Applied Mathematical Modelling 37 (2013) 3138–3151

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A production-inventory model with probabilistic deteriorationin two-echelon supply chain management

Biswajit Sarkar ⇑Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

a r t i c l e i n f o

Article history:Received 13 February 2012Received in revised form 1 July 2012Accepted 23 July 2012Available online 17 August 2012

Keywords:Supply chain managementInventoryProbabilistic deterioration

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.07.026

⇑ Address: 12D Telipara Lane, 1st Floor, DhakuriaE-mail address: bsbiswajitsarkar@gmail.com

a b s t r a c t

In this study, a production-inventory model is developed for a deteriorating item in a two-echelon supply chain management (SCM). An algebraical approach is applied to find theminimum cost related to this entire SCM. We consider three types of continuous probabi-listic deterioration function to find the associated cost. The purpose of this study is toobtain the minimum cost with integer number of deliveries and optimum lotsize for thethree different models. Some numerical examples, sensitivity analysis and graphical repre-sentation are given to illustrate the model. A numerical comparison between the threemodels is also given.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

In the real life situation, it is too difficult to preserve highly volatile items like alcohol, liquid medicines, blood, etc., for allmanufacturing sectors. These type of items may deteriorate over time. Therefore, it is very important to discuss the deteri-orating behavior of such type of items. In this direction, Ghare and Schrader [1] were the first authors who considered theeffect of deteriorating items in inventory model. They discussed the general EOQ (economic order quantity) model withdirect spoilage and exponential deterioration. Covert and Philip [2] extended the work of Ghare and Schrader [1] with Wei-bull distribution and gamma distribution. Philip [3] deduced a three parameter Weibull distribution for the deterioratingtime. Misra [4] developed optimal production lotsize model with finite production rate and different types of deteriorationrates but without any backordering. Shah [5] discussed an order-level lotsize model for both exponential and Weibull dis-tributed deterioration with backordering. Dave and Patel [6] extended ðT; SiÞ policy model with time proportional demandand deterioration. A survey of existing literature on inventory models for deteriorating items was given by Raafat [7]. While,Goyal [8] adopted economic ordering policy for deteriorating items over an infinite time horizon. In this direction, someresearchers like Goyal [9], Datta and Pal [10], Goswami and Chaudhuri [11], Hariga [12], Chang and Dye [13], Goyal and Giri[14], Skouri and Papachristos [15], Skouri et al. [16], Sarkar [17], etc., extended the inventory models with different types ofdeterioration rates.

The basic well known EOQ formula was first introduced by Harris [18] with the help of differential calculus. The sameformula was again derived by Wilson [19]. Most of the inventory costs were optimized with the help of differential cal-culus by different researchers. Since then, many researchers have discussed different types of algorithm for inventorymodel in SCM environment. SCM contains buyers and suppliers, producers and distributors, distributors and retailers,etc., in many different forms of customers. The intension, to consider a SCM, is to optimize the whole system at a time.

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B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3139

In this direction, Goyal [20] developed an integrated inventory model for a single supplier-single buyer problem. Banerjee[21] found out a joint economic lotsize model for the purchaser and the vendor. Hill [22] extended the single vendor-sin-gle buyer integrated production-inventory model as a generalized policy. Viswanathan and Piplani [23] discussed the coor-dinating supply chain inventory through common replenishment epochs. Yang and Wee [24,25] presented economiclotsize of the integrated vendor–buyer inventory system without derivatives and also for multi-item lotsize inventorymodel for deteriorating items in JIT environment. Cárdenas-Barrón [26,27] discussed a note on optimizing inventory deci-sions in a multi-stage multi-customer supply chain and presented the derivation of EOQ/EPQ inventory models with twobackordering cost using analytic geometry and algebra. In that paper, Cárdenas-Barrón [27] discussed the procedure tooptimize different types of algebraic function with the help of basic algebra. Yan et al. [28] extended the SCM model withconstant deterioration model. Khouja [29] presented optimizing inventory decisions in a multi-stage multi-customer sup-ply chain. Cárdenas-Barrón [30] developed optimum manufacturing batchsize with rework in a single-stage productionsystem. Chung and Wee [31] discussed an integrated production-inventory deteriorating model for pricing policy by con-sidering imperfect production, inspection planning, warranty period and stock-level-dependant demand. Widyadana et al.[32] derived an excellent EOQ model for planned backorder level. A note on EOQ model with imperfect quality and quan-tity discounts was developed by Cárdenas-Barrón [33]. Garc�ia-Laguna et al. [34] and Cárdenas-Barrón et al. [35] found outsome excellent ideas to find the integrality of the lotsize in the basic EOQ and EPQ model. Cárdenas-Barrón et al. [36]extended the vendor–buyer inventory model with arithmetic geometric inequality. Wee and Widyadana [37] found outEPQ models for deteriorating items with rework and stochastic preventive maintenance time. Cárdenas-Barrón et al.[38] discussed a note on EPQ model for coordinated planning with partial backlogging. Widyadana and Wee [39] devel-oped an EPQ model for deteriorating items with preventive maintenance policy and random machine breakdown. Tenget al. [40] derived the economic lotsize of the integrated vendor–buyer inventory system without using any derivative.Teng et al. [41] extended to economic order quantity for buyer–distributor–vendor supply chain with backlogging withoutderivatives. Chung and Cárdenas-Barrón [42] found out a complete solution procedure for the EOQ and EPQ inventorymodels with linear and fixed backorder costs.

