Supply-chain coordination model with insufficient production capacity and option for outsourcing

Mathematical and Computer Modelling 46 (2007) 1442–1452 www.elsevier.com/locate/mcm

Supply-chain coordination model with insufficient production capacity and option for outsourcing Santanu Sinha, S.P. Sarmah ∗ Department of Industrial Engineering and Management, IIT Kharagpur 721302, India Received 29 January 2007; accepted 1 March 2007

Abstract In this paper, a mathematical model is developed to analyze the situation of lost sales from the supply-chain coordination perspective. In a two-stage supply chain, the supplier’s production capacity is less than the annual demand of the retailer. The supplier may recover the deficit by procuring the same from an external source at a certain price and then supplying it back to the retailer. Here, the conditions have been derived when the practice of external procurement may be a viable solution to enhance the profits of both the supplier and retailer in a coordinated approach. A numerical example is carried out to illustrate the efficacy of the developed model and the procedure developed for solving the problem. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Supply-chain coordination; Insufficient production capacity; Lost sales; Outsourcing

1. Introduction and review of literature Absolute stock-outs or lost sales is one of the important issues of concern for many organizations. In today’s competitive business environment, stock-outs at a particular supplier/manufacturer will create opportunities for the other competitors and therefore the supplier/manufacturer facing the stock-outs or lost-sales situation must use effective strategies to hold their customers. There may be various reasons in an organization for stock-outs, such as, machine break-downs, labor strikes, limited space for inventory, constraints over initial investment or budget limitations, inadequate machineries/equipments or technological constraints etc. Dion et al. [8] mentioned that irrespective of the size of an organization not receiving/delivering the intended products may cause a number of issues to the firm ranging from disruption of costly production process to loss of revenue, disqualification of suppliers, and increased administrative costs. One of the strategies of the supplier/manufacturer under these situations may be to outsource the item to an external source and then to supply it back to its customers and by the process, the original supplier/manufacturer may retain its customer base. The term ‘outsourcing’ is used here since for the supplier, it is not merely an external procurement decision but rather a strategic decision. Such a type of situation is normally observed in low or medium valued items or un-branded products where all the competitors maintain the same quality and the product of a particular supplier/manufacturer cannot be distinguished and even if it is distinguished, it does not have ∗ Corresponding author. Tel.: +91 3222 283735.

E-mail addresses: santanu [email protected] (S. Sinha), [email protected] (S.P. Sarmah). c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.03.014

