Two-warehouse inventory model for deteriorating items when demand

530

Int. J. Operational Research, Vol. 7, No. 4, 2010

Two-warehouse inventory model for deteriorating items when demand is price sensitive Chandra K. Jaggi*, K.K. Aggarwal and Priyanka Verma Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi 110007, India Fax: +91 11 27666672 E-mails: [email protected]; [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: The main objective of this article is to develop a two-warehouse inventory model for deteriorating items when demand is price sensitive. It is assumed that the units are transported under a bulk release pattern from Rented Warehouse to Own Warehouse and the deterioration rates of the items are different in the two warehouses. Further, the model jointly optimises the order quantity and selling price. Depending upon the optimal order quantity, decision is made whether to rent other warehouse. The optimal shipment policy is also provided if indeed the other warehouse is needed. The results have been validated with the help of a numerical example. Sensitivity analysis on the demand parameters is also performed. Keywords: deterioration; inventory; price-dependent demand; two-warehouse system. Reference to this paper should be made as follows: Jaggi, C.K., Aggarwal, K.K. and Verma, P. (2010) ‘Two-warehouse inventory model for deteriorating items when demand is price sensitive’, Int. J. Operational Research, Vol. 7, No. 4, pp.530–551. Biographical notes: Chandra K. Jaggi is a Reader in the Department of Operational Research, Faculty of Mathematical Sciences at the University of Delhi, India. He obtained his PhD, MPhil and Masters degree (Operational Research) in the Department of Operational Research at the University of Delhi. His research interest lies on analysis of inventory system. He has more than 31 publications in Int. J. Production Economics, Journal of Operational Research Society, European Journal of Operational Research, Int. J. Systems Sciences, Canadian Journal of Pure and Applied Sciences, OPSEARCH, Investigation Operational Journal, Advanced Modelling and Optimisation, Journal of Information and Optimisation Sciences, Int. J. Mathematical Sciences, Indian Journal of Mathematics and Mathematical Sciences and Indian Journal of Management and Systems.

Copyright © 2010 Inderscience Enterprises Ltd.

Two-warehouse inventory model for deteriorating items

531

K.K. Aggarwal is a Lecturer in the Department of Operational Research, University of Delhi, India. He obtained his MSc (Operational Research) and PhD in Inventory Management from the University of Delhi. Prior to his appointment as a Lecturer in 2000, he served as a Scientist in the National Informatics Centre, India. His research interests and teaching include inventory modelling, Financial Engineering and Network Analysis. He has published more than 15 research papers in many scholarly journals. Priyanka Verma is a Research Scholar in the Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi, India. Also, she is working as a Lecturer in the Department of Mathematics, Keshav Mahavidayalya, University of Delhi, India. She completed her MSc (Operational Research) in 2003 from the University of Delhi. Currently, she is pursuing her PhD in Operational Research. Her area of interest is inventory management.

1

Introduction

Now, it is a known fact that the effect of deterioration cannot be ignored in the inventory management system. Moreover, it constitutes an important factor, which has attracted the attention of many researchers. Deterioration may be defined as decay, damage, spoilage, evaporation, obsolescence, pilferage and loss of utility or loss of marginal value of a commodity that results in decreasing usefulness from the original one. Ghare and Schrader (1963) proposed an economic order quantity model for items with an exponentially decaying inventory. Their work was extended by Covert and Philip (1973) by introducing variable rate of deterioration. A further generalisation to the above models was proposed by Shah (1977) by considering a model allowing complete backlogging of the unsatisfied demand. Extensive work has been done by different researchers to control the inventory of deteriorating items under various situations, namely, Dave (1979), Dave and Patel (1981), Hollier and Mark (1983), Kang and Kim (1983), Sachan (1984), Abad (1988), Datta and Pal (1988), Aggarwal and Jaggi (1989), Shiue (1990), Chung and Ting (1994), Hargia and Benkherouf (1994), Hargia (1995), Chakrabarti and Chaudhuri (1997), Wee (1999), Papachristos and Skouri (2003), Samanta and Roy (2004) and others. Another equally important aspect associated with inventory management is to decide where to stock the goods. Initially, lot of work has been done by many researchers for the situation of a single storage facility under different circumstances. But, looking into more practical way, there are situations that may require additional storage facility, whenever one has to procure a larger stock, which cannot be accommodated in their Own Warehouse (OW) because of its limited capacity. This additional storage capacity may be a Rented Warehouse (RW). A model considering the effect of two-warehouse was considered by Hartely (1976) in which he assumed that the holding cost in RW is greater than that in OW, therefore, items in RW are first transferred to OW to meet the demand until the stock level in RW drops to zero and then items in OW are released. Sarma (1987) extended Hartely’s model to cover the transportation cost from RW to OW that is considered to be a fixed constant independent of the quantity being transported. But he did not consider shortages in his model. Goswami and Chaudhuri (1992) further developed the model with or without shortages by assuming that the demand varies over time with linearly increasing trend and that the transportation cost from RW to OW