In this model, we try to find the optimum lotsize and the integral number of deliveries per production batch cycle whichminimizes the total cost of the entire SCM through algebraical procedure for different probabilistic deterioration functions.This paper is an extension of Yan et al. [28] with Cárdenas-Barrón’s [27] algebraical procedure under different types of prob-abilistic deterioration function. We want to find the expression of the total costs and to minimize the cost using algebraicalprocedure except using calculus. The cost of the entire SCM is found in a simplified form and it is minimized using algebraicalprocedure. There is no need to use calculus to minimize the cost of the entire SCM. The paper is designed as follows: intro-duction is given in Section 1. In Section 2, notation and assumptions are given. The model formulation and different deriva-tion for different types of deterioration function are discussed in Section 3. Numerical examples, comparisons between themodels and the sensitivity analysis are presented to illustrate the model in Section 4. Finally, the conclusion and the futureextensions of the model have been made in Section 5.

2. Notation and assumptions

The following notation and assumptions are considered to develop the model:Notation: To develop the model we use the almost similar notation as on Yan et al. [28].

P production rate (unit/unit time)C setup cost for a production batch ($/batch)Hs holding cost for the supplier ($/unit/unit time)Ds area under the supplier’s inventory levelA ordering cost for the buyer ($/order)D constant demand (units/unit time)Hb holding cost for the buyer ($/unit/unit time)Db area under the buyer’s inventory levelF constant transportation cost per delivery ($/delivery)d deterioration rateCd deterioration cost per unit ($/unit)N numbers of deliveries per production batch, N � 1Q production lotsize per batch cycle (units)q delivery lotsize (units)V the unit variable cost for order handling and receiving ð$=unitÞT duration of inventory cycleT1 production time duration for the supplierT2 non-production time duration for the supplierT3 the duration between the two successive deliveriesTC the total cost of the system

3140 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

Assumptions:

1. The production-inventory system produces single type of item.2. Demand is considered as constant and deterministic.3. The demand information and inventory position of the buyer are given to the supplier.4. Production rate of the supplier is assumed to be constant and P > D.5. No quantity and cash discount are considered.6. Deterioration follows continuous probability distribution function as (a) uniform distribution, (b) triangular distribution,

and (c) beta distribution.7. No shortage and backlogging are allowed.8. The buyer pays transportation and other handling costs.

3. Formulation of the model

In this model, we consider a single-setup-multiple delivery (SSMD) production model in which the buyer’s order quantityis manufactured at one step and is delivered at a fixed amount of products over multiple deliveries after a constant timegap. Let, T3 be the time duration between the two successive deliveries. The total time T can be divided into two timespan: one is T1 in which the supplier produces products and another is T2 in which the supplier does not produce anyproduct. We assume each of the delivery is to be assigned in such a way that each delivery arrives at that exact time whenall items from previous delivery have just been depleted. Figs. 1 and 2 represent the inventory versus time for the buyer andthe supplier.

3.1. The buyer’s inventory cost

The different types of cost are as follows:

1. Ordering cost per unit time =A=T.2. Holding cost per unit time =HbDb=T .3. Deterioration cost per unit time =CddDb=T .4. Transportation cost and handling cost per unit time =ðNF þ VNqÞ=T .

During T3, let y be the numbers of deteriorating items then,q ¼ yþ DT3.Since, deterioration rate is small, we can neglect its square and higher powers. Hence, y can be treated as the deterioration

of q units during T3.Therefore,

i:e:;

i:e:;

q ¼ T3 Dþ dq2

� �

¼ TN

Dþ dq2

� �;

1T¼ D

Nqþ d

2N;

q2¼ Nq

dT� D

d:

Fig. 1. Inventory model for the buyer’s: inventory versus time.

