ Q0

Total replenishment/produced units are ZQ w…Q; Q0 † ˆ Q0 ‡

K…q† dq:

…5†

Q0

Net revenue from the ®rst cycle is N …Q; Q0 † ˆ …s ÿ p†w ÿ sh ÿ pQ0 :

…6†

Hence, the average pro®t during T is given by Z…Q; Q0 † ˆ ‰N ÿ C3 ÿ C1 G ÿ C1 Q0 T Š=T :

…7†

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Now the problem is reduced to maximize the average pro®t function Z…Q; Q0 ) and to ®nd the optimal values of Q and Q0 which are the solutions of oZ=oQ ˆ 0 and oZ=oQ0 ˆ 0; provided o2 Z=oQ2 < 0; o2 Z=oQ20 < 0 and  2 2 o2 Z o2 Z oZ ÿ > 0: oQ2 oQ20 oQoQ0 oZ=oQ ˆ 0 and oZ=oQ0 ˆ 0 lead respectively to     oN oT oG oT oT T ÿN ÿ C1 T ÿG ‡ C3 ˆ0 oQ oQ oQ oQ oQ and

    oN oT oG oT oT ÿN ÿ C1 T ÿG ‡ C3 ˆ 0: T oQ0 oQ0 oQ0 oQ0 oQ0

…8†

…9†

We solve the non-linear Eqs. (8) and (9) by the Newton±Raphson method for optimal Q and Q0 and the corresponding optimal pro®t for particular demand and damage functions.

4. Numerical results The model is numerically illustrated with following values of inventory parameters for some special cases. C3 ˆ $100;

C1 ˆ 0:1p;

p ˆ $8;

s ˆ mp:

4.1. Variable replenishment models 4.1.1. Model 1 Both demand and damage functions are of linear form. Let D…q† ˆ aq and B…q† ˆ aq; a > 0 and 0 < a < 1. From Eqs. (2)±(5), we have T ˆ t1 ‡ t2 ; t2 ˆ

t1 ˆ

ÿ1 b ÿ …a ‡ …1 ÿ c†a†Q ; log a ‡ …1 ÿ c†a b ÿ …a ‡ …1 ÿ c†a†Q0

1 Q ; log a‡a Q0

G ˆ I1 ‡ I2 ;

where I1 ˆ

Q ÿ Q0 bt1 ‡ a…1 ÿ b† ‡ a a…1 ÿ b† ‡ a

I2 ˆ

Q ÿ Q0 ; a‡a

and h ˆ aG;

w ˆ bt1 ‡ caI1 ‡ Q0 :

With these expressions and above numerical values, we solve Eqs. (8) and (9) with the parametric values, b ˆ 180; a ˆ 3; c ˆ 0:5; a ˆ 0:1 and m ˆ 1:15. The required optimal values are Z…Q ; Q0 † ˆ $217:83; Q ˆ 112:50 units, Q0 ˆ 6:00 units and T  ˆ 5:63 units.

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4.1.2. Model 2 Demand is linear and damage function is non-linear. Let p D…q† ˆ aq; B…q† ˆ a q; a > 0; 0 < a < 1: Proceeding as in Model 1, we obtain T ˆ t1 ‡ t2 ;

p 1 b ÿ a…1 ÿ c†Q ÿ a Q p t1 ˆ log a…1 ÿ c† b ÿ a…1 ÿ c†Q0 ÿ a Q0 a …Y1 ‡ X1 †…Y1 ÿ X2 † p log ÿ ; 2 …Y1 ÿ X1 †…Y1 ‡ X2 † …1 ÿ c†a 4b…1 ÿ c†a ‡ a p p 2 a Q‡a p  ; Y1 ˆ 4b…1 ÿ c†a ‡ a2 ; t2 ˆ log a a Q0 ‡ a p p X1 ˆ 2…1 ÿ c†a Q ‡ a; X2 ˆ 2…1 ÿ c†a Q0 ‡ a; G ˆ I1 ‡ I2 ; p p p bt1 ÿ Q ÿ Q0 b ÿ a2 Q ÿ a Q Q ÿ Q0 a 2 p ÿ 2a ‡ log I1 ˆ a22 a2 a32 b ÿ a2 Q0 ÿ a Q0 a…a2 ‡ 2ba2 † …Y1 ‡ X1 †…Y1 ÿ X2 † ; ‡ 3 p2 log …Y1 ÿ X1 †…Y1 ‡ X2 † a2 4a2 b ‡ a p p p c2 ‡ Q 2 p ; I2 ˆ …Q ÿ Q0 † ÿ c2 Q ÿ Q0 ÿ c2 log c2 ‡ Q0 p p 2a… Q ÿ Q0 † 2ab …Y1 ‡ X1 †…Y1 ÿ X2 † a2 ‡ 2 p2 log ÿ t1 ; hˆÿ …Y1 ÿ X1 †…Y1 ‡ X2 † a2 a2 a2 4a2 b ‡ a w ˆ Q0 ‡ b1 t1 ‡ baI1 ; where a2 ˆ …1 ÿ c†a;