532

C.K. Jaggi, K.K. Aggarwal and P. Verma

depends upon the quantity being transported. In their model, the stock was transferred from RW to OW in an intermittent pattern. However, their work is for non-deteriorating items only. Sarma (1983) developed a two-warehouse model for deteriorating items with the infinite replenishment rate and shortages. Pakkala and Achary (1992) further considered the two-warehouse model for deteriorating items with finite replenishment rate and shortages. Bhunia and Maiti (1998) developed a two-warehouse model for deteriorating items with linearly increasing demand and shortages during the infinite period. In another article, Zhou (1998) presented a two-warehouse model for deteriorating items with time-varying demand and shortages during the finite-planning horizon. Kar, Bhunia and Maiti (2001) studied a two-warehouse inventory model for items by considering lot-size dependent replenishment cost, linearly time-dependent demand and finite time horizon. Zhou and Yang (2003) considered a two-warehouse inventory model for the items having stock-level-dependent demand rate. Later Yang (2004, 2006), Zhou and Yang (2005), Lee (2006), Chung and Huang (2007), Das, Maity and Maiti (2007), Dye, Ouyang and Hsieh (2007), Hsieh, Dye and Ouyang (2008), Niu and Xie (2008), Rong, Mahapatra and Maiti (2008), Dey, Mondal and Maiti (2008) all have worked in the area of two-warehousing. All these models and many more treated the demand rate as constant or timedependent function. However in the reality, the unit-selling price affects the demand tremendously; a low-selling price stimulates the demand, while high-selling prices decline the demand. Owing to this fact, many researchers developed models where demand is taken as a function of the selling price, the so-called optimal pricing policies. Cohen (1977) jointly determined the optimal replenishment cycle and price for inventory that is subject to continuous decay over time at a constant rate. Aggarwal and Jaggi (1989) reconsidered the Cohen’s model and provided an alternative approach to find the optimum price and production levels for exponentially decaying products. Wee (1997, 1999) extended Cohen’s model to consider a Weibull distribution for deterioration of items with shortage. Wee and Law (2001) extended Wee’s (1997) model and applied discounted cash flow approach to the finite planning horizon in which the replenishment is known. There is an extended literature concerning pricing of inventories by Kang and Kim (1983), Mukhopadhyay, Mukherjee and Chaudhuri (2004, 2005) and many more. Under various situations, lot of work has been done by many authors on twowarehouse system for deteriorating items. But unfortunately in these models, price dependent demand function has received very little or no attention. Although, pricing play a vital role for deteriorating items as almost all deteriorating items are pricesensitive. Some items are highly price sensitive and some are less. Therefore, a more practical system would be one that considers the demand to be a function of price. In this article, an attempt has been made to develop an order-level inventory model for deteriorating items with two-storage facility under a bulk release pattern. It is assumed that the deterioration rate of the items stored might be different in the two warehouses due to the difference in the environmental conditions or preserving conditions. However, the model is applicable even when the deterioration rate is same in both the warehouses. The demand is assumed to be a function of price. The optimal order size for the period has been obtained, depending upon which, decision is made whether to rent a warehouse. A computational procedure is also proposed to obtain the optimal number of shipments, optimal shipment quantity and optimal selling price, which make the average total profit of the system to be maximum. Finally, the results have been validated with the help of a numerical example. Sensitivity analysis on the demand parameters is also performed.


2

533

Assumptions and notations

The following assumptions are used in developing the model: 1

Replenishment is instantaneous.

2

The time horizon of the inventory system is infinite.

3

Shortages are not allowed to occur.

4

The time of transporting items from RW to OW is ignored.

5

The RW has unlimited capacity. Each shipment from RW to OW will restore OW to W units, which means q d W.

The notations adopted in this article are as below: D(p)

the demand rate that is a function of selling price p(D(p) = kp O) where k and O are positive constants)

W

the storage capacity of OW

T

the length of replenishment cycle

Q

the replenishment quantity per replenishment

n

the number of shipments from RW to OW

q

the quantity per shipment

T0

the fixed time interval between two successive shipments from RW to OW

Ct

the transportation cost per shipment from RW to OW

p

the selling price per unit item

c

the purchasing cost per unit item

h0

the holding cost per unit per unit time in OW

hr

the holding cost per unit per unit time in RW and hr t h0

A1/A

the fixed replenishment cost per replenishment for a two-warehouse system/for a single warehouse system, generally, A1 t A (extra cost may be included for the two-warehouse system due to transportation)

D

the deterioration rate in OW, 0 < D < 1

E

the deterioration rate in RW, 0 < E < 1 and E < D

I(t)

the inventory level at time t in OW.