Fig. 2. Inventory model for the supplier’s: inventory versus time.

B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3141

Again, the total deterioration from the buyer’s end, we obtain

dDb ¼ Nq� DT;

i:e:;Db

T¼ q

2:

Therefore, the buyer’s cost function can be derived as

TCb ¼1TðAþ HbDb þ CddDb þ NF þ VNqÞ ¼ D

Nqþ d

2N

� �ðAþ NF þ VNqÞ þ q

2½ðHb þ CddÞ:

3.2. The supplier’s inventory cost

The different types of cost are as follows:

1. Setup cost per unit time ¼ C=T.2. Holding cost per unit time ¼ HsDs=T.3. Deterioration cost per unit time ¼ CddDs=T.

Now, we assume that x represents the number of deteriorating units for the supplieri.e., x ¼ dDs which implies Ds ¼ x=d.Therefore, the total number of deteriorated units for the entire SCM is xþ dqT=2. Since, lotsize Q ¼ Nqþ x and T1 ¼ Q=P,

we can express it as

xþ dqT=2 ¼ dTQð1� D=PÞ

Pþ Dq=P

� �;

hence; Ds ¼xd

¼ qTDPþ N � 1

2� DN

2P

� �:

Thus, the total supplier’s cost function can be written as

TCs ¼1TðC þ HSDS þ CddDSÞ ¼

DNqþ d

2N

� �C þ qðHs þ CddÞ D

Pþ N � 1

2� DN

2P

� �:

3.3. Integrated inventory cost for the entire SCM

The average total cost of the production inventory model for the entire SCM is as (see for instance Yan et al. [28])

TCðq;NÞ ¼ DNqþ d

2N

� �Aþ C þ NF þ VNqð Þ þ q

2ðHB þ CddÞ þ ðHs þ CddÞ ð2� NÞD

Pþ N � 1

� �� �: ð1Þ

In our model, we consider the deterioration d follows three different types of probability distribution function as d ¼ E½f ðxÞ�,where f ðxÞ follows (1) uniform distribution, (2) triangular distribution, and (3) beta distribution and a different algebraic ap-proach to find the optimal solution. We also show a numerical comparison between the three models.

3142 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

3.3.1. d follows uniform distributionWe consider that d follows uniform distribution as d ¼ E½f ðxÞ� ¼ aþb

2 ; a > 0; b > 0; a < b.Now, from Eq. (1)

TCðq;NÞ ¼ DNqþ aþ b

4N

� �ðAþ C þ NF þ VNqÞ þ q

2HB þ

Cdðaþ bÞ2

� �þ Hs þ

Cdðaþ bÞ2

� �ð2� NÞD

Pþ N � 1

� �� �: ð2Þ

Eq. (2) can be written after some algebraic simplification as

TCðq;NÞ ¼ q4f2HB þ Cdðaþ bÞg þ f2Hs þ Cdðaþ bÞg ð2� NÞD

Pþ N � 1

� �þ ðaþ bÞV

� �þ 1

qDNðAþ C þ NFÞ

� �þ DV

þ aþ b4NðAþ C þ NFÞ: ð3Þ

For fixed N, we obtain

TCðqÞ ¼ q4f2HB þ Cdðaþ bÞg þ f2Hs þ Cdðaþ bÞg ð2� NÞD

Pþ N � 1

� �þ ðaþ bÞV

� �þ 1

qD

NqðAþ C þ NFÞ

� �þ DV

þ aþ b4NðAþ C þ NFÞ: ð4Þ

The above equation can be written in the form

TCðqÞ ¼ a1qþ a2

qþ a3 ¼

a1

qq�

ffiffiffiffiffia2

a1

r� �2

þ 2ffiffiffiffiffiffiffiffiffiffia1a2p

þ a3;

where, a1 ¼ 14 ½f2HB þ Cdðaþ bÞg þ f2Hs þ Cdðaþ bÞgfð2�NÞD

P þ N � 1g þ ðaþ bÞV �; a2 ¼ ½DN ðAþ C þ NFÞ�and

a3 ¼ DV þ aþb4N ðAþ C þ NFÞ.

