Joint optimization of price and order quantity with ... - Springer Link

Top (2008) 16: 195–213 DOI 10.1007/s11750-008-0041-8 O R I G I N A L PA P E R

Joint optimization of price and order quantity with shortages for a two-warehouse system Chandra K. Jaggi · Priyanka Verma

Received: 11 June 2007 / Accepted: 24 January 2008 / Published online: 8 February 2008 © Sociedad de Estadística e Investigación Operativa 2008

Abstract In developing the best strategy for real-world applications, the vendor must have some knowledge of the buyers’ behavior such as response to shortages and price increases. Using this knowledge, he can develop a policy that will ensure the largest net profit. Considering the fact, a two-warehousing inventory model has been developed where the demand is price-sensitive under the bulk release rule. Stockouts are allowed and are fully backlogged. Moreover, the transportation cost is taken to be dependent on the transported units. The model jointly optimizes the selling price and the order quantity by maximizing the system profit. Results have been validated with the help of a numerical example. Keywords Inventory · Two-warehouse system · Price-dependent demand · Shortages Mathematics Subject Classification (2000) 90B05

1 Introduction The general assumption in classical inventory models is that the organization owns a single-warehouse without capacity limitation. But, in practice, the capacity of any warehouse is limited. In fact, there exist many reasons due to which the organizations are motivated to hold a large stock. For example, the demand of the item may be quite high or attractive price discounts are available on the bulk purchase; and so on. Therefore, the classical models are unsuitable in such situations, due to the limited C.K. Jaggi () · P. Verma Department of Operational Research, Faculty of Mathematical Sciences, University of Delhi, Delhi 110007, India e-mail: [email protected]

196

C.K. Jaggi, P. Verma

capacity of the owned warehouse (OW). Hence, the organization may require an additional warehouse to store these excess quantities. This additional warehouse may be a rented warehouse (RW) with an unlimited capacity. Further, the holding cost in RW is usually higher than in OW due to additional cost of maintenance, material handling, etc. To reduce the inventory costs, it is economical to consume the goods of RW at the earliest. In the literature, a lot of work has been done on two-warehouse inventory system. An early discussion on the effect of two warehouse was considered by Hartely (1976) in which he assumed that the holding cost in rented warehouse (RW) is greater than that in own warehouse (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) developed a deterministic inventory model with infinite replenishment rate. 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 depends on the quantity being transported. In their model, the stock was transferred from RW to OW in an intermittent pattern. Sarma (1983) first 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. Benkherouf (1997) also considered shortages in his paper for deteriorating items with two storage facilities. Bhunia and Maiti (1998) developed a two-warehouse model for deteriorating items with linearly increasing demand and shortages during the infinite period. In another paper, Zhou (1998) presented a two-warehouse model for deteriorating items with time-varying demand and shortages during the finite-planning horizon. Further, Kar et al. (2001) studied a twowarehouse 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-leveldependent demand rate. Zhou (2003) developed a multi-warehouse inventory model for items with time-varying and shortages. Although, a lot of work has been done by many authors for two-warehouse problems, but till the recent past authors have not considered the situation where the demand is price sensitive and shortages are also allowed to occur. In fact, almost all items are price sensitive, as some items may be highly price sensitive and some may be less. For example, in the retail industry, organizations may dynamically adjust their prices in order to boost demand and enhance revenues. Many authors have considered the demand to be price-dependent in their models. Some of them are Cohen (1977), Aggarwal and Jaggi (1989), Eliashberg and Steinberg (1991), Gallego and Ryzin (1994), Wee (1997, 1999), Papachristos and Skouri (2002), Mukhopadhyay et al. (2004, 2005), and many more. Therefore, it is more realistic to consider the demand to be a function of selling price. This paper considers an inventory model for a two-warehouse system where demand is a function of price under a bulk release pattern, the transportation cost is taken to be dependent of the transported amount, and shortages are also allowed and completely backlogged. The model jointly optimizes the selling price and the order quantity by maximizing the system profit. Further, on the basis of the order quantity,

Joint optimization of price and order quantity with shortages

197

it is decided whether to rent a warehouse or not, according to which the optimal number of shipments have been obtained. Results have been illustrated with the help of a numerical example.