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much impact on the retailer. Typical examples may be various machine parts, fasteners, small stationery items etc. In this work, the term ‘supplier’ or ‘manufacturer’ is used interchangeably to represent the same entity who is delivering the items to the retailer. In the related literatures, the researchers and practitioners have considered lost sales as the cause of lesser revenue or profit and devised several methods to optimize the situation considering backorder or complete lost sales for the individual manufacturer/supplier. Hill [9] discussed a case of an intra-organizational/departmental transfer of stocks to meet the end demand in the case of a stock-out and mentioned the scope for inter-organization/departmental transfer of stocks. Van Delft and Vial [11] discussed a case where a particular type of product is subjected to fast obsolescence and have shown the stock-out situation as an incentive for saving inventory-holding cost. They also considered a strategy where it is possible to deliver the exact product to the customer if they agree to accept a late delivery. Further, Gallego and Moon [12] addressed a problem related to stock-outs in a multi-item, single facility location where initial inventories, set-up times, production and demand rates are known and with given stock-out penalties. They derived a schedule that minimizes weighted stock-out cost. Goyal and Gopalakrishnan [13] have considered a production lot-sizing model with insufficient production capacity that leads to a shortage and a model is developed to determine the optimal number of set-ups to maximize the profit of the supplier. Anupindi and Bassok [15] have considered a periodic review in a two-echelon inventory system where a part of the unsatisfied sales at the retailers is lost. Hill [16] considered replenishment policies for the continuous review model with Poisson demand, fixed leadtime on replenishment and lost sales during a stock-out. They considered a time-dependent stockholding cost and a cost per unit of lost sale but there was no cost associated with placing a replenishment order. The standard policy shown was to specify a maximum stock level and place a replenishment order whenever the demand occurs. Ng et al. [20] developed a model where the stock-out situation is managed from an alternative source with a higher cost due to higher transportation cost. Andersson and Melchiors [18] considered a single warehouse and several retailers inventory system and presented a heuristic to find out the cost for base-stock policies to optimize the lost-sales case. Chu et al. [21] have addressed a capacitated dynamic lot sizing problem for a single item and the cases of backlog or stock-outs have been taken care of by partial/complete outsourcing to minimize the total cost. Liu et al. [22] have mentioned that there are a few particular types of industries where there is high production capacity but limited inventory storage (oil drums/barrels) capacity resulting in a shortage. The authors have developed an algorithm for such capacitated lot-sizing problem where the unsatisfied part of the demand has been considered as lost sales. In the literature discussed above, it is observed that in most of the cases, strategies are developed to deal with lost-sales/stock-out situations either for the manufacturer or for the retailer by adopting suitable ordering, production or inventory reviewing policies. In this work, a lost sale is considered from the supply chain perspective where the retailer and the manufacturer together constitute the supply chain. In the competitive market for doing business, parties involved in the supply chain align their business processes to improve the performance system-wide rather than at the individual level in terms of cost, delivery etc., and in the supply chain literature this is termed supply chain coordination. Thomas and Griffin [14] have mentioned that coordination among different parties of the supply chain improves the performance of the supply chain. A coordinated policy from the supply chain perspective is more desired in today’s competitive business environment to enhance the profit/revenue for all the members of the supply chain. The integrated inventory model can be considered as one of the origins of supply chain coordination literature and there are extensive studies on the integrated inventory models that bring in a ‘win–win’ situation to all the members of the system. Readers can go through review articles carried out in this stream by authors such as Goyal and Gupta [7] and Sharafali and Co [17]. Recently, Sarmah et al. [25] have carried out a detailed review of literatures on buyer–vendor coordination models in supply chain management. Goyal [1] initiated the joint economic lot size (JELS) model and later it was reinforced by Banerjee [4]. The assumption that the manufacturer has the larger set-up cost compared to the buyer is the starting point for the joint economic lot size literature. In situations where the set-up cost of the dominant player of the channel, normally the manufacturer, is higher than the ordering cost of the buyer, the manufacturer induces the buyer to order in quantities greater than the economic order quantity of the buyer. This reduces the set-up cost of the manufacturer but at the same time increases the inventory related costs of the buyer. The manufacturer offers incentive in some form such as quantity discount for the purchases to compensate the increase in cost of the buyer. Many other authors, such as Rosenblatt and Lee [3], Lal and Staelin [2], Lee and Rosenblatt [5] and Weng and Wong [10] have established that the manufacturer’s profit can be improved by asking the buyer to order a quantity larger than the economic order quantity and subsequently compensating any increase in buyer’s cost by giving a

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quantity discount. Recently, Shin and Benton [23] have studied a joint replenishment program coupled with a channel coordination practice. Yang and Wee [24] have used a joint economic order quantity program for channel coordination for deteriorating items. In this work, the problem is formulated as a two-stage supply chain coordination problem with the lost-sales case and is based on the earlier work of Goyal and Gopalakrishnan [13]. We have incorporated the possibility of external procurement by the supplier when it faces shortages and extended the model by considering the interest of the retailer also as long term partnership is desired by the business entities in today’s business environment. The retailer purchases the item from the supplier/manufacturer and sells it in the market. In the coordinated environment, when the supplier is unable to meet the demand of the retailer, the retailer faces lost sales in the market. The penalty cost due to the lost-sales case is incorporated in the supplier’s/manufacturer’s cost equation and a strategy of external procurement or outsourcing by the supplier is considered to meet the shortages at the retailer’s level. The paper investigates the situation when given a certain unit outsource price, the manufacturer/supplier may go for external procurement of the item to maximize the channel profit as well as the individual profits. 2. Assumptions and notation In the development of the mathematical model for the problem, the following assumptions and notation are considered. (i) Demand, procurement price, and selling prices are known and constant. (ii) Replenishments are instantaneous. (iii) Both the supplier and the retailer share complete information. The following notations have been used: D Q g hb hv Cv k1 n1 β Πb Πv Πch Cb P Pb Po Pv