Now, the expression f ðqÞ ¼ a1qþ a2q þ a3 ¼ ð

ffiffiffiffiffiffiffiffia1qp Þ2 þ ð

ffiffiffiffiffiffiffiffiffiffia2=q

pÞ2 þ a3 ¼ ð

ffiffiffiffiffiffiffiffia1qp �

ffiffiffiffiffiffiffiffiffiffia2=q

pÞ2 þ 2

ffiffiffiffiffiffiffiffiffiffia1a2p þ a3 will be minimum

when q ¼ffiffiffiffia2a1

q, [see for instance Cárdenas-Barrón 27] and the minimum cost is 2

ffiffiffiffiffiffiffiffiffiffia1a2p þ a3.

Therefore, TC(q) is minimum when

q ¼ffiffiffiffiffia2

a1

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4DPðAþ C þ NFÞ

N½2ðPHB þ Hsfð2� NÞDþ PðN � 1ÞgÞ þ Cdðaþ bÞfð2� NÞDþ PNg þ ðaþ bÞVP�

sð5Þ

and the minimum cost is

TCðqÞ ¼ 2ffiffiffiffiffiffiffiffiffiffia1a2p

þ a3

¼ f2HB þ Cdðaþ bÞg þ 2Hs þ Cdðaþ bÞf g ð2� NÞDP

þ N � 1� �

þ ðaþ bÞV� �

DNðAþ C þ NFÞ

� �� �1=2

þ DV

þ aþ b4NðAþ C þ NFÞ: ð6Þ

Now, for fixed q

TCðNÞ ¼ q4

2HB þ Cdðaþ bÞf g þ 2Hs þ Cdðaþ bÞf g ð2� NÞDP

þ N � 1� �

þ ðaþ bÞV� �

þ DNqðAþ C þ NFÞ þ DV

þ aþ b4NðAþ C þ NFÞ

¼ Nq4ð2Hs þ Cdðaþ bÞÞ

n o1� D

P

� �� �þ ðAþ CÞ

Naþ b

4þ D

q

� �þ q

42HB þ

4HsDP� 2Hs þ ðaþ bÞV

� �

þ DqCdðaþ bÞ2P

þ DFqþ DV þ ðaþ bÞF

4: ð7Þ

Eq. (7) can be written in the form

B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3143

TCðNÞ ¼ a4N þ a5

Nþ a6;

where; a4 ¼q4ð2Hs þ Cdðaþ bÞÞ

n o1� D

P

� �� �;

a5 ¼ ðAþ CÞ aþ b4þ D

q

� �and

a6 ¼q4

2HB þ4HsD

P� 2Hs þ ðaþ bÞV

� �þ DqCdðaþ bÞ

2Pþ DF

qþ DV þ ðaþ bÞF

4:

Therefore, TC(N) is minimum when

N ¼ffiffiffiffiffia5

a4

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðAþ CÞðaqþ bqþ 4DÞP

q2ðP � DÞf2Hs þ Cdðaþ bÞg

sð8Þ

and the minimum cost is

TCðNÞ ¼ 2ffiffiffiffiffiffiffiffiffiffia4a5p

þ a6

¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Hs þ Cdðaþ bÞÞf g 1� D

P

� �� �ðAþ CÞ½qðaþ bÞ þ 4D�

sþ DF

qþ DV þ ðaþ bÞF

4

þ q4

2HB þ4HsD

P� 2Hs þ ðaþ bÞV

� �þ DqCdðaþ bÞ

2P: ð9Þ

3.3.1.1. Optimal interval of the lotsize. Since N, the number of deliveries per production batch cycle, must be greater than orequal to 1 by our assumption and also from the expression for optimum q, it attains its upper bound at N ¼ 1, i.e.,

q 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4DPðAþ C þ FÞ

2ðPHB þ HsDÞ þ ðaþ bÞfVP þ CdðDþ PÞg

s:

When N increases, the corresponding lotsize q decreases. Therefore, from the optimum lotsize equation, we have

q P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4PDðAþ C þ NFÞ

½2HB þ ðaþ bÞðCd þ VÞ�½2Dþ NðP � DÞ�N

s

3.3.2. d follows triangular distributionNow, we assume that d follows triangular distribution as

d ¼ E½f ðxÞ� ¼ aþ bþ c3

;

where, f ðxÞ is the probability density function of triangular distribution with lower limit a, upper limit b and mode c as wellas a < b and a 6 c 6 b.