2 Assumptions and notation The following assumptions are used in developing the model: (1) (2) (3) (4) (5)

Replenishment is instantaneous. The time horizon of the inventory system is infinite. Shortages are allowed and completely backlogged. The time of transporting items from RW to OW is built in maintenance. The rented warehouse RW has unlimited capacity. Each shipment from RW to OW will restore OW to W units, which means q ≤ W . (6) Based on the practical observation, the transportation cost for q units per shipment is assumed as if 0 < q ≤ m, and Ct , Ct (q) = Ct + b(q − m), if m < q ≤ W, where m is the maximum number of units which can be shipped under a fixed transportation cost Ct , b is the variable charge to be paid for every additional unit after m. The notations adopted in this paper are as below: D(p) The demand rate which is a function of selling price p [D(p) = kp −e , where k and e are positive constants]; W The storage capacity of OW; Q The replenishment quantity per replenishment; n The number of shipments from RW to OW; q The quantity per shipment; S The maximum number of shortages; t0 The fixed time interval between two successive shipments from RW to OW; The time at which the inventory level reaches zero in OW; t1 T The length of replenishment cycle; Ct The transportation cost per shipment from RW to OW; π The shortage cost per unit per unit time; 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 (includes the transportation cost from vendor to buyer); hr The holding cost per unit per unit time in RW and hr ≥ h0 (includes the transportation cost from vendor to buyer); A1 The fixed replenishment cost per replenishment for a two-warehouse system; A The fixed replenishment cost per replenishment for a single warehouse system, generally, A1 ≥ A (extra cost may be included for the two-warehouse system due to transportation);

198


I0 (t) The inventory level at time t in OW; Ir (t) The inventory level at time t in RW; S(t) The shortage level at time t in OW. The main objective of this paper is to answer the following questions: (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 decisionmaker should make if he needs to rent RW? (3) What should be the optimal selling price and the optimal level of shortages? To answer the first question, we first simply depict a single warehouse system.

3 Single-warehouse model An inventory system with a single warehouse can be stated as follows: Initially the inventory system receives Q (i.e. order quantity) units out of which S units are delivered towards backorders. Therefore, the initial inventory level at the beginning of each replenishment cycle is Q − S units. Then the inventory level gradually decreases because of meeting the demand. By time t1 the inventory level reaches zero and shortages build up until the end of the period T . Hence, the order quantity is Q = D(p)T = kp −e T .

(1)

The inventory level, I0 (t), in the OW in the interval (0, t1 ) satisfies the following differential equation: dI0 (t) = −D(p), dt

0 ≤ t ≤ t1 , i = 1, 2, . . . , n.

(2)

After using the boundary condition I0 (0) = Q − S, the solution to (2) is I0 (t) = −kp −e t + Q − S,

0 ≤ t ≤ t1 .

(3)

Now I0 (t1 ) = 0, and we have t1 =

(Q − S) . kp −e

(4)

The holding cost of items in OW in the interval (0, t1 ) is HCOW =

t1

h0 I0 (t) dt =

0

h0 (Q − S)2 . 2kp −e

(5)

Similarly, the shortage level S(t) during (t0 , T ) satisfies the following differential equation: dS(t) = D(p), dt

t1 ≤ t ≤ T .

(6)


199

After using the boundary condition S(t1 ) = 0, the solution to (6) is S(t) = kp −e (t − t1 ),

t1 ≤ t ≤ T .