Annual demand Lot size Good-will lost or penalty cost per unit Inventory holding cost of the retailer per unit per year Inventory holding cost of the supplier per unit per year Set-up cost of the supplier Fraction of the annual shortage that is procured externally Production lot size multiplication factor (a positive integer) Ratio of annual capacity to annual demand Profit of the retailer Profit of the supplier System profit Ordering cost of the retailer Unit production cost of the supplier Unit selling price of the retailer in the market Unit cost of procurement from the external sources Unit selling price of the supplier to the retailer.

3. Model formulation In this section, we have considered the case when both the retailer and the supplier individually determine their own policies by optimizing the individual interests and there no effort has been made to recover the lost sales by outsourcing. In Section 3.2, however, both the members take coordinated decisions and we have analyzed whether an effort should be made to encounter the lost sales by outsourcing with a focus on maximizing system profit and thereby individual profit of the members of the supply chain.

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3.1. Non-coordinated policy without outsourcing In this case, the supplier faces an annual demand of D units from the retailer and due to insufficient production capacity; the supplier annually produces only βD units. The retailer places the order at its own optimal lot size Q ∗ units to the supplier and the supplier produces n ∗1 times the retailer’s order quantity (Q ∗ ) to optimize the production set-up number. The quantity that could not be supplied to the retailer is a lost-sale case for the supplier and the supplier is subjected to a penalty cost per unit per year for the lost sales. The retailer’s net profit is given by the following expression Πb = Gross revenue − Purchase cost − Ordering cost − Holding cost   β DCb Qh b = β D (Pb − Pv ) − − . Q 2 From Eq. (1), the retailer’s individual optimal lot size Q ∗ is obtained as follows s 2β DCb ∗ . Q = hb Accordingly, the retailer’s individual optimal profit is given by   Q∗hb β DCb − Πb (Q ∗ ) = β D (Pb − Pv ) − . Q∗ 2

(1)

(2)

(3)

The supplier accepts this lot size Q ∗ and finds an optimal n ∗1 corresponding to the Q ∗ to maximize its individual profit Πv . Πv = Gross revenue − Set-up cost − Inventory holding cost − Penalty cost   h v Q ∗ (n 1 − 1) β DCv − = β D (Pv − P) − − D(1 − β)g . n1 Q∗ 2 From Eq. (4), $ n ∗1

=

1 Q∗

s

2β DCv hv

(4)

% where bxc = integer nearest to x.

Thus, the maximum profit of the supplier is given by   h v Q ∗ (n ∗1 − 1) β DCv ∗ ∗ Πv (n 1 , Q ) = β D Pv − β D P + ∗ ∗ + + D(1 − β)g , n1 Q 2

(5)

(6)

where, Q ∗ and n ∗1 are obtained from (2) and (5). Thus, in the non-coordinated situation, system profit i.e. profit of the buyer and the profit of the manufacturer with complete lost sales is given as follows  Πch = Πb (Q ∗ ) + Πv (n ∗1 , Q ∗ )   h v Q ∗ (n ∗1 − 1) β DCv Q∗hb β DCb (7) = β D (Pb − P) − − ∗ ∗ − − − (1 − β)Dg . Q∗ n1 Q 2 2 3.2. Coordinated policy with outsourcing In this section, we study the earlier non-coordinated policy in a supply chain coordination point of view where we assume that both the retailer and the supplier take decisions jointly and the supplier decides to go for outsourcing to recover lost sales partially/completely, if possible. Any shortage leading to a lost sale is always detrimental even for the coordinated system as a whole and thus a cost (penalty, good-will loss) is always associated with it. Though, it is

