A supply chain with variable demand under three level trade credit policy

Accepted Manuscript A supply chain with variable demand under three level trade credit policy Prasenjit Pramanik, Manas Kumar Maiti, Manoranjan Maiti PII: DOI: Reference:

S0360-8352(17)30068-2 http://dx.doi.org/10.1016/j.cie.2017.02.007 CAIE 4637

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

28 July 2015 30 October 2016 6 February 2017

Please cite this article as: Pramanik, P., Maiti, M.K., Maiti, M., A supply chain with variable demand under three level trade credit policy, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie. 2017.02.007

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Title Page

A SUPPLY CHAIN WITH VARIABLE DEMAND UNDER THREE LEVEL TRADE CREDIT POLICY

Prasenjit Pramanik

a,∗

, Manas Kumar Maiti b , Manoranjan Maiti

c

a

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, Paschim-Medinipur, West Bengal 721102, India. Tel: +91 9851411811. E-mail address: [email protected] b

Department of Mathematics, Mahishadal Raj College, Mahishadal, Purba-Medinipur, West Bengal 721628, India. Tel: +91 9434611765. E-mail address: [email protected] c

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, Paschim-Medinipur, West Bengal 721102, India. Tel: +91 9434944709. E-mail address: [email protected]

∗ Corresponding Author. 1

A SUPPLY CHAIN WITH VARIABLE DEMAND UNDER THREE LEVEL TRADE CREDIT POLICY In this paper, an integrated supply chain model has been developed under three level trade credit policy with price, credit period and credit amount dependent demand, where a supplier oers a credit period to his/her wholesaler to boost the demand of the item. Due to this facility, wholesaler also oers a credit period to his/her retailer and the same practice is followed by the retailer to increase the base demand of the item. Here it is assumed that, both wholesaler and retailer enjoy the same full credit facility but retailer just oers the partial trade credit to his/her customers. The main purpose of this paper is to maximize the joint pro t of the wholesaler and the retailer. Model is formulated in crisp, fuzzy and rough environments. Here, a Particle Swarm Optimization (PSO) algorithm is used to nd marketing decision for the proposed models. For fuzzy model, credibility measure of fuzzy event and for rough model, trust measure of rough event are used to compare the corresponding objectives for PSO. Models are illustrated with numerical examples and some parametric studies are performed. Abstract.

Key words : Supply Chain; Price, credit period and credit amount dependent demand; Three level trade credit; PSO.

1. Introduction

A supply chain (SC) involves several stages which directly or indirectly satisfy the customers requirements. In a supply chain, there are manufacturers, transporters, suppliers, wholesalers, retailers, nal customers etc., those are bonded by a decision in such a manner that all the players in the chain are mostly pro table. It is also well known that all players of a co-ordinated supply chain get better shares of bene t which cannot be obtained when one optimizes its own decision individually. With the advent of multi-nationals in the market of developing countries, competition among the business enterprises is very sti, and they adopt various promotional tools to sale the products eciently, which increases the base demand. Trade credit policy is one the most eective tools to push the products. There are two types of trade credit policies, one of them is full trade credit policy, which refers to the manner that a player gives the full credit to his/her customers on the total purchased amount, and on the other hand if the player gives the credit on a fraction of the total purchased amount, it is known as partial trade credit. Again if the credit period is depend on some conditions, then it is known as conditional delay in payment or conditional trade credit. If the credit period is oered to the retailer only by the wholesaler, it is called one level trade credit policy. On the other hand if both the wholesaler and the retailer oer credit period to the retailer and the customer respectively, then it is known as two level trade credit policy. Goyal(1985)[10] rst developed an EOQ model under the conditions of permissible delay in payment. Aggarwal and Jaggi(1995)[1] extended Goyal's model for a deteriorating item. Chung and Liao(2004)[8] studied a lot-sizing problem under supplier's trade credit depending on the retailer's order quantity. Huang(2007)[16] established an EOQ model in which the supplier oers a partially permissible delay in payment when the order quantity is smaller than a predetermined quantity. Ouyang (2009)[26] et al.