a c2 ˆ : a

Solving Eqs. (8) and (9) with these expressions and b ˆ 180; m ˆ 1:15; a ˆ 0:1; a ˆ 3; c ˆ 0:5, we get the optimal values: Z…Q ; Q0 † ˆ $50:66; Q ˆ 119:27 units, Q0 ˆ 1:09 units, T  ˆ 8:45 units. 4.1.3. Model 3 Demand is non-linear and damage function is linear. Let p D…q† ˆ a q; B…q† ˆ aq; a > 0; 0 < a < 1: Proceeding as above, we get T ˆ t1 ‡ t 2 ;

p 1 b ÿ …1 ÿ c†aQ ÿ a Q p t1 ˆ ÿ log …1 ÿ c†a b ÿ …1 ÿ c†aQ0 ÿ a Q0 a …Y1 ‡ X1 †…Y1 ÿ X2 † p log ÿ ; 2 …Y1 ÿ X1 †…Y1 ‡ X2 † …1 ÿ c†a 4b…1 ÿ c†a ÿ a

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p p p 2 a‡a Q p ; Y1 ˆ 4b…1 ÿ c†a ÿ a2 ; X1 ˆ 2…1 ÿ c†a Q ‡ a; t2 ˆ log a a ‡ a Q0 p a X2 ˆ 2…1 ÿ c†a Q0 ‡ a; G ˆ I1 ‡ I2 ; C2 ˆ ; a p p p 2 bt1 ÿ Q ‡ Q0 2a… Q ÿ Q0 † a b ÿ a…1 ÿ c†Q ÿ a Q p ‡ log ÿ I1 ˆ 2 2 …1 ÿ c†a b ÿ …1 ÿ c†aQ0 ÿ a Q0 …1 ÿ c† a3 …1 ÿ c† a2 a…a2 ‡ 2ba…1 ÿ c†† …Y1 ‡ X1 †…Y1 ÿ X2 † p log ; 3 …Y1 ÿ X1 †…Y1 ‡ X2 † a3 …1 ÿ c† 4a…1 ÿ C†k ‡ a2 p p p C1 ‡ Q 2 p ; I2 ˆ …Q ÿ Q0 † ÿ C2 … Q ÿ Q0 † ÿ C2 log C1 ‡ Q0 p p ÿ2a… Q ÿ Q0 † 2ba …Y1 ‡ X1 †…Y1 ÿ X2 † t 1 a2 p   log ÿ ; hˆ ‡ …Y1 ÿ X1 †…Y1 ‡ X2 † a…1 ÿ c† …1 ÿ c†a …1 ÿ c†2 a2 4…1 ÿ c†ab ‡ a2 ‡

w ˆ Q0 ‡ b1 t1 ‡ baI1 ; with b ˆ 100; a ˆ 0:3; a ˆ 1:5; m ˆ 2; c ˆ 0:3. Eqs. (8) and (9) are solved and optimal values are Z…Q ; Q0 † ˆ $128:82, Q0 ˆ 11:25 units and Q ˆ 261:82 units, T  ˆ 14:91. 4.1.4. Model 4 Both demand and damage functions are non-linear. p p Let D…q† ˆ a q; B…q† ˆ a q; a > 0; 0 < a < 1. Proceeding as above, we get T ˆ t1 ‡ t 2 ; p p p 2… Q ÿ Q0 † 2b b ÿ …a ‡ …1 ÿ c†a† Q  ; p t1 ˆ log ÿ a ‡ …1 ÿ c†a b ÿ …a ‡ …1 ÿ c†a† Q0 …a ‡ …1 ÿ c†a†2   3=2 2 Q3=2 ÿ Q0 ; G ˆ I1 ‡ I2 ; t2 ˆ 3…a ‡ a† " p # p p ÿ2 Q3=2 ÿ Q03=2 …Q ÿ Q0 †C2 Cÿ Q 2 3 p ; I1 ˆ ‡ ‡ C2 Q ÿ Q0 ‡ C2 log 3 2 a ‡ …1 ÿ c†a C ÿ Q0   3=2 3=2 2 Q ÿ Q 0 b ; I2 ˆ C2 ˆ ; a ‡ …1 ÿ c†a 3…a ‡ …1 ÿ c†a† p p a…Q ÿ Q0 † bat1 a… Q ÿ Q0 † ; I4 ˆ ‡ h ˆ I3 ‡ I4 ; I3 ˆ ; a ‡ …1 ÿ c†a a ‡ …1 ÿ c†a a ‡ …1 ÿ c†a w ˆ bt1 ‡

bct1 a ba…Q ÿ Q0 † ÿ : a ‡ …1 ÿ c†a a ‡ …1 ÿ c†a

Now, we solve Eqs. (8) and (9) with the above expressions and b ˆ 150; a ˆ 3; c ˆ 0:5; r ˆ 0:15; a ˆ 0:1 and get Z…Q ; Q0 † ˆ $133:10; Q0 ˆ 0:05 units, Q ˆ 65:84 units, T ˆ 12:05.