(7)

Now S(T ) = S, so we have T − t1 =

S . kp −e

(8)

Substituting the value of t1 from (4), we get T=

Q . kp −e

(9)

The shortage cost of items in the interval (t0 , T ) is given by T πS 2 πS(t) dt = . SC = 2kp −e t1

(10)

Therefore, the net profit per unit time of the system, P1 , can be expressed as P1 =

1 (p − c)Q − A − HCOW − SC . T

(11)

After substituting (5), (9), (10) into (11), we get the net profit per unit time, P1 (Q, p, S), which is a function of three continuous variables Q, p, and S for the single warehouse system as given below: P1 (Q, p, S) = (p − c)kp −e −

Akp −e h0 (Q − S)2 πS 2 − − . Q 2Q 2Q

(12)

The necessary conditions for maximizing P1 (Q, p, S) are ∂P1 (Q, p, S) = 0, ∂Q

∂P1 (Q, p, S) = 0, ∂p

and

∂P1 (Q, p, S) = 0, ∂S

which give A(kp −e ) h0 (Q − S)2 πS 2 h0 (Q − S) = 0, + + − Q Q2 2Q2 2Q2 −(e+1) eA −e − e(p − c) = 0, and kp + kp Q h0 (Q − S) πS − = 0. Q Q

(13) (14) (15)

P1 (Q, p, S) is a concave function (Appendix 1), but it is very difficult to get the close form solution from (13), (14), and (15); therefore, we make use of Solver software in order to determine the optimal values of Q, p, and S simultaneously. Moreover, one should notice that the single-warehouse model here does not impose the restriction (Q − S) ≤ W . Therefore, the optimal replenishment quantity

200


Q∗ may be either greater than or less than W . If (Q − S)∗ ≤ W then the decisionmaker will order only Q∗ units and will not need RW, and if (Q − S)∗ > W then the decision-maker will need a two-warehouse system. In the next section, we will show the mathematical formulation of the twowarehouse system.

4 Two-warehouse model At the beginning of each replenishment cycle, the system receives Q units out of which S units are delivered towards backorders. Therefore, the initial inventory level at the beginning of each replenishment cycle is (Q − S) units, out of which W units are kept in OW and the remaining part is kept in RW. Items in OW are first used to satisfy customers’ demand until the inventory level in OW drops to (W − q) units. At this moment, q units from RW are shipped to OW to restore the stock into the original level W . Then the process is repeated until n shipments are completed. After the nth shipment, no units are left in RW. So, the remaining W units in OW are used to meet the demand. By time t1 the inventory level reaches zero and shortages build up until the end of the period T . The diagrams of inventory level for n = 3 in RW and OW are shown respectively in Figs. 1 and 2. As described above, it is clear that the order quantity for each cycle is Q = S + W + nq.

(16)

And the inventory level, I0 (t), in the ith shipment cycle in OW satisfies the following differential equation: dI0 (t) = −D(p), dt

(i − 1)t0 ≤ t ≤ it0 ,

i = 1, 2, . . . , n.

(17)

After using the boundary condition I [(i − 1)t0 ] = W , the solution to (17) is I0 (t) = W − kp −e t − (i − 1)t0 , (i − 1)t0 ≤ t ≤ it0 , i = 1, 2, . . . , n.

(18)

Thus, the inventory level in OW at the end of the ith shipment cycle becomes I0 (it0 ) = W − t0 kp −e .

(19)

Fig. 1 Inventory level for n = 3 in RW


201

Fig. 2 Inventory level for n = 3 in OW

And the amount of items transported from RW to OW in each shipment cycle is q = W − I (iT0 ) = t0 kp −e . (20) From (19), we have t0 =

q , (kp −e )

(21)

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 kp −e t02 . h0 I0 (t) dt = h0 W t0 − 2 (i−1)T0

iT0

Similarly, inventory level, I0 (t), in OW in the interval [nt0 , t1 ] is given by I0 (t) = W − kp −e (t − nt0 ), nt0 ≤ t ≤ t1 .