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a channel/system penalty cost, here it is assigned to the supplier for its linear additive property. If k1 is the fraction of the demand shortfall that may be recovered by outsourcing or external procurement, the modified profit equations of the retailer and the supplier are respectively as follows:   Qh b [β D + k1 (D − β D)] Cb 0 − Πb = [β D + k1 (D − β D)] . (Pb − Pv ) − Q 2  β DC h (n Q − Q) v v 1 Πv0 = [β D + k1 (D − β D)] Pv − β D P + + n 1 Q 2 + k1 (D − β D)Po + (1 − k1 )(D − β D)g . Total channel/system profit is given by,  [β + k1 (1 − β)] .DCb 0 Πch = [β + k1 (1 − β)] .D Pb − β D P + Q +

 β DCv h v Q(n 1 − 1) Q.h b + + + k1 D(1 − β).Po + (1 − k1 )(1 − β)Dg . 2 n1 Q 2

If k1 = 0 i.e. when no item is outsourced, the coordinated channel profit equation reduces to (9),   β DCv Qh b h v Q(n 1 − 1) β DCb − − − − (1 − β)Dg . Πch = β D (Pb − P) − Q n1 Q 2 2

(8)

(9)

The objective is to find out the conditions when outsourcing or external procurement may be a better option to improve channel profit and thereby to increase the profits of both the retailer and the supplier beyond their noncoordinated policies. Thus, 0 Maximize Πch , 0 Πch

 [β + k1 (1 − β)] .DCb = [β + k1 (1 − β)] .D Pb − β D P + Q  β DCv h v Q(n 1 − 1) Q.h b + + + k1 D(1 − β).Po + (1 − k1 )(1 − β)Dg . + 2 n1 Q 2

(10)

Subjected to, (i) 0 ≤ k1

(11)

(ii) k1 ≤ 1

(12)

(iii) n 1 Q ≤ D (iv) n 1 ε I [Integer]

(13)

(v) Q > 0.

(14)

3.3. Solution procedure In the above problem n 1 , k1 and Q are decision variables and it is a non-linear optimization problem. To solve the problem, we first consider n 1 to be constant and find out the values of k1 and Q. Then we go for selecting a wide range of possible values of n 1 and compute the values of the objective function. Finally, we select the best solution in a direct search method. The objective function (10) is concave in nature for a particular value of n 1 (see Appendix A) and all the constraints are linear and thereby satisfy the conditions of the Karush–Kuhn–Tucker (KKT) sufficiency theorem [6] and if there exists a solution, it will be optimal. Four possible sets of solutions with k1 = 0 (i.e. the case when outsourcing is not possible at all) and k1 = 1 are obtained (i.e. the case where complete outsourcing is possible). The results are shown in Table 1 (derivation is given in Appendix B). 0 ) ∗ ∗ ∗ The following stepwise procedure is used to determine the value of (Πch Max = f (Q , k1 , n 1 ).

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k1

Q r

2β D {n 1 Cb +Cv } n 1 {h b +h v (n 1 −1)}

1

k1 = 0

Q=

2

k1 = 0

Q = nD 1

Condition(s) i h C Po > Pb − Qb + g i h C Po > Pb − Qb + g and r 2β D {n Cb +Cv } Q < n {h +h1 (n −1)} 1

r 3

k1 = 1

Q=

4

k1 = 1

Q = nD

2D {Cb n 1 +βCv } n 1 {h b +h v (n 1 −1)}

h

C

Pb − Qb + g > Po

i C Pb − Qb + g > Po and r 2D {Cb n 1 +βCv } Q < n {h +h (n −1)} 1

5

k1 = 1

Q=

1

i

h

1

r

v

b

2D {Cb n 1 +βCv } n 1 {h b +h v (n 1 −1)}

h

b

v

1

i

C Pb − Qb − Po + g = 0

Step 1 Set n 1 = 1. Step 3 Step 4 Step 5

q

2D{Cb n 1 +βCv } D 0 n 1 {h b +h v (n 1 −1)} and Q 0 = n 1 . Cb If Q 0 > Q 00 > (Pb −P , then Q = Q 00 and k1 = 1; 0 +g) Cb Else, if Q 00 > Q 0 ≥ (Pb −P , then Q = Q 0 and k1 = 1. 0 +g) q D{Cb n 1 +Cv } and Qˆ 0 If step (3) is false, compute Qˆ = n2β 1 {h b +h v (n 1 −1)} If Qˆ > Qˆ 0 , then Q = Qˆ 0 and k1 = 0;

Step 2 Compute Q 0 =

=

D n1 .