1

2

Supply chain model under three level trade credit policy

developed an EOQ model for deteriorating items with partially permissible delay in payments linked to order quantity. In these studies, it is usually assumed that the supplier would oer a credit-period to the retailer but in turn, the retailer would not oer any credit-period to its customers, which is unrealistic, because in real practice retailer might oer a credit-period to its customers in order to stimulate his own demand. Huang(2003)[14] rst explores an EOQ model under two level trade credit policy considering that the customer purchases one item from the retailer at the time t belonging to [0,N], then the customer has a trade credit period (N-t) and must make the payment at time N. A major drawback of the model is that here credit period of the customer is not same for all customers, who purchase earlier will get more credit period than those who purchase later. The inventory/supply chain models under two level trade credit policy using the viewpoint of Huang(2003)[14] can be found in several papers such as Huang and Hsu(2008)[17], Chung(2008[4], 2011[5], 2013[6]), Huang(2006[15], 2007[16]), Teng (2007)[31] and others. Teng(2009)[29] established an EOQ model under two level trade credit considering that credit period given to the all customers are same. From the view point of Teng(2009)[29] there are several research papers for deteriorating items such as Teng and Chang(2009)[30], Teng (2013)[32], Chen and Kang(2010)[3], Ho(2011)[13], Chung (2014)[7] and many others. Mahata(2012)[21] developed an EPQ model to investigate the optimal retailer's replenishment decisions for a deteriorating item, where retailer avails full credit period from the wholesaler but, oers a partial credit to his/her customers. According to the authors' best knowledge, all the inventory models/supply chain models under trade credit policy are developed from the retailer's point of view only. But in today's competitive business, decision has to be made from the supply chain point of view where retailer's as well as supplier's and wholesaler's costs/pro ts should be considered. Also, trade credit policy in business is observed in dierent levels- Manufacturer, Distributor, Supplier, Wholesaler, Retailer, Customer etc. But till now, all the models are developed under single level or two level trade credit policies only. Min (2010)[25] developed an inventory model for deteriorating item under two level trade credit with stock dependent demand, but their analysis is imposed a terminal condition of zero ending-inventory. However with a stock dependent demand, it may be desirable to order large quantities. Recently, Soni(2013)[28] extended their model to optimize replenishment policies for deteriorating items with stock sensitive demand under two level trade credit with limited capacity, considering the non- zero ending inventory and used an upper limitation on the stock items, as too much stock leaves a negative expression to the buyer. Huang(2007)[16] developed an EPQ Model under two level trade credit, but in this model, retailers credit period has no eect on the demand of the item. To remove these shortcomings Jaggi (2008)[18] developed an inventory model under two-level trade credit policy, where demand of the item depends on retailer's credit-period. Maiti(2011)[22] developed an inventory model under two level trade credit where credit period oered to customers for few cycles to boost the demand. When demand of the item reaches a certain level, the credit period is withdrawn by the retailer. Guchhait (2014)[12] developed an inventory model for a deteriorating item with selling price and customer's credit period linked demand. From the above discussion it is shown that researchers considered the demand either constant in nature or stock dependent or selling price linked or customer's credit period linked. But, none considered the demand as a function of the customer's credit amount. As in the real practice the customer's credit amount is proportional to the base demand, i.e., if the customer's credit amount increases/decreases, then the base demand of the item also will be increase/decrease, so customer's credit amount is an eective argument for the demand function. et al.

et al.

et al.

et al.

et al.

et al.

Supply chain model under three level trade credit policy

3

In the recent years, due to the uncertainty of inventory parameters in the cost functions, the cost function becomes imprecise in nature i.e., in fuzzy sense. Now-a-days, this phenomenon is well established by some researchers. After the introduction of fuzzy set (Zadeh(1965)[34]), it has been well developed and applied widely in dierent areas of science and technology including inventory control problems. Maiti and Maiti(2007)[23] developed a multi-item inventory model with stock dependent demand and two storage facilities in fuzzy environment where processing time of each unit is fuzzy. Lee and Yao(1998)[19], Maiti(2011)[22], Guchhait (2010[11], 2014[12]) and many others developed their models in fuzzy environment. Due to vague description of the parameters in a cost function, it became a crucial issue. After fuzzy set theory, rough set theory is another attempt to this problem. After the introduction of rough set theory (Pawlak(1982)[27]), it attracted the attention of many researchers and practitioners all over world. It is well developed and applied widely in computer science and in the eld of arti cial intelligence. Now-a-days, many researchers used rough set theory in developing their models in the eld of inventory control system. Recently, Guchhait (2014)[12] developed an inventory model allowing two-level trade credit assuming the parameters in rough nature. From the above discussions, some shortcomings in the formation of the inventory models may be summarized as below:  No inventory model/supply chain model considered the customer's credit amount linked demand.  None of the investigators considered supply chain models with the trade credit in more than two levels (Wholesaler ! Retailer ! Customer).  Very few supply chain models are formulated considering the cost/pro t from both wholesaler's and retailer's point of view. Incorporating the above mentioned phenomenon, here a supply chain model with selling price, customer's credit period and customer's credit amount dependent demand has been developed under three level trade credit policy. The supplier oers a xed grace period to the wholesaler and consequently wholesaler also oers a credit period to the retailer. Due to this facility, retailer also oers a partial trade credit period to the customers on a fraction of amount purchased to boost the base demand. Obviously the demand depends on the duration of the credit period oered. Demand also depends on the amount of item for which credit period is oered. The supplier and wholesaler both charges the same interest to the wholesaler and to the retailer respectively, if the grace period is over, but there is no penalty is to be considered for the case of customers. Here it is assumed that the wholesaler and the retailer both pay the total amount to the supplier and to the wholesaler respectively just after the end of the credit period by taking a loan from a bank, with the same interest. And then wholesaler and the retailer both repay the loan amount to the bank immediately as the items are sold. The proposed model is formulated in crisp, fuzzy and rough environment. Here, some parameters like ordering cost, interest rate etc. are considered as fuzzy/rough in nature. Models (Crisp, Fuzzy, Rough) are formulated following pro t maximization principle and depending upon dierent parameters, dierent models are obtained. The Non-linear optimization problems for dierent models are solved by a PSO algorithm developed for this purpose. In the fuzzy/rough model, credibility (x2.4)/trust (x2.6) measure is used to compare the fuzzy/rough objectives of the problem directly without transferring the objectives into the equivalent crisp objectives and using this PSO is followed to optimize the objectives. et al.