(22)

(23)

Noting that I0 (t1 ) = 0, we have t1 = nt0 +

W . kp −e

Substituting the value of t0 from (21), we get t1 =

W + nq . kp −e

(24)

The holding cost of items in the interval [nt0 , t1 ] in OW is t 2 n2 h0 I0 (t) dt = h0 W (t1 − nt0 ) + kp −e nt0 t1 − 1 − t02 . 2 2 nt0

t1

(25)

202


So the holding cost of items in OW can be given by kp −e t02 t2 1 + n2 + W t1 + kp −e nt0 t1 − 1 . (26) HCOW = h0 W t0 (1 − n) − 2 2 Now the shortage level, S(t), in the OW in interval [t1 , T ] satisfies the following differential equation: dS(t) = D(p), t1 ≤ t ≤ T . dt After using the boundary condition S(t1 ) = 0, the solution to (27) is S(t) = kp −e (t − t1 ),

t1 ≤ t ≤ T .

(27)

(28)

Noting that S(T ) = S, we have T=

S + W + nq . kp −e

Therefore, the shortage cost in the interval [t1 , T ] is T πS 2 SC = πS(t) dt = . 2kp −e t1

(29)

(30)

Furthermore, the holding cost of items in RW is

1 HCRW = hr nqt0 + (n − 1)qt0 + · · · + 2qt0 + qt0 = n(n + 1)hr qt0 . 2

(31)

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

1 (p − c)Q − A1 − nCt (q) − HCOW − HCRW − SC . T

(32)

After substituting (26), (29), (30), (31) into (32), we get the net profit per unit time,P2 (n, p, q, S), which is a function of four variables n, p, q, and S, where n is discrete and p, q, and S are continuous, for the two-warehouse system as given below kp −e (p − c)(S + W + nq) − A1 − nCt P2 (n, q, p, S) = (S + W + nq) W q(1 − n) q 2 (1 + n2 ) W (W + nq) − h0 − + kp −e 2kp −e kp −e

n(n + 1)hr q 2 (W + nq)(nq − W ) πS 2 − . (33) + − 2kp −e 2kp −e 2kp −e After noting assumption (6), we can rewrite the average total profit function, P2 (n, p, q, S), as the following segmented function: kp −e P21 (n, q, p, S) = (p − c)(S + W + nq) − A1 − nCt (S + W + nq)


203

W q(1 − n) q 2 (1 + n2 ) W (W + nq) − + kp −e 2kp −e kp −e

n(n + 1)hr q 2 (W + nq)(nq − W ) πS 2 − + − 2kp −e 2kp −e 2kp −e

= X1 (n, p, q, S) X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) − πX5 (p, S) , (34a) − h0

for 0 < q ≤ m, where X1 (n, p, q, S) =

kp −e , (S + W + nq)

X2 (n, p, q, S) = (p − c)(S + W + nq), X3 (n, p, q) =

W q(1 − n) q 2 (1 + n2 ) W (W + nq) (W + nq)(nq − W ) − + + , kp −e 2kp −e kp −e 2kp −e

X4 (n, p, q) =

n(n + 1)q 2 , 2kp −e

X5 (p, S) =

πS 2 , 2kp −e

P22 (n, q, p, S) =

for m < q ≤ W,

and

kp −e (p − c)(S + W + nq) − A1 − nCt − nb(q − m) (S + W + nq) W q(1 − n) − h0 kp −e

q 2 (1 + n2 ) W (W + nq) (W + nq)(nq − W ) − + + 2kp −e kp −e 2kp −e n(n + 1)hr q 2 πS 2 , − − 2kp −e 2kp −e

= Y1 (n, p, q, S) Y2 (n, p, q, S) − A − nCt − nb(q − m) − h0 Y3 (n, p, q) − hr Y4 (n, p, q) − πY5 (p, S) ,

(34b)

where Y1 (n, p, q, S) =

kp −e , (S + W + nq)