Else, Q = Qˆ and k1 = 0. 0 ) = f (Q, k , n ). Step 6 Calculate (Πch 1 1 1 Step 7 Repeat steps (1)–(6) with n 1 = 2 to n 1 = j (where j = an arbitrary possible high value of n 1 ). 0 ) ∗ ∗ ∗ Step 8 Find (Πch Max = f (Q , k1 , n 1 ). Following the above-mentioned steps, the maximum channel/system profit may be derived in both cases, where, (i) outsourcing is possible and (ii) outsourcing is not possible, for a given unit outsource price P0 . 4. Analysis of coordinated channel profit considering complete outsourcing If we take the first derivative of (10) with respect to k1 for a certain value of Q,   0 ∂Πch Cb = (1 − β)D. Pb + g − − Po ∂k1 Q   Cb − Po > 0 Now, if Pb + g − Q 0 ∂Πch > 0. ∂k1 0 = f (k , Q) is an increasing function of k for a certain This shows, for (Pb + g − CQb − Po ) > 0, the function Πch 1 1 value of Q and the maximum will occur if we take the highest value of k1 possible for a corresponding optimal value of Q and the possible k1 max = 1. Thus, it is found that for a particular value of n 1 and Q, as long as [Pb + g − CQb − Po ] > 0, system profit increases with increasing k1 and reaches the maximum value when a complete outsourcing is implemented. On the other hand, if [Pb + g − CQb − Po ] < 0, it is better not to choose the outsourcing strategy. The underlying mechanism is to manipulate lot size Q in such a way that the overall system cost is decreased and the savings can be utilized to go for outsourcing even at a higher price till a particular limiting value.

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In the solution procedure, we focused on either k1 = 0 or k1 = 1 and the proposed algorithm yields the best solution for the coordinated policy in both cases. It is also shown numerically that a coordinated policy has a better system profit in both cases. 5. Numerical example In the numerical example, we have considered the same data as used by Goyal and Gopalakrishnan [13] for the supplier. Here, data for the retailer is assumed (see Table 2). Table 2 Data for supplier and retailer Supplier

Retailer

D = 10 000 β = 0.9 Cv = 200 P = 30 Pv = 40 hv = 6 g=5

Cb = 80 Pb = 45 hb = 5

5.1. When the retailer and the supplier take decisions individually (non-coordinated situation) The retailer selects the optimal Q ∗ = 536.65. Accordingly, the supplier has determined the lot size multiplier n ∗1 = 1.5 ≈ 2. The individual profit of the retailer and the supplier are given as: Πb∗ (Q ∗ ) = 42 316.72. Πv∗ (n ∗1 , Q ∗ ) = 81 712.98. ∗ = 124 029.70. Total system profit, Πch Case 1: Coordinated policy with complete outsourcing The value of P0 = 47 is assumed and the problem is solved using the solution procedure described in Section 3.3. The results are tabulated in Table 3.

Table 3 Coordinated policy n1

Q

k1



1* 2 3 4 5 6 7 8 9 10

1019.8* 556.0 405.8 329.7 282.8 250.7 227.1 208.8 194.2 182.3

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

2.921 2.856 2.803 2.757 2.717 2.681 2.648 2.617 2.588 2.561

C

Pb + g − Qb − Po



System profit 127 901.0* 126 884.4 126 100.7 125 417.1 124 797.6 124 225.0 123 689.5 123 184.2 122 704.4 122 246.4

The result shows that maximum system profit is obtained for the coordinated policy when k1 = 1.0, Q ∗ = 1019.8, n ∗1 = 1. Further, it is observed that [Pb + g − CQb − Po ] = 2.921 > 0 and therefore it has become possible to outsource the entire shortage beneficially. From the results, it is also observed that as the value of n 1 i.e. the lot size multiplier increases, the system profit has decreased. The profit components of different members of the supply chain are shown in Table 4.