Therefore, the equation of TCðq;NÞ can be written as

TCðq;NÞ ¼ DNqþ aþ bþ c

6N

� �ðAþ C þ NF þ VNqÞ

þ q2

HB þ Cdaþ bþ c

3

� �� �þ Hs þ

Cdðaþ bþ cÞ3

� �ð2� NÞD

Pþ N � 1

� �� �

¼ DNqðAþ C þ NFÞ þ DV þ aþ bþ c

6NðAþ C þ NFÞ þ Vqðaþ bþ cÞ

6

þ q2

3HB þ Cdðaþ bþ cÞ3

þ 3Hs þ Cdðaþ bþ cÞ3

� �ð2� NÞD

Pþ N � 1

� �� �: ð10Þ

For fixed N, the above equation is in the form

TCðqÞ ¼ q6½f3HB þ Cdðaþ bþ cÞg þ f3Hs þ Cdðaþ bþ cÞg ð2� NÞD

Pþ N � 1

� �þ Vðaþ bþ cÞ� þ DðAþ C þ NFÞ

Nq

þ DV þ aþ bþ c6N

ðAþ C þ NFÞ: ð11Þ

The equation can be written in the symbolic notation as

3144 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

TCðqÞ ¼ a7qþ a8

qþ a9;

where; a7 ¼16½f3HB þ Cdðaþ bþ cÞg þ f3Hs þ Cdðaþ bþ cÞg ð2� NÞD

Pþ N � 1

� �þ Vðaþ bþ cÞ�;

a8 ¼DðAþ C þ NFÞ

Nand

a9 ¼ DV þ aþ bþ c6N

ðAþ C þ NFÞ:

Therefore, TC(q) is minimum, when

q ¼ffiffiffiffiffia8

a7

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6DPðAþ C þ NFÞ

N½3fPHB þ Hsfð2� NÞDþ PðN � 1Þgg þ ðaþ bþ cÞfCdfð2� NÞDþ PNg þ VPg

sð12Þ

and the minimum cost is

TCðqÞ ¼ 2ffiffiffiffiffiffiffiffiffiffia7a8p

þ a9

¼ 2DðAþ C þ NFÞ

6N3HB þ Cdðaþ bþ cÞf g þ 3Hs þ Cdðaþ bþ cÞf g ð2� NÞD

Pþ N � 1

� �þ Vðaþ bþ cÞ

� �� �1=2

þ DV þ aþ bþ c6N

ðAþ C þ NFÞ:

ð13Þ

For, fixed q

TCðNÞ ¼ Nq6

3Hs þ Cdðaþ bþ cÞf g 1� DP

� �� �þ Aþ C

Naþ bþ c

6þ D

q

� �þ q

63 HB þ

2HsDP� Hs

� �þ ðaþ bþ cÞV

� �

þ qCdðaþ bþ cÞD3P

þ DFqþ DV þ ðaþ bþ cÞF

6ð14Þ

which is in the form

TCðNÞ ¼ a10N þ a11

Nþ a12;

where; a10 ¼q6f3Hs þ Cdðaþ bþ cÞgð1� D

PÞ

� �;

a11 ¼ ðAþ CÞ aþ bþ c6

þ Dq

� �;

and

a12 ¼q6

3 HB þ2HsD

P� Hs

� �þ ðaþ bþ cÞV

� �þ qCdðaþ bþ cÞD

3Pþ DF

qþ DV þ ðaþ bþ cÞF

6:

Now, using the same type of algebraic procedure,TC(N) is minimum when

N ¼ffiffiffiffiffiffiffia11

a10

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðAþ CÞPfqðaþ bþ cÞ þ 6Dg

q2f3Hs þ Cdðaþ bþ cÞgðP � DÞ

sð15Þ

and the minimum cost is

TCðNÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffia10a11p

þ a12

¼ 13

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif3Hs þ Cdðaþ bþ cÞg 1� D

P

� �� �ðAþ CÞ½qðaþ bþ cÞ þ 6D�

s

þ q6

3 HB þ2HsD

P� Hs

� �þ ðaþ bþ cÞV

� �þ qCdðaþ bþ cÞD

3Pþ DF

qþ DV þ ðaþ bþ cÞF

6: ð16Þ

3.3.2.1. Optimal interval of the lotsize. Since N, the number of deliveries per production batch cycle, must be greater than orequal to 1 by our assumption and also from the expression for optimum q, it attains its upper bound at N ¼ 1, i.e.,

B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3145

q 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6DPðAþ C þ FÞ

½3fPHB þ HsDg þ ðaþ bþ cÞfCdðDþ PÞ þ VPg�

s:

When N increases, the corresponding lotsize value q decreases. Therefore, from the optimum lotsize equation, we have

q P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6PDðAþ C þ NFÞ

½3HB þ ðCd þ VÞðaþ bþ cÞ�½2Dþ NðP � DÞ�N

s:

3.3.3. d follows beta distributionNow, we consider that d follows beta distribution as

d ¼ E½f ðxÞ� ¼ aaþ b

where, f ðxÞ follows beta distribution which is a continuous probability distributions defined on the interval (0, 1) parame-terized by two positive parameters, denoted by a and b.

