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International Journal of Electronic Business Management, Vol. 8, No. 3, pp.231-238 (2010)

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A COLLABORATIVE DETERIORATING INVENTORY SYSTEM WITH IMPERFECT QUALITY AND SHORTAGE BACKORDERING Jonas C.P. Yu Logistics Management Department Takming University of Science and Technology Taipei (114), Taiwan

ABSTRACT In this study, we develop a collaborative inventory system consisting of one supplier and one buyer. The objective is to maximize the total profit of the whole system when shortage caused by the imperfect quality can be completely backordered. To ensure mutual benefits to every player, a negotiation mechanism is incorporated to share the profit between both players according to their contribution. A deteriorating inventory system with finite replenishment rate and price sensitive demand is set to simulate a real situation of the high-tech, short life cycle and perishable fashion product. A numerical example and sensitivity analysis are carried out to illustrate the model. Keywords: Collaborative System, Imperfect Quality, Deterioration Rate, Price-Sensitive Demand

*

1. INTRODUCTION

Supply chain collaboration is where members in a supply chain have a common objective, namely, to construct a collaborative network and share mutual information, such as sales and stock-level. Undoubtedly, good collaboration within the supply chain will enhance performance of the entire supply chain. Recently, due to rising costs, globalization, shrinking resources, shortening product life cycles and quicker response time, increasing attention has been placed on the collaboration of the whole supply chain systems. An effective supply chain network requires a cooperative relationship between the vendor and the buyer. Based on mutual trust, cooperation includes the sharing of information, resources and profit. The result of close cooperation is a mutually beneficial environment, which increases the joint profit as well as enables quicker response to customer demand. For the reason, the trade of between quality and partner negotiation in a supply chain system is seriously implemented especially on the moment that shortage is caused by the imperfect quality and whole system tends to ensure mutual benefits to every player according to their contribution. One of the most common strategies of a collaborative system is to develop an optimal replenishment and mutually beneficial policy acceptable to both the vendor and the buyer. Clark *

Corresponding author: [email protected]

and Scarf [5] presented the concept of serial multi-echelon structures to determine the optimal policy. Banerjee [1] derived a joint economic lot size model for a single vendor, single buyer system where the vendor has a finite production rate. Goyal [6] extended Banerjee’s model by relaxing the lot-for-lot production assumption. Ha and Kim [9] used a graphical method to analyze the integrated vendor–buyer inventory status and derive an optimal solution. Yang and Wee [15] developed algebraically an optimal policy of the integrated vendor-buyer inventory system without using differential calculus. Yu et al. [16] developed a mathematical model of deteriorating items which considers the vertical integration of the producer, the distributor and the retailer and the horizontal integration of the producers. Recently, Gunasekaran et al. [8] developed a competitive strategy in a collaborative framework of responsive supply chain by comparing their characteristics and objectives, reviewing the selected literature, and analyzing some case experiences. Lyu et al. [10] focused on the store-level retailer’s replenishment problems and proposed three collaborative replenishment mechanism models in the collaborative supplier and store-level retailer environment. As a result of the imperfect production and transportation, arriving stocks often contain some defective items. The arrival of defective items may lead to massive losses for the buyer. Therefore, the suppliers have to offer compensation to their loss. Conventional inventory models may be inappropriate

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for the case with defective items. Other factors, such as damages and breakages during the handling process may also result in defective items. These considerations were discussed by Salameh and Jaber [12] who are among the few authors to consider imperfect quality. Later, Cardenas-Barron [2] corrected a mistake in the final formulae of Salameh and Jaber’s model. Goyal and Cardenas-Barron [7] then reconsidered the work done in Salameh and Jaber’s paper [12] and presented a practical approach to determine the optimal lot size. They assumed that poor items are withdrawn from stock and no shortages are allowed. Wee et al. [14] developed an optimal inventory model for items with imperfect quality and shortage backordering. Then, Chang and Ho [3] revisit their study and applied the well-known renewal- reward theorem to obtain a new expected net profit per unit time and derived the exact closed-form solutions, specifically without differential calculus. Chen and Kang [4] developed an integrated vendor–buyer model that considered a permissible delay in payment and imperfect quality to determine the optimal solutions of the buyer’s order quantity and the frequency for each vendor’s production run. This study discusses a high-priced item for a monopolistic market channel. Shortages of items may occur due to imperfect items. Due to our monopolistic market assumption, there is only one source. The supplier has a close vender-buyer relationship with the buyer. Thus, complete backorder for the defective items is assumed. This study develops a win-win collaborative inventory model for deteriorating items with finite replenishment rate and price sensitive demand. To ensure mutually beneficial strategy, a negotiation factor is incorporated to enable profit sharing between both players. Sensitivity analysis is carried out to show the relationship between the extra profit and the demand and deterioration rate. The remainder of this paper is organized as follows. Two mathematical models with different assumptions are developed in section 3. Section 4 describes the solution procedures for the two models, the numerical example and sensitivity analysis. The last section is the conclusions.