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Table 4 Net profit components of the different members Member

Net profit

Improvement with respect to non-coordinated policy

Retailer Supplier System

46 666.03 81 234.95 127 901.0

4349.31 −478.03 3871.3

The apparent loss of the supplier may be completely compensated by the retailer’s larger gain and still the system has a coordination benefit of 3871.3 with outsourcing, which may be further shared to raise the individual profit higher than that of their earlier non-coordinated approach. 5.2. Case 2: Coordinated policy without outsourcing The value of P0 = 52 is considered here and the same problem is solved using the procedure described in Section 3.3. The results are tabulated in Table 5. Table 5 Coordinated policy n1

Q

k1



1* 2 3 4 5 6 7 8 9 10

1004.0* 542.7 394.1 318.9 272.9 241.4 218.3 200.5 186.3 174.7

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

−2.078 −2.147 −2.203 −2.251 −2.293 −2.331 −2.366 −2.399 −2.429 −2.458

C

Pb + g − Qb − Po



System profit 124 980.0* 124 030.1 123 300.7 122 663.8 122 085.5 121 550.1 121 048.7 120 575.0 120 124.8 119 694.7

In this case, [Pb + g − CQb − Po ] = −2.078 < 0 and for the coordinated policy: k1 = 0, Q ∗ = 1004, n ∗1 = 1 gives the optimal value. The result implies that P0 = 52 is too high and outsourcing is not the feasible option for better channel profit. The profit component of supplier, retailer and the total system profit are shown in Table 6, which shows that the system profit is still higher than that of the earlier non-coordinated approach in Section 5.1. Table 6 Net profit components of the different members Member

Net profit

Improvement with respect to non-coordinated policy

Retailer Supplier System

41 772.88 83 207.16 124 980.04

−543.84 1494.18 950.34

The result in Table 6 shows that the retailer incurs a profit less than his earlier non-coordinated profit by the amount 543.84. However, this amount may be compensated by the larger gain of the supplier during the process. The system is still left with a coordination benefit of 950.34. 6. Conclusions In this study, the lost-sales case from the supply-chain perspective has been analyzed. In most of the previous works, lost-sales cases have only been studied at the individual business entity level. In this paper, considering insufficient production capacity of the supplier as the reason for shortages, it has been shown that total system profit may be improved by outsourcing. The conditions have been derived when the supplier may go for complete outsourcing

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to enhance system profit for a particular value of outsource price. The concept is that joint economic lot size may be manipulated in such a way that the overall system cost is decreased and the savings can be utilized to afford outsourcing even at a higher price, till a particular limiting value. The numerical results show the potential benefits of outsourcing in the coordinated channel. It has been shown that in a coordinated policy total system profit increases to compensate the loser and the coordination benefit can be further shared between the two parties by some predetermined static rule to enhance even their individual profit to a better extent. The work may further be extended towards a more dynamic environment considering a price sensitive demand or variable/discrete demand situation. Appendix A. Test for concavity of the objective function Z( Q, k1 ) It is sufficient to show that Z (Q, k1 ) is concave for Q, k1 > 0. The Hessian Matrix H (Q, k1 ) is given as  2    ∂2 Z ∂ Z −2 [β + k1 (1 − β)] .DCb + β DCv  ∂ Q2 3 ∂ Q∂k1  Q n1 Q3  =   H (Q, k1 ) =   ∂2z ∂ 2 Z  (1 − β).DCb Q2 ∂k1 ∂ Q ∂k12  2 (1 − β)DCb =− < 0. Q2