Y2 (n, p, q, S) = (p − c)(S + W + nq), Y3 (n, p, q) =

W q(1 − n) q 2 (1 + n2 ) W (W + nq) (W + nq)(nq − W ) − + + , kp −e 2kp −e kp −e 2kp −e

204


Y4 (n, p, q) = Y5 (p, S) =

n(n + 1)q 2 , 2kp −e

πS 2 . 2kp −e

It is seen that, for a specified n-value, the maximum of the average profit function, P2∗ , ∗ and P ∗ , each determined under its respective q-range. We can find is the larger of P21 22 ∗ and P ∗ without considering their respective q-range constraints. maximums of P21 22 The necessary conditions for maximizing P21 (n, p, q, S) (for fixed n) are ∂P21 (n, p, q, S) = 0, ∂q

∂P21 (n, p, q, S) = 0, ∂p

and

∂P21 (n, p, q, S) = 0, ∂S

which give ∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂q ∂X2 (n, p, q, S) − hr X4 (n, p, q) − πX5 (p, S) + X1 (n, p, q, S) ∂q h0 ∂X3 (n, p, q) hr ∂X4 (n, p, q) − = 0, − ∂q ∂q

(35)

∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) ∂p ∂X2 (n, p, q, S) h0 ∂X3 (n, p, q) − πX5 (p, S) + X1 (n, p, q, S) − ∂p ∂p hr ∂X4 (n, p, q) π∂X5 (p, S) − = 0, (36) − ∂p ∂p ∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) ∂S ∂X2 (n, p, q, S) π∂X5 (p, S) − πX5 (p, S) + X1 (n, p, q, S) − = 0. (37) ∂S ∂S P21 (n, p, q, S) is a concave function for a fixed n (Appendix 2), but it is very difficult to get the close form solution from (35), (36), and (37); therefore, we make use of Solver in order to determine the optimal values of q, p, and S for fixed n. Similarly, we can show that P22 (n, p, q, S) is also a concave function for fixed n. By using the following algorithm, we can find the optimal values of n, p, q, S, T , Q, and P21 . Algorithm 1. Put n = 1. 2. Determine the optimal values of q, p, and S by solving (34a) using Solver and then calculate P21 (n, p, q, S).


205

3. If P21 (n, p, q, S) > P21 (n − 1, p, q, S), 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, S, ∗ (n, p, q, S) can be calculated. and P21 ∗ (n, p, q, S) by solving 4. Similarly, determine the optimal values n, p, q, S, and P22 (34b) by repeating the steps 1, 2, and 3. 5. The maximum of average total profit function, P2∗ , can be obtained by comparing ∗ and P ∗ . The corresponding values would be optimal values the maximums of P21 22 of n, p, q, and S. 6. The optimal values of the order quantity (Q) and the cycle length (T ) can be determined by substituting the values of n, p, q, and S into (16) and (29), respectively. Special case When shortages are not allowed, then taking S = 0 and t1 = T in (24), (26), and (30), we get W + nq , kp −e kp −e t02 T2 HCOW = h0 W t0 (1 − n) − 1 + n2 + W T + kp −e nt0 T − , 2 2

T=

and SC = 0. So the expression for the average total profit function reduces to −e W q(1 − n) nCt kp −e Akp −e − − h0 − P3 (n, q, p) = (p − c) kp (W + nq) (W + nq) (W + nq) 2 2 2 n(n + 1)hr q q (1 + n ) (W + nq) + − , − 2(W + nq) 2 2(W + nq) for 0 < q ≤ m, and nCt kp −e nb(q − m)kp −e Akp −e P3 (n, q, p) = (p − c) kp −e − − − (W + nq) (W + nq) (W + nq) W q(1 − n) q 2 (1 + n2 ) (W + nq) n(n + 1)hr q 2 − + − , − h0 (W + nq) 2(W + nq) 2 2(W + nq) for m < q ≤ W . 5 Numerical examples The model developed above is illustrated by the following numerical example. Let A1 = Rs 200/order, A = Rs 150/order, c = Rs 10/item, W = 400 units, B = Rs 40/shipment, b = 0.1/unit, hr = Rs 0.3/unit/month, h0 = Rs 0.15/unit/month. The optimal values of n, q, p, S, T , Q, and P have been computed for different values of demand parameters. The computed results are given in Table 1.