The main aims of this article are: 1

What should be the optimal order lot-size and how the decision-maker knows whether to rent or not to rent RW to hold more items?

2

What should be the corresponding shipment policy from RW to OW, the decision-maker should make if he needs to rent RW?

For answering the first question, we first simply depict the single warehouse system.

534

3


Single warehouse model

The inventory system with a single warehouse can be stated as follows: the inventory level of the system is Q (i.e. order quantity) at the beginning of each replenishment cycle. Let I(t) be the inventory level at any time t, (0 d t d T). Depletion due to demand and deterioration will occur simultaneously. The differential equation that describes the instantaneous state of I(t) in the interval (0, T) is given by

dI D I (t ) dt

D( p), 0 d t d T .

(1)

The solution of above differential equation with boundary condition I(T) = 0 is D( p)

I (t )

D

(eD (T t ) 1), 0 d t d T .

(2)

Using I(0) = Q, we get

1

T

D

§ DQ · log ¨1 ¸, ( p) ¹ D ©

(3)

The holding cost of items in the interval (0,T) is T

HC

³ h I (t )dt 0

(4)

0

h0 Q

D

h0 D( p )

D2

§ DQ · log ¨1 ¸. ( p) ¹ D ©

Therefore, the net profit per unit time can be expressed as P

1 {( p c)Q A HC}. T

(5)

After substituting Equations (3) and (4), the net profit per unit time, P1 (Q, p) which is a function of two variables Q and p, becomes P1 (Q, p)

h0Q ½ h0 D ( p) ®( p c)Q A ¾ D ¿ D § DQ · ¯ log ¨ 1 ¸ © D( p) ¹

D

D f (Q, p ) ®( p c)Q A ¯

where f (Q, p)

1 § DQ · log ¨ 1 ¸ D ( p) ¹ ©

h0Q ½ h0 D ( p) ¾ D ¿ D

(6)

(7)


535

Since P1(Q, p) is a concave function in both Q and p (Appendix A), therefore, the necessary conditions for maximising P1(Q, p) are wP1 (Q, p ) wP (Q, p) 0 and 1 0 wQ wp which gives

D

h0 Q ½ h0 · wf (Q, p ) § ®( p c)Q A ¾ D f (Q, p) ¨ p c ¸ wQ ¯ D ¿ D¹ ©

D

h0 Q ½ O kh0 p (O 1) wf (Q, p) ( p c ) Q A Qf ( Q , p ) D ® ¾ wp ¯ D ¿ D

(8)

0

and 0.

(9)

As it is very difficult to get the close form solution from Equations (8) and (9), therefore, we make use of software Solver in order to determine the optimal values of Q and p. Moreover, one should notice that the single-warehouse model here does not impose the restriction Q d W. Therefore, the optimal replenishment quantity Q* may be either greater than or less than W. If Q* d W then the decision-maker will order only Q* units and does not need RW and if Q* ! W then the decision-maker will need a two-warehouse system. The purpose is to see, which system would be more profitable for the decision maker, a single warehouse system or a two-warehouse system. It has been demonstrated with the help of examples under which situations it is optimum for the decision maker to use RW. In Section 4, the mathematical formulation of the two-warehouse system has been discussed.

4

Two-warehouse model

The two-warehouse inventory system discussed here goes like this. At the beginning of each replenishment cycle, the system receives Q units out of which W units are kept in OW and the remaining part is kept in RW. Items in OW depletes due to demand and due to deterioration. When the inventory level in OW drops to (W q) units, then, q units from RW are shipped to OW to restore the stock into the original level W. This process is repeated until n shipments are completed. During this time, deterioration of units also continues in RW. So after the nth shipment and due to deterioration, no units are left in RW. The remaining W units in OW also depletes to zero up to the end of the replenishment cycle due to both demand and deterioration. The diagrams of inventory level for n = 3 in RW and OW are shown, respectively, in Figures 1 and 2. As described above, it is clear that the order quantity for each cycle is Q W nq z

(10)

where z are the number of units deteriorated in RW. And the inventory level, I(t) in the ith shipment cycle in OW satisfies the following differential equation:

536

C.K. Jaggi, K.K. Aggarwal and P. Verma dI (t ) D I (t ) dt

D ( p);

(i 1)T0 d t d iT0 , i 1, 2, ! , n

(11)

After using the boundary condition I [(i 1)T0 ] W , the solution to Equation (11) is I (t )

D( p)

D

eD

(i 1)T0 t

1 WeD (i 1)T0 t ;

Figure 1

Inventory level for n = 3 in RW

Figure 2

Inventory level for n = 3 in OW

(i 1)T0 d t d iT0 , i 1, 2, ! , n (12)


537

Thus, the inventory level in OW at the end of the ith shipment cycle becomes I (iT0 )

D( p)

D

(e DT0 1) WeD T0 .