et al.

4

Supply chain model under three level trade credit policy

Numerical experiments are performed to illustrate the models and some parametric studies are made. 2. Mathematical Prerequisite 2.1 Triangular Fuzzy Number(TFN) : A TFN a~ = (a1 ; a2 ; a3 ) has three parameters a1 , a2 ,a3 , where a1 < a2 < a3 and is characterized by the membership function a~ , given by

8 < a~ (x) = :

for a1  x  a2 for a2  x  a3 (1) 0 otherwise. A TFN a~ = (a1; a2; a3) is represented pictorially in Figure-1. Please insert Figure-1 here 2.2 Fuzzy Extension Principle : If a~, ~b  < and c~ = f(~a; ~b), where f : <  < ! < is a binary operation, membership function c~ of c~ is de ned as For each z 2 <, c~(z) = supfmin(a~ (x),~b(y)), x,y 2 < and z = f(x,y)g (2) where, < represents set of real numbers and symbol ` e' on any variable represents fuzzy number. 2.3 Possibility and Necessity : Let a~ and ~b be two fuzzy numbers in < with membership functions a~ (x) and ~b(x) respectively. Then according to Dubois and Prade (1980)[9] and Zadeh (1978)[35]; P os(~a  ~b) = supfmin(a~ (x); ~b (y)); x; y 2 <; x  yg (3) Nes(~a  ~b) = 1 P os(~a  ~b); (4) where `Pos' and `Nes' represents possibility and necessity respectively.  is any one of the relations >; <; =; ; . Similarly, possibility and necessity measures of a~ with respect to ~b are denoted by ~b(~a) and N~b (~a) respectively and are de ned as ~b(~a) = supfmin(a~ (x); ~b(x)); x 2 B~ ) is given by 1 Cr(A~ > B~ ) = [P os(A~ > B~ ) + Nes(A~ > B~ )] (8) 2 where, 8 < a b 0 for a3  b1 for a3  b1 & a2  b2 P os(A~ > B~ ) = (9) : b b +a a1 for a2  b2 2

x a2 a3 a3

a1 a1 x a2

3

1

1

3

2

Supply chain model under three level trade credit policy

and

8 < Nes(A~ > B~ ) = :

5

0 for b2  a2 for b2  a2 & a1  b3 (10) b 1 for a1  b3 2.5 Rough variable : A rough variable  is a measurable function from the rough space (;
; ; ) to the set of real numbers. That is for every Borel set B of <, we have f 2  j () 2 B g 2 : The lower and upper approximations of the rough variable are de ned as = f() j  2
g and = f() j  2 g respectively. The symbol `'on any parameter represents rough quantity. Example : A rough variable f[a; b][c; d]g with c  a  b  d is a measurable function from a rough space (;
; ; ) to the real line, where = fx j c  x  dg,
= fx j a  x  bg and (x) = x for all x 2 . 2.6 Trust Measure (Liu(2002)[20]) : Let (;
; ; ) be a rough space. The trust measure of event A is denoted by TrfAg and de ned by T rfAg = 12 (T rfAg + T rfAg), where T rfAg = fAf
\
g g and T rfAg = ffAgg are the lower and upper trust measure of the event A respectively. Let  = f[a; b][c; d]g, c  a  b  d be a rough variable and lebesgue measure is used for trust measure of an rough event associated with   t. Then the trust measure of the rough event   t is denoted by Trf  tg and de ned by 8 0 for d  t > > d t > < 2(d c) for b  t  d 1 d t  T rf  tg = (11) ( d c + bb at ) for a  t  b 2 > 1 ( d t + 1) for c  t  a > > : 2 d c 1 for t  c Let, Z1 = f[a1; b1][c1; d1]g and Z2 = f[a2; b2][c2; d2]g be two rough variables, then trust measure of the rough event Z1 > Z2 is given by 8 0 for d1 c2  0 > d c > > < 2(d c c +d for b1 a2  0  d1 c2 d c b a 1   T rfZ1 > Z2 g = ( + ) for a1 b2  0  b1 a2 (12) 2 d c c + d b a > 1 ( d c a ++b1) for c1 d2  0  a1 b2 > > 2 d c c +d : 1 for 0  c1 d2 a2 b2 b2 +a2 a1

3

1

1

1

1

2

2

1

2

1

1

2

1

2

2

2

1

1

1

2

2

1

2

2

2

3. Notations and assumptions for the proposed model

The following notations and assumptions are used in developing the models. 3.1 Notations

(i) q(t), the inventory level of the retailer at any time t. (ii) i, subscript indicating the supply chain level, where i=w for the wholesaler, i=r for the retailer and i=c for the customer. (iii) hi , holding cost in $ for unit item per unit time for level i.