206


Table 1 Optimal solution of the model for different values of demand parameters k 400000

500000

600000

e

n

q

Q

t1

T

p

S

Profit

Use RW?

2.8

1

84

540.9

3

3.3

16.3

56.9

903.6

Yes

2.9

1

56

505.1

3.5

3.9

16

49.1

674.9

Yes

3

1

475.6

4.2

4.6

15.8

42.6

3.1

–

–

33

–

–

–

–

–

–

503.5

No

Yes

3.2

–

–

–

–

–

–

–

–

No

2.8

1

115

581.2

2.5

2.8

16.2

66.2

1140.1

Yes

2.9

1

84

540.9

3

3.3

16

56.9

853.3

Yes

3

1

56

505.1

3.5

3.9

15.7

49.1

638.1

Yes

3.1

1

33

475.5

4.2

4.7

15.5

42.5

476.4

Yes

3.2

–

–

–

–

–

–

–

–

No

2.8

1

145

619.9

2.2

2.5

16.2

74.9

1377.6

Yes

2.9

1

109

573.3

2.6

2.9

15.9

64.3

1032.3

Yes

3

1

78

533.3

3.1

3.4

15.7

55.3

773.3

Yes

3.1

1

51

498.7

3.7

4.1

15.5

47.7

578.6

Yes

3.2

1

29

470.4

4.4

4.8

15.3

41.4

432

Yes

Table 1 shows that for a fixed value of k (demand parameter), an increase in e (demand parameter) will result in an decrease in the demand and so it will result in a decrease in the order quantity (Q), selling price (p), number of shortages (S), profit (P ), and an increase in the cycle length (T ). For a fixed value of e (demand parameter), an increase in k (demand parameter) will result in an increase in the demand and so it will result in an increase in the order quantity (Q), selling price (p), number of shortages (S), profit (P ), and a decrease in the cycle length (T ).

6 Conclusion The present paper jointly optimizes the price and the order quantity for a twowarehouse inventory system where the demand is a function of price. Shortages are also allowed and fully backlogged. The units are transported from rented warehouse (RW) to own warehouse (OW) under a bulk release pattern and the transportation cost of transporting items from RW to OW is taken to be dependent on the shipped lot-size. The model determines the optimal order quantity for a single-warehouse system depending on which decision is made whether to rent a warehouse or not. The optimal number of shipments and the optimal shipment size are also provided if indeed the other warehouse is needed. Results have been illustrated with the help of a numerical example. Acknowledgement

The authors would like to thank Dr. Satish K. Goel for his valuable suggestions.


207

Appendix 1 The necessary conditions for maximizing P1 (Q, p, S) are ∂P1 (Q, p, S) = 0, ∂Q

∂P1 (Q, p, S) = 0, ∂p

and

∂P1 (Q, p, S) = 0, ∂S

which imply A(kp −e ) h0 (Q − S)2 πS 2 h0 (Q − S) + + − = 0, Q Q2 2Q2 2Q2

(1.1)

eA − e(p − c) = 0, kp −e + kp −(e+1) Q

(1.2)

and

h0 (Q − S) πS − = 0. Q Q

(1.3)

Now (1.1), (1.2), and (1.3) can be solved simultaneously to find the values of Q, p, and S. For a sufficient condition, we determine the following: ∂ 2 P1 (Q, p, S) (πS 2 + 2A(kp −e ) + h0 (Q − S)2 ) 2h0 (Q − S) h0 = − + − , (1.4) Q ∂Q2 Q3 Q2 ∂ 2 P1 (Q, p, S) −(e+1) −(e+2) eA − e(p − c) , = −2ekp − (e + 1)kp Q ∂p 2