From Eq. (1), we have

TCðq;NÞ ¼ DNqþ a

2Nðaþ bÞ

� �ðAþ C þ NF þ VNqÞ þ q

2ðHB þ

Cdaaþ b

Þ þ ðHs þCdaaþ b

Þ ð2� NÞDP

þ N � 1� �� �

¼ q2ðaþ bÞP ðaþ bÞfPHB þ Hsfð2� NÞDþ PðN � 1Þgg þ Cdaf2Dþ NðP � DÞg þ aVP½ � þ DðAþ C þ NFÞ

Nq

þ DV þ aðAþ C þ NFÞ2Nðaþ bÞ : ð17Þ

For fixed N, TC(q) can be written in the form

TCðqÞ ¼ a13qþ a14

qþ a15;

where; a13 ¼1

2ðaþ bÞP ½ðaþ bÞfPHB þ Hsfð2� NÞDþ PðN � 1Þgg þ Cdaf2Dþ NðP � DÞg þ aVP�;

a14 ¼DðAþ C þ NFÞ

Nand

a15 ¼ DV þ aðAþ C þ NFÞ2Nðaþ bÞ :

Applying similar argument, TC(q) is minimum when

q ¼ffiffiffiffiffiffiffia14

a13

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DðAþ C þ NFÞPðaþ bÞ

N½ðaþ bÞfPHB þ Hsð2� NÞDþ PðN � 1Þg þ aCdf2Dþ NðP � DÞg þ aVP�

s

and the minimum cost is

TCðqÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffia13a14p

þ a15

¼ 2DðAþ C þ NFÞNPðaþ bÞ ½ðaþ bÞfPHB þ Hsfð2� NÞDþ PðN � 1Þgg þ Cdaf2Dþ NðP � DÞg þ aVP�

� �1=2

þ DV

þ aðAþ C þ NFÞ2Nðaþ bÞ : ð18Þ

Now, for fixed q

TCðNÞ ¼ DNqþ a

2Nðaþ bÞ

� �ðAþ C þ NF þ VNqÞ þ q

2HB þ

Cdaaþ b

� �þ Hs þ

Cdaaþ b

� �ð2� NÞD

Pþ N � 1

� �� �

¼ N ð1� DPÞ Cdaq

2ðaþ bÞ þqHs

2

� �� �þ Aþ C

NDqþ a

2ðaþ bÞ

� �

þ ðF þ VqÞ Dqþ a

2ðaþ bÞ

� �þ qDCda

Pðaþ bÞ þq2

Hs2DP� 1

� �þ HB

� �� �ð19Þ

which can be written as

3146 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

TCðNÞ ¼ a16N þ a17

Nþ a18;

where; a16 ¼ 1� DP

� �Cdaq

2ðaþ bÞ þqHs

2

� �� �;

a17 ¼ ðAþ CÞ Dqþ a

2ðaþ bÞ

� �

and

a18 ¼ ðF þ VqÞ Dqþ a

2ðaþ bÞ

� �þ qDCda

Pðaþ bÞ þq2

Hs2DP� 1

� �þ HB

� �� �:

TC(N) is minimum when

N ¼ffiffiffiffiffiffiffia17

a16

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðAþ CÞf2Dðaþ bÞ þ aqg

q2ðP � DÞfCdaþ Hsðaþ bÞg

sð20Þ

and the minimum cost is

TCðNÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffia16a17p

þ a18 ¼1

ðaþ bÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� D

P

� �Cdaqþ qHsðaþ bÞð ÞðAþ CÞ 2Dðaþ bÞ þ aq

q

� �s

þ ðF þ VqÞ Dqþ a

2ðaþ bÞ

� �þ qDCda

Pðaþ bÞ þq2

Hsð2DP� 1Þ þ HB

� �� �: ð21Þ

3.3.3.1. Optimal interval of the lotsize. Since N, the number of deliveries per production batch cycle, must be greater than orequal to 1 by our assumption and also from the expression for optimum q, it attains its upper bound at N ¼ 1, i.e.,

q 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PDðaþ bÞðAþ C þ FÞ

½ðaþ bÞfPHB þ DHsg þ aCdðDþ PÞ þ aVP�

s:

When N increases, the corresponding lotsize value q decreases. Therefore, from the optimum lotsize equation, we obtain

q P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PDðaþ bÞðAþ C þ NFÞ

fðaþ bÞHB þ Cdaþ Vagf2Dþ NðP � DÞgN

s:

3.4. Solution procedure

In this model, we have to find the optimal range and value of q and integral value of N. We first calculate the optimalrange of q for each of the model. Then, we start to find the optimal value of q and N from the different set of values of eachmodel. If N� in each of the model (as in Eq. (8), (15) and (20)) is not an integer, we choose N which gives upminfTCðNþÞ; TCðN�Þg in each of the three models (as in Eq. (3), (10) and (17)) where Nþ and N� represent the closest inte-gers larger or smaller than the optimal N�. Substituting the value of N� and q� in TCðq�;N�Þ, we obtain the optimal minimalcost.

3.5. Numerical experiments

In this section, we consider numerical examples, comparison of the three models by graphical presentations and sensi-tivity analysis of the key parameter of the models.

3.6. Numerical example

We use Mathematica 7 version to find the numerical results. One can get results from TC.

Example 1. The values of the following parameters are to be taken in appropriate units: a ¼ 0:15; b ¼ 0:25; P ¼ 10000units/year, C ¼ $800/batch, HB ¼ $7/unit/year, HS ¼ $6/unit/year, D ¼ 4800 units/year, A ¼ $25/order, F ¼ $50/delivery,V ¼ $1/unit, Cd ¼ $50/unit. Then, the optimal solution is fTC ¼ $15757:9 =year;N ¼ 6=production batch cycle;q ¼ 164:547units} (see Fig. 3.)

Fig. 4. Total cost versus number of deliveries per production and lotsize for the deterioration which follows triangular distribution.

Fig. 3. Total cost versus number of deliveries per production and lotsize for the uniformly distributed deterioration.

B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3147

Example 2. The values of the following parameters are to be taken in appropriate units: a ¼ 0:15; b ¼ 0:35; c ¼ 0:25;P ¼ 10000 units/year, C ¼ $800/batch, HB ¼ $7/unit/year, HS ¼ $6/unit/year, D ¼ 4800 units/year, A ¼ $25/order, F ¼ $50/delivery, V ¼ $1/unit, Cd ¼ $50/unit. Then, the optimal solution is fTC ¼ $16575:6 =year;N ¼ 6=production batch cycle;q ¼ 153:162 units} (see Fig. 4.)

Example 3. The values of the following parameters are to be taken in appropriate units: a ¼ 0:15; b ¼ 0:35; P ¼10000 units/year, C ¼ $800/batch, HB ¼ $7/unit/year, HS ¼ $6/unit/year, D ¼ 4800 units/year, A ¼ $25/order, F ¼ $50/deliv-ery, V ¼ $1/unit, Cd ¼ $50/unit. Then, the optimal solution is fTC ¼ $17340:7 =year;N ¼ 6=production batch cycle;q ¼ 143:85 units} (see Fig. 5.)

Number of deliveries

TC (uniform) TC (triangular) TC (beta)

1

32459.1 34523.7 36456.2 2 21069.3 22281.4 23416. 3 17729.1 18691.8 19592.9 4 16401.2 17265.4 18074.1 5 15878.3 16704.2 17476.9 6� 15757.9 16575.6 17340.7 7 15867.4 16694.3 17467.8 8 16120.7 16967.5 17759.4 9 16469.8 17343.7 18160.8 10 16886. 17792. 18639.1 11 17351. 18292.7 19173.1 12 17852.6 18832.7 19749.1

We obtain the optimal lotsize as q ¼ 164:547 units (for uniform distribution), q ¼ 153:162 units (for triangular distribu-tion) and q ¼ 143:85 (for beta distribution) and optimal production batch size for each of the model as N ¼ 6. We considerthe table for different values of the TC.

Fig. 5. Total cost versus number of deliveries per production and lotsize for the deterioration which follows beta distribution.

3148 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

3.7. Comparison between the three models by the graphical representations

The comparison between the three probabilistic deteriorated models is done with the help of graphical representations.The following plots are due to the change of the three probabilistic deterioration functions. There are three figures and eachof the three figures contains a combination of three figures in which the above plot is for beta distribution, middle-plot is fortriangular distribution and last (downside) plot is for uniform distribution. Fig. 6–8.

Fig. 6. When q = 164.547 units then the total cost versus number of deliveries per production and lotsize for the three types of deterioration distribution. Inbetween the three graphs, the above one is for beta distribution, the middle graph is for triangular distribution and the lower graph is for uniformdistribution.