(13)

And the amount of items transported from RW to OW in each shipment cycle is q W I (iT0 ) D( p) · § (1 eD T0 ) ¨ W . D ¸¹ ©

(14)

From Equation (14), we have 1

T0

D

§ DW D ( p ) · log ¨ ¸ © D (W q) D ( p) ¹

(15)

which indicates that the shipment cycle length depends on variables q and p. The holding cost of items in OW in the ith shipment period is iT0

h0 I (t )dt

³

(i 1)T0

h0 ° § DW D( p ) · ½° D( p) log ¨ ®q ¸¾. D °¯ D © D (W q) D ( p ) ¹ °¿

(16)

Similarly, inventory level, I(t), in OW in the interval [nT0,T] is given by

D( p)

I (t )

D

D( p) · § eD ( nT0 t ) ¨ W , nT0 d t d T . D ¸¹ ©

(17)

Noting that I(T) = 0, we have T

(DW D( p )) n 1 ° °½ log ® . n¾ D ¯° ( D( p))(D (W q ) D ( p)) ¿° 1

(18)

The holding cost of items in the interval [nT0, T] in OW is T

³ h I (t )dt 0

nT0

°W D ( p) § DW D( p) · ½° h0 ® 2 log ¨ ¸¾ . D © D( p ) ¹ ¿° ¯° D

(19)

Therefore, the holding cost of items in OW can be given by HCow

§ · °½ (DW D( p)) 2 ° q W D ( p) 2 log ¨¨ h0 ® ¸¸ ¾ . D °¯ D © D ( p)(D (W q) D ( p )) ¹ °¿

(20)

Now total inventory level in RW is (Q W )T0 ((Q W )e E T0 q )T0 ((Q W )e2 E T0 qe E T0 q)T0 " ((Q W )e ( n 1) E T0 qe ( n 2) E T0 qe ( n 3) E T0 ! q )T0 .

(21)

538


After simplifying (21), we get § T0 ¨ ET © 1 e 0

§ qe nE T0 ·§ n E T0 )¨ ¸ ¨¨ (Q W )(1 e ¨ 1 e E T0 ¹© ©

· § qe E T0 ¸¸ q(n 1) ¨¨ E T0 ¹ ©1 e

·· ¸¸ ¸¸ . ¹¹

(22)

Total number of deteriorated units in RW is (Q W )(1 e E T0 ) ((Q W )(1 e E T0 )e E T0

z

q(1 e E T0 )) " (Q W )(1 e E T0 )e ( n 1) E T0 q(1 e ( n 1) E T0 ) § 1 e ( n 1) E T0 (Q W )(1 e nE T0 ) (n 1)q qe E T0 ¨ ¨ 1 e E T0 ©

· ¸¸ . ¹

(23)

(24)

Now substituting the value of z from Equation (24) in Equation (10), we get Q W

q (1 e nE T0 ) (1 e E T0 )

.

(25)

After expanding the terms, Equation (25) becomes, § § DW D( p ) · · nE log ¨ ¨ 2 ¸¸ D © D (W q) D( p ) ¹ ¸ . Q W nq ¨ ¨ § DW D( p ) · ¸ E ¨¨ 2 log ¨ ¸ ¸¸ D © D (W q) D( p) ¹ ¹ ©

(26)

Putting Equation (25) in Equation (22), the total inventory level in RW becomes T0 q (1 e

E T0 2

)

((e nE T0 1) n(e E T0 1)).

(27)

Therefore, using the expression (27) and expanding the terms, the holding cost of items in RW becomes, HC rw

2n(n 1)hr q

§ DW D ( p ) · log ¨ ¸. © D (W q) D( p) ¹ § § DW D ( p ) · · E D ¨ 2 log ¨ ¸¸ D © D (W q) D( p) ¹ ¹ © 2

(28)

Therefore, the net profit per unit time of the system, P2 can be expressed as P2

1 {( p c)Q A nCt HCow HC rw }. T

(29)

After substituting Equations (18), (20), (26) and (28) into Equation (29), we get the net profit per unit time, P2 (n, p, q) which is a function of three variables n, p and q where n is discrete and p and q are continuous, for the two-warehouse system as given below


P2 (n, q, p)