2. ASSUMPTIONS AND NOTATIONS 2.1 Assumptions The mathematical models in this paper are developed on the basis of the following assumptions: (a) Single vendor and single buyer are considered. (b) Buyer’s replenishment rate is instantaneous, and vendor’s replenishment rate is finite. (c) Demand rate is linear decreasing with retail price.

(d) (e) (f) (g) (h) (i)

The players have complete knowledge of each other’s information. Shortage is completely backordered. A single item with a constant rate of deterioration is considered. A constant deterioration rate for the on-hand stock. There is no replacement or repair of deteriorated units. Holding cost applies to good units only.

Two models are discussed. The first model does not consider the vendor-buyer collaboration. The second model considers the vendor-buyer collaboration. The buyer’s related notations are as follows: Buyer’s inventory level for model i, i=1, 2 1 Buyer’s inventory level in the non-shortage I bi (t) stage for model i, i=1, 2 Buyer’s inventory level in the shortage I 2bi (t) stage for model i, i=1, 2 Co Ordering cost for buyer, $ per order b Unit purchasing price υ Percentage holding cost per year and per dollar Qbi Lot size per delivery for model i qbi The highest inventory level for model i δ The defective percentage in Qbi S the maximum backordering quantity in units (S=Qbi－qbi) Cb Backordering cost, $ per unit TPbi Annual total profit for model i EPb Extra profit sharing for model 2 as compared to model 1 (EPb=TPb2-TPb1). I bi (t)

The vendor’s related notations are: Annual replenishment rate Vendor’s actual inventory level 1 Vendor’s total inventory level in I vi (t ) production period Vendor’s total inventory level in I vi2 (t ) non-production period Cs Setup cost, $ per cycle v Unit cost μ Percentage holding cost per year and dollar TPvi Annual total profit for model i EPv Extra profit sharing for model 2 compared to model 1 (EPv=TPv2-TPv1)

p I vi (t )

the the

per as

The other related notations for both the vendor and the buyer: a Demand scale parameter b Demand price-sensitive parameter  Deterioration rate of on-hand-stock TPi Annual total profit (TPvi & TPbi) for model

C. P. Yu: A Collaborative Deteriorating Inventory System with Imperfect Quality and Shortage Backordering i Negotiation factor of extra profit sharing between the vendor and the buyer

ζ

Tbi ni

r di Ti Ti1 Ti 2

The decision variable notations are as follows: Buyer’s replenishment period for model i Number of deliveries from the vendor to the buyer per cycle for model i End customer’s retail price Annual price-sensitive demand rate, where d i  a  b r Vendor’s cycle time for model i Vendor’s production period per cycle for model i Vendor’s non-production period per cycle for model i

3. MATHEMATICAL MODELING AND ANALYSIS

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The vendor’s total inventory systems for the production period and the non-production period are as follows: dI vi1 (t ) 1  I vi1 (t )  p  d i , 0  t  Ti dt

(2)

dI vi2 (t )  I vi2 (t )   d i , 0  t  Ti 2 dt

(3)

and

The boundary and other related conditions are: I bi (Tbi1 )  0

(4)

I (0)  

(5)

I (Ti 2 )  

(6)

I vi1 (Ti1 )  I vi2 (0)

(7)

Ti  Ti  Ti

(8)

1 vi 2 vi

1

2

and

The vendor’s total inventory level, the buyer’s inventory level and the vendor’s actual inventory level are illustrated in Figure 1. The actual vendor’s average inventory level in the integrated two-echelon inventory model is the difference between the vendor’s total average inventory level and the buyer’s average inventory level. Since the inventory level is depleted by a constant deterioration rate to the on-hand stock, the buyer’s inventory level is represented by the following differential equation