Fig. 7. When q = 153.162 units then the total cost versus number of deliveries per production and lotsize for the three types of deterioration distribution. Inbetween the three graphs, the above one is for beta distribution, the middle graph is for triangular distribution and the lower graph is for uniformdistribution.

B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151 3149

3.8. Sensitivity analysis

The sensitivity analysis of the key parameter has been discussed below:

Parameters change (%)

TC ðuniformÞ (%) TC ðtriangularÞ (%) TC ðbetaÞ (%)

P

�50 �29.24 �29.93 �30.48 �25 �5.99 �6.12 �6.24 +25 3.17 3.24 3.29 +50 4.77 4.88 4.97

Parameters change (%)

TCðuniformÞ (%) TCðtriangularÞ (%) TCðbetaÞ (%)

C

�50 �14.49 �14.82 �15.10 �25 �6.67 �6.82 �6.95 +25 6.01 6.14 6.26 +50 11.31 11.56 11.78

HB

�50 �1.83 �1.62 �1.45 �25 �0.91 �0.81 �0.73 +25 0.91 0.81 0.73 +50 1.83 1.62 1.45

HS

�50 �4.94 �4.27 �3.83 �25 �2.41 �2.13 �1.92 +25 2.41 2.27 1.92 +50 4.77 4.27 3.83

D

�50% �30.86 �30.44 �30.08 �25 �13.79 �13.55 �13.34 +25 10.99 10.69 10.43 +50 18.61 17.93 17.35

A

�50 �0.39 �0.40 �0.40 �25 �0.19 �0.20 �0.20 +25 0.19 0.20 0.20 +50 0.39 0.40 0.40

F

�50 �4.64 �4.75 �4.83 �25% �2.32 �2.37 �2.42 +25 2.32 2.37 2.42 +50 4.64 4.75 4.83

Parameters change (%)

TC ðuniformÞ (%) TC ðtriangularÞ (%) TC ðbetaÞ (%)

V

�50 �15.28 �14.54 �13.90 �25 �7.64 �7.27 �6.95 +25 7.64 7.27 6.95 +50 15.28 14.54 13.90

Cd

�50 �11.31 �12.57 �13.58 �25 �5.33 �5.93 �6.42 +25 5.33 5.90 6.32 +50 10.06 11.06 11.87

Fig. 8. When q = 143.85 units then the total cost versus number of deliveries per production and lotsize for the three types of deterioration distribution. In

3150 B. Sarkar / Applied Mathematical Modelling 37 (2013) 3138–3151

� If the production rate of the system increases, the total cost of the system increases for fixed time interval.

� For each setup, if the setup cost increases, the total cost increases. Thus, the profit is less if the selling price and the total

revenue earning are fixed.� In this model, it is considered that the holding cost of the supplier is less than the holding cost of the buyer, thus, the

buyer always tries to use multiple delivery from the supplier such that he has to face less holding cost. With the increas-ing value of the holding cost, the total cost for the whole system increases.� If demand, ordering cost of the buyer, transportation cost, handling and receiving cost as well as deterioration cost

increase then, the total cost of the system increases.

4. Concluding remarks

In this model, we represent an algebraical procedure to obtain the minimum cost of the entire SCM. The model extendsthe existing literature in SCM with algebraical procedure and the probabilistic deterioration function. The deterioration func-tion follows probability distribution like (a) uniform distribution, (b) triangular distribution, and (c) beta distribution. In eachcase, we find the minimum total cost associated with the system and the range of the delivery lotsize. The main contributionof the model is to find the minimum cost with integer number of deliveries and optimal lotsize using algebraic procedure. Anumerical comparison between the three models is shown graphically. The proposed procedure is simple and does not re-quire tedious computation effort. To the author’s best knowledge, such type of model has not yet been discussed in the exist-ing literature. There are several extensions of this work that could constitute future research related in this field. Oneimmediate possible extension could be to discuss the effect of inflation. The model may be extended inventory to multi-itemEOQ model. These are, among others, some models of ongoing future research.

Acknowledgements

I would like to thank the anonymous referees for their very helpful comments and corrections suggested in revising thepaper. I take this favorable chance to express my indebtedness to the Honorable Editor-in-Chief and his Editorial Board fortheir helpful support. I wish to express my heartiest gratitude to my parents, wife and son without whose inspiration, itwould have been impossible to make the research a reality. I am also grateful to Vidyasagar University, West Bengal, Indiafor infrastructural assistance to carry out the research.

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