539

§ § § § DW D( p) · · · nE log ¨ ¨ ¨ ¨ 2 ¸ ¸¸ D D ¨ ( p c) ¨W nq ¨ © D (W q) D( p) ¹ ¸ ¸ ¨ ¨ § ·¨ E § DW D( p) · ¸ ¸ (DW D( p))n1 ¨ ¨¨ ¨¨ 2 log ¨ ¸¸ log ¨¨ n¸ ¸ ¨ D © D (W q) D( p) ¸¹ ¸¹ ¸¹ © © © D( p)(D (W q) D( p)) ¹ © ·°½ (DW D( p))2 °§ q W · D( p) § A nB h0 ®¨ ¸¾ ¸ 2 log ¨¨ e ¸ © (kp )(D (W q) D( p)) ¹¿° ¯°© D ¹ D · ¸ § DW D( p) · ¸ 2hr n(n 1)q log ¨ ¸ ¸. 2 © D (W q) D( p) ¹ ¸ § E § DW D( p) · · D ¨ 2 log ¨ ¸¸ ¸ © D (W q) D( p) ¹ ¹ © D ¹

(30)

q W ¯ D

ª

D g1(n, p, q) «( p c)(W g2 (n, p, q)) A nB h0 ® ¬

(31)

kp j

º ½° 2hr g n p q g n p q ( , , ) ( , , ) » ¾ 3 4 D2 »¼ ¿° D

where

g1 (n, p, q)

1 (DW kp O )n 1 ° °½ log ® O O n ¾ ¯° (kp )(D (W q) kp ) ¿°

g 2 ( n, p , q )

§ DW kp O · °½ nE ° nq ® 2 log ¨¨ O ¸ ¸¾ D °¯ © D (W q) kp ¹ °¿ ° § DW kp O · ½° E 2 log ¨ ® ¨ D (W q) kp O ¸¸ ¾ © ¹ ¿° ¯° D

g 3 ( n, p , q )

° ½° (DW kp O ) 2 log ® O O ¾ °¯ (kp )(D W q kp ) °¿

g 4 ( n, p , q )

n(n 1)q § DW kp O E ° ®2 log ¨¨ O © D (W q) kp ¯° D

§ DW kp O log ¨ 2 ¨ D (W q) kp O © · °½ ¸¸ ¾ ¹ ¿°

· ¸¸ ¹

Since P2 (n, p, q) is a concave function of p and q for fixed n (Appendix B), therefore, the necessary conditions for maximising P2 ( n, p, q) (as n is discrete) are wP2 (n, p, q) wq

0 and

wP2 (n, p, q) wp

0

540

C.K. Jaggi, K.K. Aggarwal and P. Verma which gives

D

§ q W kpO · wg1 (n, p, q) ° 2 g3 (n, p, q) ¸ ®( p c)(W g2 (n, p, q)) A nB h0 ¨¨ ¸ wq D °¯ © D ¹

2hr

wg (n, p, q) ½ g4 (n, p, q) ¾ D g1 (n, p, q) ®( p c) 2 wq D ¿ ¯

(32)

§ 1 kpO wg3 (n, p, q) · 2hr wg4 (n, p, q) ½ h0 ¨¨ 2 ¸¸ ¾ 0 wq wq ¿ ©D D ¹ D and D

§ q W kp O · wg1 (n, p, q) ° 2 g3 (n, p, q) ¸¸ ®( p c)(W g 2 ( n, p, q)) A nB h0 ¨¨ wp D D © ¹ ¯°

2hr

wg (n, p, q) ½ g4 (n, p, q) ¾ D g1 (n, p, q) ®( p c) 2 W g 2 (n, p, q) wp D ¿ ¯

(33)

§ O kp (O 1) kp O wg3 (n, p, q) · 2hr wg 4 (n, p, q) ½ g3 (n, p, q) 2 h0 ¨¨ ¸¸ ¾ 0. 2 wp wp D D ¿ © D ¹

Again, it is very difficult to get the close form solution from Equations (32) and (33), therefore, determine the optimal values of q and p for fixed n with the help of Software Solver. By using the following algorithm, we can find the optimal values of n, p, q, T, Q and P2.

Algorithm 1

Put n = 1.

2

Determine the optimal values of q and p by solving Equations (32) and (33) simultaneously using the Solver and then calculate P2(n, p, q).

3

If P2(n, p, q) > P2(n 1, p, q), increment the value of n by 1 and go to step 2 else current value of n is optimal and the corresponding values of q, p and P2(n, p, q) can be calculated.

4

The optimal values of the order quantity (Q) and the cycle length (T) can be determined by substituting the values of n, p and q in Equations (26) and (18), respectively.