6

Supply chain model under three level trade credit policy

(iv) Ai , ordering cost per order in $ for level i. (v) Ci , unit purchasing cost in $ for level i. (vi) Si , unit selling price in $ for level i. (vii) m , mark up of selling price for the retailer, i.e, Sr = m:Cr = m:Sw (viii) Mi , delay period obtained by the i-th player. (ix) Ti , inventory cycle length for the i-th player. (x) Q , ordered quantity by the retailer. (xi)  , a positive integer, multiplier of the retailer's order quantity, which is equal to the wholesaler lot-size, i.e., wholesaler ordered quantity is Q. (xii) D, demand per unit time. (xiii)  , a positive real number between 0 to 1. (xiv) Ip , rate of interest charged per $ by the bank. (xv) Ie , rate of earned interest per $ by the bank. 3.2 Assumptions

(i) Supplier oers a delay period Mw to the wholesaler on the total purchasing cost for settling his/her account. After the delay period, interest will be charged at a rate Ip per $ on the amount of the unsold items. This interest rate is same as bank interest. (ii) Wholesaler oers a delay period Mr to the retailer on the total purchasing cost for settling his/her account. After the delay period, interest will be charged at a rate Ip per $ on the amount of the unsold items. This interest rate is same as bank interest. (iii) Retailer oers a delay period Mc to his/her customers on a (1 )-fraction of the total purchasing amount and the -fraction of the total purchasing amount to be paid at the time of receiving items. (iv) Mc < Mr < Mw (v) Demand D per unit time of the item depends on the selling price for the customers (Sr ), credit period oers by the retailer(Mc) and the credit amount and is of the form  c Mc a be D= , where a, b, c, ( 1) and (0    1), are ve parameters so Sr chosen to best t the demand function. Pictorial representation of demand function is presented in Figure-2. Please insert Figure-2 here (1

)

4. Model development and analysis

In this model, it is assumed that in each cycle, the supplier and wholesaler both oer permissible delay periods Mw and Mr in payments to the wholesaler and the retailer respectively. At the same time retailer also oers partial permissible delay period Mc (< Mr < Mw ) to his customers, i.e, delay period is oered on a fraction (1 ) of purchased units. Supply is instantaneous, i.e., time lag between order and receipt of the item is assumed as zero. During the time interval [0; Tr ] the inventory level at retailer decreases owing to demand. Thus instantaneous state of retailer's inventory level q(t) at any time t can be represented by the following dierential equation : dq = D(Sr ; Mc; 1 ) (13) dt With the boundary condition q(Tr ) = 0 and q(0) = Q, we have q(t) = D (Tr t) and Q = D Tr (14)

Supply chain model under three level trade credit policy

7

The total pro t for the retailer consists of the following: (i) Sales Revenue = QRSr = D Tr Sr . R(ii) Cost of purchasing = Q Cr = D Tr Cr . (iii) Holding cost = hr 0Tr q(t) dt= hr 0Tr D(Tr t) dt= hr D T2r . (iv) Ordering cost = Ar . (v) Interest payable and earned: Here three scenarios arise Scenario-1: Mc < Mr < Tr . Scenario-2: Mc < Tr < Mr . Scenario-3: Tr < Mc < Mr . In each of these scenarios, total pro t for the wholesaler consists of the following: (i) Sales Revenue =  Q Sw =  D Tr Sw . (ii) Cost of purchasing =  Q Cw =  D Tr Cw . (iii) Holding cost = hw [( 1) Q Tr + ( 2) Q Tr + ::: + Q Tr ] = hw Q Tr (2 1) . (iv) Ordering cost = Aw . (v) Interest payable and earned. 2

Scenario-1:

Mc < Mr < Tr

The scenario-1 is pictorially represented in Figure-3. Please insert Figure-3 here

Interest payable and earned from Retailer's point of view : Retailer obtained a credit period Mr from the wholesaler, so after the end of the grace period(Mr ), retailer have to pay interest for the outstanding amount i.e., for the credit amount of customers and for the amount of unsold units. Customers are initially paid the - fraction of the total purchased amount at the receiving item t(say) and the remaining amount ((1 )fraction of the total purchased amount) should be paid just after Mc time from the purchasing time i.e., at the time (t + Mc). Thus retailer have to pay interest for the customers credit amount of the sold units during [Mr Mc; Mr ] and [Mr ; Tr ] for Mc time and consequently interest have to pay for the unsold units during [Mr ; Tr ]. (A) Interest to be paid for the amount, IPR(say), where IP R = Cr (IP1 + IP2 + IP3 ); (15)