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4.2. Finite replenishment models As a particular case, the optimal values for ®nite replenishment models are obtained from the above expression putting C ˆ 0. The results are: Model 1: Z…Q ; Q0 † ˆ $69:03; Q ˆ 58:06 units, Q0 ˆ 1:81 units, T ˆ 5:35. Model 2: Z…Q ; Q0 † ˆ $29:99; Q ˆ 59:74 units, Q0 ˆ 0:78 units, T ˆ 5:20. Model 3: Z…Q ; Q0 † ˆ $24:13; Q ˆ 217:19 units, Q0 ˆ 1:03 units, T ˆ 13:19. Model 4: Z…Q ; Q0 † ˆ $69:03; Q ˆ 56:07 units, Q0 ˆ 0:007 units, T ˆ 16:25. 5. Conclusion With the development of information technology, transport system, distribution arrangement, infrastructural facilities, etc., inventory practitioners in developed countries like Japan, USA, UK, Canada, etc., now-a-days, use simpler inventory control systems like just-in-time, heuristic approach, etc., rather than the complex inventory systems like lot size reorder point model, etc. But, in the developing countries or third world countries, the above mentioned facilities are either developing or yet to be developed. For this reason, in these countries, inventory practitioners follow either the age old crude method like ABC analysis or the usual complex inventory models with necessary modi®cations. Therefore, present inventory models in which the production rate depends upon the demand whereas demand and damage functions are stock-dependent of polynomial forms will be much useful for the second category of inventory practitioners. Here, the ordering point i.e. stock at hand at the time of placing order is also taken as variable. These models can be extended to include fully or partially backlogged shortages, inventory dependent production rate, ®xed time horizon, etc. Although the models presented here have been formulated in crisp environment, these may also be considered in fuzzy and probabilistic environments. References [1] G. Hadley, T.M. Whitin, Analysis of Inventory systems, prentice-Hall, Englewood Cli€s, NJ, 1963. [2] E. Naddor, Inventory Systems, Wiley, New York, 1966. [3] K. Arrow, J. Karlin, H. Scarf (Eds.), Studies in the Mathematical Theory of Inventory and Production, Stanford, 1958. [4] A. Goswami, K.S. Chaudhuri, An EOQ model for deteriorating items with shortages and linear trend in demand, J. Opl. Res. Soc. 42 (1991) 1105±1110. [5] A. Bhunia, M. Maiti, Deterministic inventory models for variable production, J. Opl. Res. Soc. 48 (1997) 221±224. [6] Z.T. Balkhi, L. Benkherouf, A production lot size inventory model deteriorating items and arbitrary production and demand rates, Eur. J. Oper.Res. 92 (1998) 302±309. [7] B.N. Mandal, S. Phaujdar, An inventory model for deteriorating items and stock-dependent consumption rate, J. Opl.Res. Soc. 40 (1989) 483±488. [8] T.K. Datta, A.K. Pal, Order-level inventory system with power demand pattern for items with variable rate of deterioration, Indian J. Pure Appl. Math. 19 (1) (1988) 1043±1053. [9] M. Mandal, M. Maiti, Inventory model for damageable items with stock-dependent demand and shortages, Opsearch. 34 (3) (1997) 155±166. [10] T.M. Whitin, Inventory control and price theory, Mgmt. Sci. 2 (1955) 61±68. [11] H.B. Wolfe, A model for control of style merchandise, Industrial Mgmt. Rev. 9 (1968) 69±82. [12] R.I. Levin, C.P. Mclaughlin, R.P. Lamone, J.F. Kothas, Production/Operations Management: Contemporary policy for Managing Operating Systems, McGraw-Hill, New York, 1972.

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[13] T.L. Urban, An inventory model with an inventory level dependent demand rate and relaxed terminal conditions, J. Opl. Res. Sco. 43 (1992) 721±724. [14] S. Pal, A. Goswami, K.S. Chaudhuri, A deterministic inventory model for deteriorating items with stockdependent demand rate, Inter. J. Prod. Econ. 32 (1993) 291±299. [15] B.C. Giri, S. Pal, A. Goswami, K.S. Chaudhuri, An inventory model for deteriorating items with stock-dependent demand rate, Eur, J. Oper. Res. 95 (1996) 604±610.