A special case Let us consider a situation when the deterioration rate in OW (D) is equal to the deterioration rate in RW (E) in a two-warehouse system, then the profit function is given by

Two-warehouse inventory model for deteriorating items P2 (n, q, p )

541

D § · (DW D ( p ))n 1 log ¨¨ n ¸ ¸ © D ( p )(D (W q ) D ( p )) ¹ § § § § DW D ( p ) · · · ¨ ¨ ¨ 2 n log ¨ ¸ ¸¸ ¨ ( p c) ¨ W nq ¨ © D (W q ) D ( p ) ¹ ¸ ¸ ¨ ¨ ¨ § DW D ( p ) · ¸ ¸ ¨ ¨¨ ¨¨ 2 log ¨ ¸ ¸¸ ¸ ¨ © D (W q ) D ( p ) ¹ ¹ ¸¹ © © © °§ q W A nB h0 ®¨ ¯°© D

(34)

§ · ½° (DW D ( p )) 2 · D( p) ¸ 2 log ¨¨ ¸¸ ¾ ¹ D © D ( p )(D (W q ) D ( p )) ¹ ¿°

· ¸ 2hr n(n 1)q § DW D ( p ) · ¸ log ¨ ¸¸ 2 © D (W q ) D ( p ) ¹ ¸ § § DW D ( p ) · · D ¨ 2 log ¨ ¸¸ ¸ © D (W q ) D ( p ) ¹ ¹ © ¹

q W ¯ D

ª

D g1(n, p, q) «( p c)(W g2 (n, p, q)) A nB h0 ® ¬

D( p)

D2

2hr

½º g3 (n, p, q) g4 (n, p, q)¾» D ¿¼

where g1 (n, p, q)

(DW D ( p))n 1 ° °½ log ® n¾ ¯° D ( p )(D (W q) D( p)) ¿°

g 2 ( n, p , q )

§ DW D ( p ) · ½ nq ®2 n log ¨ ¸¾ © D (W q) D( p) ¹ ¿ ¯ § DW D( p ) · ½ ®2 log ¨ ¸¾ © D (W q) D ( p ) ¹ ¿ ¯

g 3 ( n, p , q )

(DW D( p)) 2 ° °½ log ® ¾ D D p W q D p ( )( ( ) ( )) ¯° ¿°

g 4 ( n, p , q )

5

1

n(n 1)q

§ DW D ( p ) · log ¨ ¸. © D (W q) D( p) ¹ § DW D( p ) · ½ ®2 log ¨ ¸¾ © D (W q) D ( p ) ¹ ¿ ¯ 2

Numerical examples

The model developed above is illustrated by the following numerical example.

(35)

542


Let A1 = Rs 200/order, A = Rs 150/order, c = Rs 10/item, W = 400 units, B = Rs 35/shipment, hr = Rs0.6/unit/month, h0 = Rs0.3/unit/month, D = 0.01, E = 0.006. Table 1 k 3,00,000

3,50,000

4,00,000

Table 2 k 3,00,000

3,50,000

4,00,000

Optimal solution of the model for different values of demand parameters

O

n

p

q

Q

T

Profit

Use RW?

2.5

2

17.6

89.3

579.3

2.5

1583.8

Yes

2.6

2

17.2

74.5

549.5

2.9

1175.4

Yes

2.7

2

16.9

61.9

524.3

3.5

871.2

Yes

2.8

–

16.3

–

393.6

3.2

669.4

No

2.9

–

16.1

–

347.7

3.6

495.9

No

2.5

2

17.5

100.9

602.6

2.2

1858.6

Yes

2.6

2

17.2

84.6

569.8

2.6

1381.3

Yes

2.7

2

16.8

70.1

540.7

3.1

1025.8

Yes

2.8

2

16.5

58.2

516.8

3.7

760.3

Yes

2.9

–

16

–

377.1

3.3

585.9

No

2.5

2

17.5

112.2

625.2

1.9

2134.1

Yes

2.6

2

17.1

93.7

588.1

2.3

1587.7

Yes

2.7

2

16.8

78

556.6

2.8

1180.8

Yes

2.8

2

16.5

64.8

530

3.3

876.9

Yes

2.9

2

16.2

53.7

507.8

4

649.5

Yes

Optimal solution of the model for different values of demand parameters when the deterioration rate is same in OW and RW

O

n

p

q

Q

T

Profit

Use RW?