= Interest to be paid due to units sold during [Mr Mc; Mr ] Z Mr 2 = (1 )D(t + Mc Mr ) dt = (1 )D M2c ; (16) Mr Mc IP2 = Interest to be paid due to units sold during [Mr ; Tr ] Z Tr = Mc (1 ) D dt = (1 ) D Mc (Tr Mr ); (17) Mr IP3 = Interest to be paid due to the stock units during [Mr ; Tr ] Z Tr D = q(t) dt = (Tr Mr )2 (18) 2 Mr In case of earned interest, retailer earned interest for the initial payment(- fraction of the total purchased amount) of the customers of the sold units during [0; Mr ] and for the remaining amount((1 )- fraction of the total purchased amount) of the sold units during [0; Mr Mc]. (B) Interest earned for the amount, IER(say), where IER = Cr (IE1 + IE2 ); (19) IP1

8

Supply chain model under three level trade credit policy

IE1 IE2

= Interest earned due to units sold during [0; Mr Mc] Z Mr Mc = (1 )D(Mr t Mc) dt = D2 (1 )(Mr Mc)2; (20) 0 = Interest earned due to the initial payments of customers for sold items Z Mr D during [0; Mr ] =  D (Mr t) dt = Mr2 (21) 2 0

Interest payable and earned from Wholesaler's point of view :

Let, [ MTrw ] = n, where n is a positive integer. Here two cases arise: Case-1 : n Tr + Mr  Mw For this case, the relation between wholesaler's inventory level and retailer's inventory level is described pictorially in Figure-4. Please insert Figure-4 here (A) Interest to be paid for the amount, IP W 1 (say), where IP W 1 = Cw IP; (22) = f Q (n + 1) Qgf(n + 1) Tr + Mr Mw g + f Q (n + 2) Qg Tr + ::: +f Q ( 1) Qg Tr = Q( n 1)[ T2r ( + n) + Mr Mw ] (23) (B) Interest earned for the amount, IEW 1 (say), where IEW 1 = Cw IE; (24)

IP

= Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw n Tr = Q (n + 1)) [Mw Mr nT2 r ] Case-2 : n Tr + Mr > Mw This case is represented pictorially in Figure-5. Please insert Figure-5 here (A) Interest to be paid for the amount, IP W 2 (say), where IP W 2 = Cw IP; IE

IP

Mr )

(25)

(26)

= f Q n Qgfn Tr + Mr Mw g + f Q (n + 1) Qg Tr + ::: + f Q ( 1) Qg Tr = Q( n)[ T2r ( + n 1) + Mr Mw ] (27) (B) Interest earned for the amount, IEW 2 (say), where IEW 2 = Cw IE; (28)

Supply chain model under three level trade credit policy

9

= Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw (n 1) Tr Mr ) = n Q [Mw Mr (n 21) Tr ] (29) Detail derivation about wholesaler's payable and earned interest i.e., the derivation of the equations (23), (25), (27) & (29) are given in Appendix B. IE

Scenario-2 :

Mc < Tr < Mr

Scenario-2 is represented pictorially in Figure-6. Please insert Figure-6 here

Interest payable and earned from Retailer's point of view : (A) Interest to be paid for the amount, IPR(say), where IP R = Cr IP1 ; Case-1 : Mr Mc  Tr IP1 = 0 Case-2 : Mr Mc < Tr IP1 = Interest to be paid for the sold units during [Mr Mc ; Tr ] Z Tr = (1 ) D (t + Mc Mr ) dt = D (1 ) (Tr + Mc Mr )2

2

Mr Mc

(B) Interest earned for the amount, IER(say), where IER = Cr (IE1 + IE2 ); Case-1 : Mr Mc  Tr IE1 IE2

Case-2 : Mr

IE2

(31) (32) (33)

= Interest earned for the initial payments of the customers for the sold units Z Tr during [0; Tr ] = D  (Mr t) dt = D2  [Mr2 (Mr Tr )2]; (34) 0 = Interest earned forZ the rest payments of the customers for the sold units Tr during [0; Tr ] = D(1 )(Mr t Mc) dt = D2 (1

IE1

(30)

0

)[(Mr

Mc )2

(Mr

Mc

Tr ) 2 ]

(35)

Mc < Tr

= Interest earned for the initial payments of the customers for the sold units Z Tr during [0; Tr ] = D  (Mr t) dt = D2  [Mr2 (Mr Tr )2]; (36) 0 = Interest earned Zfor the due payments of the customers for the sold units during Mr Mc D [0; Mr Mc] = D(1 )(Mr t Mc ) dt = (1 )(Mr Mc )2 (37) 2 0

10

Supply chain model under three level trade credit policy

Interest payable and earned from Wholesaler's point of view :