(1 − β).DCb Q2

0



Since, H (Q, k1 ) < 0 and the members of the principal diagonal are negative/zero, the objective function is a concave function (Ref. [19]). Appendix B The optimization problem with the constraints (11)–(14) can be transformed to the following unconstrained maximization problem:   {β + k1 (1 − β)} DCb Qh b β DCv h v Q (n 1 − 1) L = [β + k1 (1 − β)] D Pb − β D P + + + + Q 2 n1 Q 2 − [k1 D (1 − β) P0 + (1 − k1 ) (1 − β) Dg] − λ1 (k1 − 1) − λ2 (n 1 Q − D) , With k1 , Q > 0 the Karush–Kuhn–Tucker (KKT) conditions can be derived as follows:   {β + k1 (1 − β)} .DCb β DCv 1 Q + − {h b + h v (n 1 − 1)} − n 1 λ2 = 0 2 Q2 n1 Q2   Cb (D − β D) k1 Pb (D − β D) − − Po (D − β D) + g(D − β D) − λ1 = 0 Q

(λ1 , λ2 ≥ 0) .

(B.1)

(B.2) (B.3)

λ1 (k1 − 1) = 0

(B.4)

λ2 (n 1 Q − D) = 0.

(B.5)

Now, four possible combinations for the Lagrange’s multipliers λ1 , λ2 ≥ 0 are possible and the combinations are shown in Table B.1. Table B.1 Possible combinations for the Lagrange’s multipliers Case

1

2

3

4

λ1 λ2

>0 =0

>0 >0

=0 >0

=0 =0

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Case 1. λ1 > 0, λ2 = 0. From (B.4), k1 = 1. From (B.3), λ1 = [Pb − CQb − Po + g](D − β D) > 0 or,   Cb + g > Po . Pb − Q From (B.2), s Q=

2D {Cb n 1 + βCv } . n 1 {h b + h v (n 1 − 1)}

Case 2. λ1 > 0, λ2 > 0, From (B.4) and (B.5), k1 = 1 and Q = From (B.3),   Cb Pb − + g > Po . Q From (B.2), s Q<

1451

(B.6)

(B.7)

D n1 .

2D {Cb n 1 + βCv } . n 1 {h b + h v (n 1 − 1)}

(B.8)

(B.9)

Case 3. λ1 = 0, λ2 > 0. From (B.3)–(B.5), D . n1   1 {β + k1 (1 − β)} .DCb β DCv 1 λ2 = + − {h b + h v (n 1 − 1)} > 0 n1 2 Q2 n1 Q2   Cb − Po + g (D − β D) = 0. k1 Pb − Q • If k1 = 0, from (B.11), s 2β D {n 1 Cb + Cv } . Q< n 1 {h b + h v (n 1 − 1)} Q=

• If k1 = 1, from (B.11), s 2D {Cb n 1 + βCv } . Q< n 1 {h b + h v (n 1 − 1)} And, from (B.12),   Cb Pb − − Po + g = 0. Q

(B.10) (B.11) (B.12)

(B.13)

(B.14)

(B.15)

Case 4. λ1 = 0, λ2 = 0. This case implies, {β + k1 (1 − β)} .DCb β DCv 1 + − {h b + h v (n 1 − 1)} = 0 2 Q2 n1 Q2   Cb k1 Pb − − Po + g (D − β D) = 0. Q

(B.16) (B.17)

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• If k1 = 0, from (B.16), s 2β D {n 1 Cb + Cv } Q= . n 1 {h b + h v (n 1 − 1)} • If k1 = 1, from (B.16), s 2D {Cb n 1 + βCv } . Q= n 1 {h b + h v (n 1 − 1)} And, from (B.17),   Cb − Po + g = 0. Pb − Q

(B.18)

(B.19)

(B.20)