2.5

2

17.6

94.2

589.6

2.5

1585.2

Yes

2.6

2

17.2

78.4

557.8

2.9

1176.3

Yes

2.7

2

16.9

65.1

531.1

3.6

871.8

Yes

2.8

–

16.3

–

393.6

3.2

669.4

No

2.9

–

16.1

–

347.7

3.6

495.9

No

2.5

2

17.5

106.4

613.9

2.2

1860.3

Yes

2.6

2

17.2

88.6

578.3

2.6

1382.4

Yes

2.7

2

16.8

73.6

548.2

3.1

1026.6

Yes

2.8

2

16.5

61.1

522.9

3.8

760.9

Yes

2.9

–

16

–

377.1

3.3

585.9

No

2.5

2

17.5

118.1

637.5

2

2136.1

Yes

2.6

2

17.1

98.5

598.2

2.4

1589.1

Yes

2.7

2

16.8

81.9

564.8

2.8

1181.8

Yes

2.8

2

16.5

67.9

536.7

3.4

877.6

Yes

2.9

2

16.2

56.3

513.3

4.1

649.9

Yes


543

The optimal values of n, q, p, T, Q and P have been computed for different values of demand parameters. When the deterioration rate in OW is greater than the deterioration rate in RW then the observations are as follows (Table 1): 1

as the demand parameter k increases, profit also increases

2

as the demand parameter O increases, profit decreases.

When the deterioration rate is same in OW and RW then we observe that the profit also increases and we observe that (Table 2): 1

as the demand parameter k increases, profit also increases

2

as the demand parameter O increases, profit decreases.

6

Conclusions

In this article we have investigated that under what situation it is optimal to use a RW when the end demand is price sensitive for deteriorating items under a bulk release pattern. It is assumed that the holding cost in RW is greater than the OW due to better preserving facilities; therefore, the deterioration rate is less in RW than that of OW. The transportation cost of transporting items from RW to OW is taken to be independent of the shipped lot-size. The model jointly optimises the selling price and the order quantity, which ultimately helps the decision-maker to decide whether to rent other warehouse. The optimal shipment policy has also been provided, if indeed the other warehouse is needed.

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546


Appendix A The necessary conditions for maximising P1(Q, p) are wP1 (Q, p ) wQ which implies

0 and

wP1 (Q, p ) wp

0

D

h0 Q ½ h0 · wf (Q, p ) § ®( p c)Q A ¾ D f (Q, p) ¨ p c ¸ D ¿ D¹ wQ ¯ ©

D

O kh0 p (O 1) h0 Q ½ wf (Q, p) ®( p c)Q A ¾ D Qf (Q, p ) D ¿ D wp ¯

(A1)

0

and 0.

(A2)

Now, Equations (A1) and (A2) can be used to find the approximated values of Q and p. For sufficient condition, we determine the following: w 2 P1 (Q, p ) wQ

2

w 2 P1 (Q , p ) wp

2

w 2 P1 (Q, p) wQwp

D

D

h0 Q ½ h0 · wf (Q, p) w 2 f (Q, p) § ®( p c)Q A ¾ 2D ¨ p c ¸ D D ¹ wQ wQ 2 ¯ ¿ ©

h0 Q ½ w 2 f (Q , p ) wf (Q , p ) O ( O 1) kh0 p ( O 2) (A4) ® ( p c )Q A ¾ 2D Q 2 D ¿ wQ D wp ¯

w 2 P1 (Q, p) wpwQ

D

h0Q ½ w 2 f (Q , p ) ®( p c)Q A ¾ wpwQ ¯ D ¿

h · wf (Q, p) wf (Q, p) § D ¨ p c 0 ¸ DQ D f (Q, p). w wQ D p © ¹ The sufficient conditions for maximising P1(Q, p) are w 2 P1 (Q, p ) wQ 2 w 2 P1 (Q, p ) wp 2

(A3)

0

0

and § w 2 P1 (Q, p) ·§ w 2 P1 (Q, p) · § w 2 P1 (Q, p) · ¨¨ ¸¨ ¸¸ ¨¨ ¸¸ ! 0. 2 ¸¨ wp 2 wp 2 © wQ ¹© ¹ © ¹ at optimal values of Q and p.

(A5)


547

Graphically, we have shown below that the profit function is concave with respect to both the order quantity and the price.

548


Appendix B The necessary conditions for maximising P2 (n, p, q) (as n is discrete) are wP2 (n, p, q) wq

0 and

wP2 (n, p, q) wp

0

which gives

D

§ q W kp O · wg1 (n, p, q) ° 2 g3 (n, p, q) ¸¸ ®( p c)(W g2 (n, p, q)) A nB h0 ¨¨ wq D D © ¹ ¯°

2hr

wg (n, p, q) ½ g4 (n, p, q) ¾ D g1 (n, p, q) ®( p c) 2 wq D ¿ ¯

(B1)

§ 1 kp O wg3 (n, p, q) · 2hr wg4 (n, p, q) ½° h0 ¨ 2 ¸¸ ¾ 0 ¨D D wq wq © ¹ D ¿°

and

D

§ q W kp O · wg1 (n, p, q) ° 2 g3 (n, p, q) ¸¸ ®( p c)(W g 2 (n, p, q)) A nB h0 ¨¨ wp D °¯ © D ¹

2hr

D

wg (n, p, q) ½ g 4 (n, p, q) ¾ D g1 (n, p, q) ®( p c) 2 W g 2 (n, p, q) wp ¿ ¯

§ O kp ( O 1) kp O wg3 (n, p, q) · 2hr wg 4 (n, p, q) ½° g n p q ( , , ) h0 ¨¨ ¸¸ ¾ 3 2 D wp wp D2 °¿ © D ¹

(B2) 0.