Let, [ MTrw ] = n, where n is a positive integer. Here two cases arise: Case-1 : (n 1) Tr + Mr  Mw

(A) Interest to be paid for the amount, IP W 1 (say), where IP W 1 = Cw IP;

(38) 1) Qg Tr (39)

= f Q n Qgfn Tr + Mr Mw g + f Q (n + 1) Qg Tr + ::: + f Q ( = Q( n)[ T2r ( + n 1) + Mr Mw ] (B) Interest earned for the amount, IEW 1 (say), where IEW 1 = Cw IE; (40) IE = Q fMw Mr g + Q fMw Tr Mr g + ::: + Q fMw (n 1) Tr Mr g = n Q [Mw Mr (n 21) Tr ] (41) Case-2 : (n 1) Tr + Mr > Mw IP

(A) Interest to be paid for the amount, IP W 2 (say), where IP W 2 = Cw IP; IP

(42)

= f Q (n 1) Qgf(n 1) Tr + Mr Mw g + f Q n Qg Tr + ::: +f Q ( 1) Qg Tr = Q( n + 1)[ ( + n2 2) Tr + Mr Mw ]

(B) Interest earned for the amount, IEW 2 (say), where IEW 2 = Cw IE; IE

= Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw (n 2) Tr = (n 1) Q [Mw Mr (n 22) Tr ]

Scenario-3 :

Tr < Mc < Mr

The pictorial representation of scenario-3 is represented in Figure-7. Please insert Figure-7 here

Interest payable and earned from Retailer's point of view :

Same as Scenario-2.

Interest payable and earned from Wholesaler's point of view :

Same as Scenario-2.

(43) (44)

Mr )

(45)

Supply chain model under three level trade credit policy

11

According to the above discussions, the total Retailer's pro t for all the scenarios (Scenario-1,2,3) represented by the following non-linear equation:  Zr1 if Mr  Tr Zr = (46) Zr2 otherwise where, T2 Zr1 = D Tr Sr D Tr Cr hr D r Ar Ip IP R + Ie IER; (47) 2 Retailer's pro t :

Zr2 = D Tr Sr

Tr2

(48) 2 Ar Ip IP R + Ie IER: IPR and IER are given by equations (15) and (19) respectively in Zr1, and by the equations (30) and (33) respectively in Zr2. Hence, the Retailer's pro t per unit time for the scenarios (Scenario-1,2,3) can be written as  Zr1a if Mr  Tr Zra = (49) Zr2a otherwise D Tr Cr

hr D

where, Zr1a = ZTrr and Zr2a = ZTrr . Zr1 and Zr2 are given by (47) and (48) respectively. Wholesaler's pro t : Following above discussions, the total Wholesaler's pro t for three scenarios are represented by the following non-linear equation:  Zw1 if Mr  Tr Zw = (50) Zw2 otherwise where  Zw11 if n Tr + Mr  Mw Zw 1 = (51) Zw12 otherwise and  Zw21 if (n 1) Tr + Mr  Mw Zw 2 = (52) Zw22 otherwise Zw11 , Zw12 , Zw21 and Zw22 are represented by the followings: ( 1) Zwj 1 =  Q Sw  Q Cw hw Q Tr Aw Ip IP W 1 + Ie IEW 1 ; 8j = 1; 2, 2 and Zwj2 =  Q Sw  Q Cw hw Q Tr (2 1) Aw Ip IP W 2 + Ie IEW 2; 8j = 1; 2 For j = 1; IP W 1, IEW 1, IP W 2 and IEW 2 are given by (22), (24),(26) and (28) respectively. And for j = 2; IP W 1, IEW 1, IP W 2 and IEW 2 are given by (38), (40), (42) and (44) respectively. Hence, Wholesaler's pro t per unit time for three scenarios can be written as  Zw1a if Mr  Tr Zwa = (53) Zw2a otherwise 1

2

where, Zw1a = ZwTr and Zw2a = ZwTr . Zw1 and Zw2 are given by (51) and (52) respectively. 1

2

12

Supply chain model under three level trade credit policy

Total pro t :

The Retailer's and Wholesaler's total pro t for three scenarios is given by  r1 + Zw1 if Mr  Tr Z= Z (54) Zr2 + Zw2 otherwise

And hence, the Retailer's and Wholesaler's total pro t per unit time for three scenarios represented as  r1a + Zw1a if Mr  Tr Za = Z (55) Zr2a + Zw2a otherwise Crisp Model (CM) :

be formulated as

When all the parameters are crisp in nature, average pro t, Za can Za =



Zr1a Zr2a

+ Zw1a if Mr  Tr + Zw2a otherwise

(56)