Tbi 

Ti  Tbi1  Tbi2 ni

The solutions of the above equations using Spiegel (1960) are: I bi1 (t ) 

di



I vi2 (t ) 

  Inventory levels

p  di



 S t , 0  t  T2 bi Tbi2

(10) (11)

1 (1  exp(  t ))   , 0  t  Ti

(12)

2 [exp( (Ti 2  t ))  1]   , 0  t  Ti

(13)



di

differential

1 [exp( (Tbi1  t ))  1] , 0  t  Tbi

I bi2 (t )  I vi1 (t ) 

(9)

qbi 

di



[exp(Tbi1 )  1]

S

qbi   1 

(14) (15)

The values of β and α in Figure 1 are derived as: 

Time  SB 



   Tbi1   Tbi2 Ti1

Ti 2  

2

(16)

and

Ti  

Figure 1: Vendor’s and buyer’s inventory levels for the case n= 4, where Vendor’s total inventory level, I vi1 (t ) Vendor’s total inventory level, I vi2 (t ) Buyer’s inventory level, Ibi(t) Vendor’s actual inventory level dI bi (t )  I bi (t )  d i , 0  t  Tbi , i= 1, 2, 3 dt

d i Ti (2ni  Ti )  2ni2 d i S 2 2ni p



1    ln 1      d i

 d i 2Ti (2ni  Ti )  2ni2 d i S    2   2ni p  

(17)

respectively. If the product of deterioration rate and replenishment interval is much smaller than one, the buyer’s and the vendor’s actual average inventory levels, I bi and I vi are

(1)

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International Journal of Electronic Business Management, Vol. 8, No. 3 (2010) 1 I bi  1 Tbi  

Tbi1

0

di

T  1 bi

TPbi   r d i  (

 I bi (t )dt 2

(18)

1 i bi

dT 2

  b I bi   b I bi 

C o 1 Cb S 2 )  Tbi 2 d iTbi

(29)

TPvi 

 b (qbi  S ) Ti

(

 v pTi1 Ti

  C   v I vi   v I vi  s ) Ti

(30)

respectively. From (7), one can derive the following condition:

T2

i 1 i I vi  [  I vi1 (t )dt   I vi2 (t )dt ]  I bi Ti 0 0



Tbi

and [ 1  Tbi1  exp(Tbi1 )]

and T1

 b (qbi  S )

2 2 2 d T1 1 ( pTi1  d i Ti1  2Ti1  2Ti 2  d i Ti 2 )  i bi 2Ti 2

(19)

( p  d i )(1  exp( Ti1 ))   d i (1  exp(Ti 2 ))

By Taylor’s series expansion, (31) is derived

respectively.

as:

The annual total holding cost for the buyer and the vendor are: Buyer’s holding cost = b I bi

(20)

Vendor’s actual holding cost = v  I vi

(21)

1 1 ( p  d i )Ti1 (1  Ti1 )  d iTi 2 (1  Ti 2 ) 2 2

Ti1 

respectively. The annual deterioration costs for the buyer and the vendor are: Buyer’s deterioration cost = b I bi

(22)

Vendor’s deterioration cost =  v I vi

(23)

The annual backordering cost for the buyer is: 2 Buyer’s backordering cost = 1 Cb S (24) 2 d iTbi The annual setup costs for the buyer and the vendor are: Buyer’s setup cost =

Co Tbi

(25)

Ti 

(26)

respectively. Therefore, the annual purchased costs for the buyer and the vendor are:  b (qbi  S ) Tbi

(27)

v pTi

Vendor’s purchased cost = T i

Ti 2 1 ( p  d iTi 2 ) p  di 2

Maximize TPb1  TPb1 (Tb1 ,  r )

(34)

(35)

where TPb1 is a function of two variables, Tb1 and r . The optimal solution must satisfy the following two conditions simultaneously:

and

TPb1 0 Tb1

(36)

TPb1 0  r

(37)

After the value of Tb1 is determined, the vendor’s second step decision is: Maximize TPv1  TPv1 (n1 )

and

(33)

Model 1: Inventory model without collaboration In this model, the buyer makes the first step decision, and then the vendor makes the second step decision. The first step decision is:

and Cs Vendor’s setup cost = T i

di 1 Ti 2 (1  Ti 2 ) p  di 2

From (8), one can derive:

and respectively.