Now Equations (B1) and (B2) can be used to find the approximated values of q and p. For sufficient condition, we determine the following

w 2 P2 (n, p, q) wq 2

D

w 2 g1 (n, p, q) § q W ®( p c)(W g 2 (n, p, q)) A nB h0 ¨ © D wq 2 ¯

kp O

D2

· 2h wg (n, p, q) ½ g3 (n, p, q) ¸ r g 4 (n, p, q) ¾ 2D 1 wq D ¹ ¿

§ 1 kp O wg3 (n, p, q) · wg 2 (n, p, q ) ( p c) h0 ¨ 2 ¸¸ ® ¨D D wq wq ¯ ¹ ©

° w 2 g 2 ( n, p , q ) 2hr wg 4 (n, p, q) ½ ¾ D g1 (n, p, q) ®( p c ) wq D wq 2 °¯ ¿

h0

kp O w 2 g3 (n, p, q)

D2

wq 2

2hr w 2 g 4 (n, p, q) ½° ¾ D wq 2 °¿

(B3)

Two-warehouse inventory model for deteriorating items w2 P2 (n, p, q)

D

wp2

w2 g1 (n, p, q) wp2

549

^( p c)(W g2 (n, p, q)) A nB h0

½° § q W kpO · 2h wg (n, p, q) 2 g3 (n, p, q) ¸¸ r g4 (n, p, q)¾ 2D 1 ¨¨ D D wp D °¿ © ¹ § Okp(O 1) wg2 (n, p, q) ( p c ) W g ( n , p , q ) h g3 (n, p, q) ¨ ® 2 0 wp © D2 ¯ kpO wg3 (n, p, q) · 2hr wg4 (n, p, q) ½ 2 ¸ ¾ D g1 (n, p, q) wp wp D ¿ ¹ D

(B4)

° 2wg2 (n, p, q) § O(O 1)kp(O 2) w2 g2 (n, p, q) ( p c) h g3 (n, p, q) ¨¨ ® 0 wp wp2 D2 © ¯°

2Okp(O 1) wg3 (n, p, q) kpO w2 g3 (n, p, q) · 2hr w2 g4 (n, p, q) °½ 2 ¸¸ ¾ wp wp2 wp2 D2 D °¿ ¹ D

w2 P(n, p, q) wqwp

w2 P(n, p, q) wpwq

D

O · w2 g1(n, p, q) § q W kp 2 g3 (n, p, q) ¸¸ ®( p c)(W g2 (n, p, q)) A nB h0 ¨ D wpwq D © ¯ ¹

wg (n, p, q) wg2 (n, p, q) § 1 kpO wg3 (n, p, q) · ½ g4 (n, p, q)¾ D 1 ¸ ¨ 2 ®( p c) ¸ D wp wq wq ¿ ¯ ©D D ¹

2hr

wg1(n, p, q) wg2 (n, p, q) 2hr wg4 (n, p, q) ½ ¾ D ®W g2 (n, p, q) ( p c) D wq wq wp ¿ ¯

§ Okp(O 1) kpO wg3 (n, p, q) · 2hr wg4 (n, p, q) ½ ( , , ) h0 ¨¨ g n p q ¸¸ ¾ 3 2 wp wp D2 ¿ © D ¹ D wg (n, p, q) w2 g2 (n, p, q) D g1 (n, p, q) ® 2 ( p c) wq wpwq ¯ § Okp(O 1) wg3 (n, p, q) kpO w2 g3 (n, p, q) · 2hr w2 g4 (n, p, q) ½° h0 ¨ 2 ¸¸ ¾ 2 ¨ wpwq wpwq wq D °¿ © D ¹ D

(B5) The sufficient conditions for maximising P2 (n, p, q) are w 2 P2 (n, p, q) wq 2

0,

w 2 P2 (n, p, q ) wp 2

0

and 2

§ w 2 P2 (n, p, q ) · § w 2 P2 (n, p, q) · § w 2 P2 (n, p, q ) · ¨¨ ¸¸ ¨¨ ¸¸ ¨¨ ¸¸ ! 0 wqwp wq 2 wp 2 © ¹© ¹ © ¹ at optimal values of q and p.

550


Graphically, we have shown below that the profit function is concave with respect to both, the shipment quantity and the price.


551