So, the problem reduces to Determine Tr ; Mr ; Mc ; m ;  and  to  (57) Maximize Za where Za is given by the Eq. (56). This problem is solved by a PSO discussed in the next section. Fuzzy Model (FM) : Due to volatile marketing condition, now-a-days, bank interests uctuates throughout the years. So it is more practical to consider the interest rates as fuzzy numbers. In the fuzzy model, it is assumed that payable interest rate (Ip) and earned interest rate (Ie) are fuzzy numbers. And also for the same reason, ordering cost, for both retailer and wholesaler are estimated as fuzzy numbers. Here it is considered that ordering cost for retailer (Ar ), ordering cost for wholesaler (Aw ), payable interest rate (Ip) and earned interest rate (Ie) are TFNs and represented as A~r =(Ar ; Ar ; Ar ), A~w =(Aw ; Aw ; Aw ), I~p=(Ip ; Ip ; Ip ), I~e =(Ie ; Ie ; Ie ). According to the extension principle of fuzzy numbers, total pro t per unit time of retailer and wholesaler for three scenarios are TFNs Z~ra =(Zra ; Zra ; Zra ) and Z~wa =(Zwa ; Zwa ; Zwa ), where Tr Ar IP R:Ip IER:Ie Zra = D Sr D Cr hr D + ; 2 T T T 1

1

2

2

3

1

2

3

1

1

2

3

Zra

2

Zra

1

Zwa

2

= =

(

3

2

3

3

1

Zwa

3

3

1

(

2

= D Sr

= D Sr

D Cr D Cr

Tr

r

2

hr D

2

hr D

Tr

Ar Tr Ar Tr

2

1

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

3

r

IP R:Ip Tr IP R:Ip Tr

2

1

1

r

e + IER:I ; Tr e + IER:I ; Tr 2

3

f Aw + IP W 1:Ip  Tr f Aw + IP W 2 :Ip 3

3

3

3

f Aw + IP W 1:Ip  Tr f Aw + IP W 2 :Ip 2

2

2

2

1 1 1 1

IEW 1 :Ie IEW 2 :Ie

g if mTr + Mr  Mw g otherwise,

IEW 1 :Ie IEW 2 :Ie

g if mTr + Mr  Mw g otherwise,

1 1

2 2

Supply chain model under three level trade credit policy

(

13

D Sw D Cw hw D Tr ( 2 1)  1Tr f Aw + IP W 1 :Ip D Sw D Cw hw D Tr ( 2 1)  1Tr f Aw + IP W 2 :Ip with, m=n for Mr  Tr and m=(n 1) for Mr  Tr respectively. Hence the joint pro t of retailer and wholesaler per unit time, Za Z~a =(Za ; Za ; Za ), where 9 Za = Zra + Zwa = Za = Zra + Zwa Za = Zra + Zwa ; Zwa

3

=

1

1

1

2

IEW 1 :Ie g; IEW 2 :Ie g

1

3

1

3

if mTr + Mr  Mw otherwise,

also becomes a TFN,

3

1

1

1

2

2

2

3

3

3

(58)

Hence, the problem for the fuzzy model is to Determine Tr ; Mr ; Mc ; m ;  and  to  (59) Maximize Z~a where Z~a is given by (58). Rough Model (RM): Due to the vagueness of the various type of costs related to the objective function, the proposed model is developed in fuzzy environment. Also there is an another approach to handle the vague description of the objects by introducing the rough set theory. In this paper, considering the retailer's ordering cost(Ar ), wholesaler's ordering cost(Aw ), payable interest rate(Ip) and earned interest rate(Ie) as the rough variables, a rough model has been developed. Here, it is assumed that Ar = f[Ar1; Ar2][Ar3; Ar4]g, Aw = f[Aw1 ; Aw2 ][Aw3 ; Aw4 ]g, Ip = f[Ip1 ; Ip2 ][Ip3 ; Ip4 ]g, and Ie = f[Ie1 ; Ie2 ][Ie3 ; Ie4 ]g. Then for all three scenarios, the pro t of the retailer and wholesaler per unit time can be written as Zra = f[Zra ; Zra ][Zra ; Zra ]g and Zwa = f[Zwa ; Zwa ][Zwa ; Zwa ]g respectively, where Tr IPr :Ip IEr :Ie Ar Zra = D Sr D Cr hr D + ; 2 T T T 1

2

3

4

1

2

3

4

2

1

Zwa

1

Zwa

2

Zwa

3

Zwa

4

= = = =

( ( ( (

Zra

2

= D Sr

D Cr

Zra

3

= D Sr

D Cr

= D Sr

D Cr

Zra

4

Tr

r

2

hr D

2

hr D

2

hr D

Tr Tr

Ar Tr Ar Tr Ar Tr

1

4

3

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

D Sw D Sw

D Cw D Cw

hw D Tr ( 2 1) hw D Tr ( 2 1)