References [1] S.K. Goyal, An integrated inventory model for a single supplier single customer problem, International Journal of Production Research 15 (1) (1977) 107–111. [2] R. Lal, R. Staelin, An approach for developing an optimal discount pricing policy, Management Science 30 (12) (1984) 1524–1539. [3] M.J. Rosenblatt, H.L. Lee, Improving profitability with quantity discounts under fixed demand, IIE Transaction 17 (4) (1985) 388–395. [4] A. Banerjee, A joint economic-lot-size model for purchaser and supplier, Decision Sciences 17 (3) (1986) 292–311. [5] H.L. Lee, M.J. Rosenblatt, A generalized quantity discount pricing model to increase supplier’s profits, Management Science 32 (9) (1986) 1177–1185. [6] A. Ravindran, D.T. Phillips, J.J. Solberg, Operations Reserch: Principles and Practice, John Wiley & Sons, Inc., NY, 1987. [7] S.K. Goyal, Y.P. Gupta, Integrated inventory model: The buyer vendor co-ordination, European Journal of Operational Research 41 (3) (1989) 261–269. [8] P.A Dion, L.M. Hasey, P.C. Dorin, J. Lundin, Consequences of inventory stockouts, Industrial Marketing Management 20 (1) (1991) 23–27. [9] R.M. Hill, Using inter-branch stock transfers to meet demand during a stock-out, International Journal of Retail & Distribution Management 20 (3) (1992) 27–33. [10] Z.K. Weng, R.T. Wong, General models for the supplier’s all unit quantity discount policy, Naval Research Logistics 40 (7) (1993) 971–991. [11] Ch. Van Delft, J.P. Vial, Discounted costs, obsolescence and planned stock-outs with the EOQ formula, International Journal of Production Economics 44 (3) (1996) 255–265. [12] G. Gallego, I. Moon, How to avoid stockouts when producing several items on a single facility? What to do if you can’t? Computers & Operations Research 23 (1) (1996) 1–12. [13] S.K. Goyal, M. Gopalakrishnan, Production lot sizing model with insufficient production capacity, Production Planning and control 7 (2) (1996) 222–224. [14] D.J. Thomas, P.J. Griffin, Coordinated supply chain management, European Journal of Operational Research 94 (1) (1996) 1–15. [15] R. Anupindi, Y. Bassok, Centralization of stocks: Retailers vs. manufacturer, Management Science 45 (1999) 178–191. [16] R.M. Hill, On the sub-optimality of (s, 1 − s) lost sales inventory policies, International Journal of Production Economics 59 (1–3) (1999) 387–393. [17] M. Sharafali, H.C. Co, Some models for understanding the cooperation between the supplier and the buyer, International Journal of Production Research 38 (15) (2000) 3425–3449. [18] J. Andersson, P. Melchiors, A two-echelon inventory model with lost sales, International Journal of Production Economics 69 (3) (2001) 307–315. [19] F.S. Hillier, G.J. Lieberman, Introduction to Operations Research, seventh ed., Tata McGraw-Hill, New Delhi, 2001, pp. 1161–1163. [20] C.T. Ng, Y.O. Li Leon, K. Chakhlevitch, Coordinated replenishments with alternative supply sources in two-level supply chains, International Journal of Production Economics 73 (3) (2001) 227–240. [21] F. Chu, C. Chu, X. Liu, Lot sizing models with backlog or out-sourcing, IEEE International Conference: Systems, Man and Cybernetics 5 (2004) 4342–4347. [22] X. Liu, C. Wang, F. Chu, C. Chu, A forward algorithm for capacitated lot sizing problem with lost sales, in: WCICA 2004. Fifth World Congress on Intelligent Control and Automation, vol. 4, 2004, pp. 3192–3196. [23] H. Shin, W.C. Benton, Quantity discount based inventory coordination: Effectiveness and critical environmental factors, Production and Operations Management 13 (1) (2004) 63–76. [24] P.C. Yang, H.M. Wee, A win–win strategy for an integrated vendor–buyer deteriorating inventory system, in: Proceedings of the 10th International Conference MMA2005 & CMAM2, Trakai, Mathematical Modelling and Analysis (2005) 541–546. [25] S.P. Sarmah, D. Acharya, S.K. Goyal, Invited review: Buyer vendor coordination models in supply chain management, European Journal of Operational Research 175 (1) (2006) 1–15.