(32)

From Misra [11], one has:

and

Buyer’s purchased cost =

(31)

(38)

1

(28)

The annual total profits for the buyer and the vendor are:

where TPv1 is a function of one variable, n1. The optimal solution of n1, denoted as n1* , must satisfy the following condition:

C. P. Yu: A Collaborative Deteriorating Inventory System with Imperfect Quality and Shortage Backordering *

*

*

TPv1 (n1  1)  TPv1 (n1 )  TPv1 (n1  1)

(39)

For model 1 without considering collaboration, the buyer and the vendor makes strategic decision independently. The total profit is a function of three decision variables. The first two decision variables, Tb1 and  r , are optimized by the buyer. The third decision variable, n1 , is then optimized by the vendor. Model 2: Inventory model with collaboration In this model, the buyer and the vendor jointly make decision. The problem is a nonlinear programming, which is stated as

Maximize TP2 (  r , n2 , T22 )  TPb 2 ( r , Tb 2 (T22 ))  TPv 2 (n2 , r , T21 (T22 ),T2 (T22 ),T22 )

(40)

Subject to ( EPv )    ( EPb ) ,   0 where Tb 2 , T21 and T2 are functions of T22 from (9), (33) and (34). TP2 is a function of three variables,  r , n2 and T22 . For model 2, the vendor-buyer collaboration is considered. The three variables, T22 ,  r and n2 are optimized jointly rather than independently as in model 1.

4. SOLUTION PROCEDURE AND NUMERICAL EXAMPLE For model 1, the problem is to determine the values of  r , Tb1 and n1. The solution procedure is stated in (36), (37) and (39). For model 2, the problem is to determine the value of n2 that maximizes TP2. Since the number of delivery per order, n2, is a discrete variable, one can derive the value of n2 by the following procedure: (a) For a range of n2 values, determine the partial derivative of TP2 with respect to  r and T22 and equate them to zero. The optimal value of  r and T22 , for each n2, are denoted by  r (n2 ) and T22 (n2 ) respectively. (b) Derive the optimal value of n2, denoted by n 2 * , such that TP2 (T22 (n2  1), n2  1,  r (n2  1)) *

*

*

 TP2 (T22 (n2 ), n2 ,  r (n2 )) *

*

*

(45)

and TP2 (T22 ( n2 ), n2 ,  r (n2 )) *

*

*

 TP2 (T22 ( n2  1), n2  1,  r ( n2  1)) *

*

*

(46)

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4.1 Numerical Example The preceding theory can be illustrated by the following numerical example where the parameters are given as follows: Price-sensitive demand rate, di  a  b r units per year Scale parameter, a=3000; price sensitive parameter, b=35 Buyer’s purchased cost, b = $35 Buyer’s backordering cost per unit, Cb= $20 Buyer’s percentage holding cost per year per dollar, ν= 0.2 Buyer’s ordering cost per order, Co= $100 Vendor’s setup cost, Cs= $6000 Vendor’s replenishment rate per year, p=3000 Vendor’s percentage holding cost per year per dollar, μ= 0.2 Vendor’s unit cost, v = $20

Negotiation factor, ζ= 1 Deterioration rate,  = 0.10 Table 1 illustrates the optimal solution for various models when ζ= 1 and  = 0.10. For model 1, the optimal retail price is $60.69. The corresponding annual demand is 875.85. The buyer’s optimal interval in model 1 is 0.1662 years. The buyer’s total profit is $21298. There are thirteen deliveries from the vendor to the buyer per cycle, the vendor’s annual total profit is $5605, and the total annual profit is 26903. For model 2, the vendor and the buyer are collaborated. The optimal retail price is $53.88. The corresponding annual demand is 1114.20. The buyer’s optimal collaborated profit interval is 0.1407 years, the buyer’s, the vendor’s and the collaborated total profit are TPb 2  $19410 TPv 2  $9151 and TP2 = $28561 respectively. The increase in the collaborated total extra profit in model 2 with respect to model 1 is $1658. Since the vendor benefits $3546 and the buyer loses $1888, the buyer will resist the change from model 1 to model 2. The results of the sensitivity analysis are shown in Table 2. Table 1: The optimal solution at various models Scenario i i= 1 i=2 r 60.69 53.88 di 875.85 1114.20 ni 10 11 qbi 86 143 S 60 15 Tbi1 0.0977 0.1272 Tbi2 0.0685 0.0135 TPbi 21298 19410 TPvi 5605 9151 TPi 26903 28561 ω2 6.16% Note: Percentage extra total profit for model 2

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International Journal of Electronic Business Management, Vol. 8, No. 3 (2010) compared to model 1,  2  TP2  TP1 . TP1

a

Table 2: Sensitivity analysis for demand scale parameter d3 r 3 n3 Tb3 TP1 TP3 ω3 (%)

4.