(1.5)

∂ 2 P1 (Q, p, S) (h0 + π) , =− Q ∂S 2

(1.6)

∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) −eA(kp −(e+1) ) = = , ∂Q∂p ∂p∂Q Q2

(1.7)

∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) πS h0 (Q − S) h0 = = 2− + , ∂Q∂S ∂S∂Q Q Q Q2

(1.8)

∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) = = 0. ∂p∂S ∂S∂p

(1.9)

Sufficient conditions for maximizing P1 (Q, p, S) are ∂ 2 P1 (Q, p, S) < 0, ∂Q2

∂ 2 P1 (Q, p, S) < 0, ∂p 2

∂ 2 P1 (Q, p, S) < 0, ∂S 2

and

∂ 2 P1 (Q, p, S) ∂Q2

2 ∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) ∂ P1 (Q, p, S) 2 − ∂p∂S ∂p 2 ∂S 2

208


∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) − ∂Q∂p ∂p∂Q ∂S 2 ∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) − ∂p∂S ∂S∂Q ∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) ∂ 2 P1 (Q, p, S) + ∂Q∂S ∂p∂Q ∂S∂p 2 2 ∂ P1 (Q, p, S) ∂ P1 (Q, p, S) >0 − ∂S∂Q ∂p 2 at optimal values of Q, p, and S. Graphically we have shown below that the profit function is concave with respect to the order quantity, the price, and the number of shortages.


209

Appendix 2 The necessary conditions for maximizing P21 (n, p, q, S) (as n is discrete) are ∂P21 (n, p, q, S) = 0, ∂q

∂P21 (n, p, q, S) = 0, ∂p

and

∂P21 (n, p, q, S) = 0, ∂S

which give ∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) ∂q ∂X2 (n, p, q, S) − πX5 (p, S) + X1 (n, p, q, S) ∂q h0 ∂X3 (n, p, q) hr ∂X4 (n, p, q) − = 0, (2.1) − ∂q ∂q ∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) ∂p ∂X2 (n, p, q, S) − πX5 (p, S) + X1 (n, p, q, S) ∂p h0 ∂X3 (n, p, q) hr ∂X4 (n, p, q) π∂X5 (p, S) = 0, (2.2) − − − ∂p ∂p ∂p ∂X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) − hr X4 (n, p, q) ∂S ∂X2 (n, p, q, S) π∂X5 (p, S) − = 0. (2.3) − πX5 (p, S) + X1 (n, p, q, S) ∂S ∂S Now (2.1), (2.2), and (2.3) can be solved simultaneously to find the values of q, p, and S. For a sufficient condition, we determine the following ∂ 2 P21 (n, p, q, S) ∂ 2 X1 (n, p, q, S)

= X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂q 2 ∂q 2 − hr X4 (n, p, q) − πX5 (p, S) 2∂X1 (n, p, q, S) ∂X2 (n, p, q, S) h0 ∂X3 (n, p, q) + − ∂q ∂q ∂q 2 ∂ X2 (n, p, q, S) hr ∂X4 (n, p, q) − + X1 (n, p, q, S) ∂q ∂q 2 h0 ∂ 2 X3 (n, p, q) hr ∂ 2 X4 (n, p, q) , (2.4) − − ∂q 2 ∂q 2 ∂ 2 P21 (n, p, q, S) ∂ 2 X1 (n, p, q, S)