3600 61.83 1435.95 0.2230 13 0.1169 48408 50422 4.16% 3300 57.85 1275.25 0.3905 12 0.1269 36971 38803 4.96%

5.

3000 53.88 1114.20 0.5983 11 0.1407 26903 28561 6.16% 2700 49.97 951.05 0.6314 10 0.1594 18223 19707 8.14% 2400 46.09 786.85 0.6657 10 0.1692 10981 12253 11.58%

5. CONCLUSION We developed two models for a deteriorating inventory system with imperfect items to determine the optimal ordering policy when the demand is price sensitive and the replenishment rate is finite. The numerical example demonstrates that the buyer-vendor collaboration results in an extra profit gain of approximately 6.16%. The win-win collaboration strategy including the profit-sharing policy is most beneficial for both players. The profit-sharing policy is used to ensure mutually beneficial strategy. Sensitivity analysis shows that the consideration of collaboration strategy and credit term for an inventory system with backordering is significant. When the vendor’s replenishment rate decreases, collaboration results in more profits. By using an integrated approach that takes into account the perspectives of both players, the global optimum is found. Our results indicate that the percentage of the extra total profit is significant when both the collaborative strategy and deterioration factor are considered. The results of this study provide managerial insights to efficient supply chain management.

ACKNOWLEDGEMENTS The author would like to thank the anonymous referees for their helpful suggestions and the National Science Research Council of Taiwan for financing this research project NSC 99-2410-H-147-012.

6. 7.

8.

9.

10.

11.

12.

13. 14.

15.

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Banerjee, A., 1986, “A joint economic-lot-size model for purchaser and vendor,” Decision Sciences, Vol. 17, pp. 292-311. Cardenas-Barron, L. E., 2000, “Observation on: Economic production quantity model for items with imperfect quality,” International Journal of Production Economics, Vol. 67, pp. 201-201. Chang, H. C. and Ho, C. H., 2010, “Exact closed-form solutions for optimal inventory

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C. P. Yu: A Collaborative Deteriorating Inventory System with Imperfect Quality and Shortage Backordering

ABOUT THE AUTHOR Jonas C.P. Yu received his PhD in the Institution of Industrial Management at the National Central University in 2006 and his MSc degree from Chung Yuan Christian University in 1995. After a long career in several international firms, he is now an Assistant Professor of Logistics Management at Takming University of Science and Technology. His research interests are in the field of production & material control, inventory and supply chain management, optimization and yield management.

(Received May 2010, revised August 2010, accepted September 2010)

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損耗性商品因不良缺貨補貨的協同存貨管理系統 游兆鵬 德明財經科技大學物流管理系 台北市內湖區環山路一段五十六號

摘要 隨著經濟的快速成長，高科技的進步，產品生命週期越來越短，使得市場上的競爭越 來越激烈，為了滿足市場上的需求，如何有效的控制存貨對企業而言是很重要的。然 而，近期隨著原物料短缺、物價上漲，導致企業之經營成本提高，如何有效降低成本、 提高利潤，已是企業能否永續發展的重要課題。在本研究中，發展一個考量單一供應 商與單一買方的存貨系統，商品具有損耗之特性，以致於商品損壞造成缺貨問題之發 生，而目標在於得到最大化之總利潤。為了補償零售商之缺貨損失，因此，供應商提 供給買方利潤分享以鼓勵購買的策略，此策略主要是加入協商因子來平均分享雙方之 利潤，以達到雙贏之效果。而本存貨系統在有限的補貨率和價格敏感需求的情況下， 並設定為高科技、生命週期短以及具有損耗性的商品。最後經由數值範例分析、敏感 度分析，以了解各個參數對本模式之影響，並做出結論。 關鍵詞：協同系統、品質不良、損耗率、受價格影響的需求 （*聯絡人：[email protected]）