= X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂p 2 ∂p 2

210


− hr X4 (n, p, q) − πX5 (p, S) 2∂X1 (n, p, q, S) ∂X2 (n, p, q, S) h0 ∂X3 (n, p, q) + − ∂p ∂p ∂p hr ∂X4 (n, p, q) π∂X5 (p, S) − − ∂p ∂p 2 ∂ X2 (n, p, q, S) h0 ∂ 2 X3 (n, p, q) + X1 (n, p, q, S) − ∂p 2 ∂p 2 hr ∂ 2 X4 (n, p, q) π∂ 2 X5 (p, S) , (2.5) − − ∂p 2 ∂p 2 ∂ 2 P21 (n, p, q, S) ∂ 2 X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) = ∂S 2 ∂S 2 − hr X4 (n, p, q) − πX5 (p, S) 2∂X1 (n, p, q, S) ∂X2 (n, p, q, S) π∂X5 (p, S) − + ∂S ∂S ∂S 2 ∂ X2 (n, p, q, S) π∂ 2 X5 (p, S) , (2.6) + X1 (n, p, q, S) − ∂S 2 ∂S 2 ∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) = ∂q∂p ∂p∂q ∂ 2 X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂q∂p ∂X1 (n, p, q, S) − hr X4 (n, p, q) − πX5 (p, S) + ∂q ∂X2 (n, p, q, S) h0 ∂X3 (n, p, q) hr ∂X4 (n, p, q) − − × ∂p ∂p ∂p ∂X1 (n, p, q, S) ∂X2 (n, p, q, S) π∂X5 (p, S) + − ∂p ∂p ∂q h0 ∂X3 (n, p, q) hr ∂X4 (n, p, q) − − ∂q ∂q 2 ∂ X2 (n, p, q, S) h0 ∂ 2 X3 (n, p, q) − + X1 (n, p, q, S) ∂q∂p ∂q∂p 2 hr ∂ X4 (n, p, q) − , (2.7) ∂q∂p

=

∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) = ∂q∂S ∂S∂q


211

∂ 2 X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂S∂q ∂X1 (n, p, q, S) − hr X4 (n, p, q) − πX5 (p, S) + ∂S ∂X2 (n, p, q, S) ∂X3 (n, p, q) ∂X4 (n, p, q) × − h0 − hr ∂q ∂q ∂q ∂X1 (n, p, q, S) ∂X2 (n, p, q, S) π∂X5 (p, S) + − ∂q ∂S ∂S 2 2 ∂ X2 (n, p, q, S) π∂ X5 (p, S) + X1 (n, p, q, S) − , (2.8) ∂S∂q ∂S∂q

=

∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) = ∂p∂S ∂S∂p ∂ 2 X1 (n, p, q, S)

X2 (n, p, q, S) − A − nCt − h0 X3 (n, p, q) ∂p∂S ∂X1 (n, p, q, S) − hr X4 (n, p, q) − πX5 (p, S) + ∂p ∂X1 (n, p, q, S) ∂X2 (n, p, q, S) π∂X5 (p, S) − + × ∂S ∂S ∂S ∂X2 (n, p, q, S) h0 ∂X3 (n, p, q) − × ∂p ∂p hr ∂X4 (n, p, q) π∂X5 (p, S) − − ∂p ∂p 2 ∂ X2 (n, p, q, S) π∂ 2 X5 (p, S) − . (2.9) + X1 (n, p, q, S) ∂p∂S ∂p∂S

=

Sufficient conditions for maximizing P21 (n, p, q, S) are ∂ 2 P21 (n, p, q, S) < 0, ∂q 2

∂ 2 P21 (n, p, q, S) < 0, ∂p 2

∂ 2 P21 (n, p, q, S) < 0, ∂S 2

and

2 ∂ 2 P21 (n, p, q, S) ∂ P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) 2 ∂q ∂p 2 ∂S 2 2 ∂ 2 P21 (n, p, q, S) ∂ P21 (n, p, q, S) 2 − − ∂p∂S ∂q∂p 2 ∂ P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) × − ∂p∂q ∂p∂S ∂S∂q ∂S 2

212


∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) + ∂q∂S ∂p∂q ∂S∂p ∂ 2 P21 (n, p, q, S) ∂ 2 P21 (n, p, q, S) >0 − ∂S∂q ∂p 2 at optimal values of q, p, and S. We have shown below graphically that the profit function is concave with respect to the shipment quantity, the price, and the number of shortages.

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