 Tr

2

r

IPr :Ip Tr IPr :Ip Tr IPr :Ip Tr

1

r

1

4

3

+ IETr :Ie ; r IEr :Ie + T ; r IEr :Ie + T ; r 2

3

4

f Aw + IP w1:Ip  Tr f Aw + IP w2 :Ip 2

2

2

2

f Aw + IP w1:Ip  Tr f Aw + IP w2 :Ip 1

1

1

1

f Aw + IP w1:Ip  Tr f Aw + IP w2 :Ip 4

4

4

4

f Aw + IP w1:Ip  Tr f Aw + IP w2 :Ip 3

3

3

3

1 1 1 1 1 1 1 1

IE w1 :Ie IE w2 :Ie

g if mTr + Mr  Mw g otherwise,

IE w1 :Ie IE w2 :Ie

g if mTr + Mr  Mw g otherwise,

IE w1 :Ie IE w2 :Ie

g if mTr + Mr  Mw g otherwise,

IE w1 :Ie IE w2 :Ie

g if mTr + Mr  Mw g otherwise,

1 1

2 2

3 3

4 4

14

Supply chain model under three level trade credit policy

with m=n for Mr  Tr and m=(n 1) for Mr  Tr . Hence the joint pro t of retailer and wholesaler per unit time can be written as Za = f[Za ; Za ][Za ; Za ]g, where 9 Za = Zra + Zwa > > Za = Zra + Zwa = (60) Za = Zra + Zwa > Za = Zra + Zwa ; So the problem is to Determine Tr ; Mr ; Mc ; m ;  and  to  (61) Maximize Za where, Za is given by (60). 1

2

3

4

1

1

1

2

2

2

3

3

3

4

4

4

5. Solution Methodology

Though crisp model can be solved by a numerical non-linear optimization technique, fuzzy and rough models can not be solved using those methods. Due to this reason, a soft computing technique, PSO is developed and used for solution. This technique requires only comparison of objectives due to dierent potential solution to nd the optimal decision. The technique is brie y presented here. 5.1 Particle Swarm Optimization(PSO) : During the last decade, intelligence is inspired by nature, more speci cally the behaviour of social animal inspired the intelligence. Now-a-days, this type of intelligence becomes heavily popular through the development and utilization of intelligent paradigms in advance information system design. Among the most popular nature inspired approaches, PSO technique drawn much attention in the eld of optimization problem within in a complex decision of data and information. PSO, introduced in 1995 by Kennedy and Eberhart, based on a social psychological model of social in uence and social learning. A PSO algorithm maintains a swarm of particles, where each particle represents a potential solution. In analogy with evolutionary computation paradigms, a swarm is similar to a population, while a particle is similar to an individual. Each individual in a PSO follows a simple behaviour: \emulate the success of neighbouring individuals". A PSO normally starts with a set of solutions(swarm) of the decision making problem under consideration. The position of the each particle is known as individual solution and food is analogous to the optimal solution. The particles are own through a multi-dimensional search space, where the position of each particle is adjusted according to its own experience. Let, Xi(t) and Vi(t) be the position vector and velocity vector of the i-th particle respectively and the position vector at which the best tness of the i-th particle Xbesti(t) encountered in the t-generation. Also let, Xgbest(t) be the best position vector of all particles at the current generation( t-generation). In the (t+1)th generation, the position vector and velocity vector of the particle are changed to Xi(t + 1) and Vi(t + 1) using the following rules:  Vi (t + 1) = ! Vi (t) + 1 r1 fXbesti (t) Xi (t)g + 2 r2 fXgbest (t) Xi (t)g (62) Xi (t + 1) = Xi (t) + Vi (t + 1) The parameter !, known as the inertia weight basically controls the momentum of the particle by weighing the contribution of the previous velocity. The value of ! is extremely

Supply chain model under three level trade credit policy

15

important to ensure convergent behaviour, here the value of ! is taken as 0.7298. The parameters 1 and 2 are set to constant value, which are normally taken as 2, in this paper the values of 1 and 2 are considered as 1.49618. r1 and r2 are to random values uniformly distributed in [0,1]. The algorithm to implement the optimization technique is given below, where it is assumed that n is the number of particles. spv, epv are the corresponding lower and upper bounds for searching the position vectors of the particles, and svv, evv are the respective lower and upper bounds for searching the velocity vectors. f (Xi(t)) is the value of the objective function for the particle Xi(t). f (Xi(t)) is taken as the tness of the particle Xi (t). avgfit denotes the average tness of all particles and maxit denotes the number of maximum iteration, used to nd the best solution. Check constraint(Xi(t)) function returns 1 if (Xi(t)) satis es constraint of the problem otherwise return 0. PSO Algorithm : 1: Initialise !; 1 ; 2 , n, spv; epv; svv; evv;  and maxit.

Set iteration counter t=0, and randomly generate initial swarm(set of solutions) P (t), such that 8Xi(t) 2 P (t), Check constraint(Xi(t)) = 1. 3: Evaluate tness of each solution Xi (t) and nd Xgbest (t). 4: Set initial velocity vector Vi (t) 8 Xi (t) 2 P (t) and set Xbesti (t)=Xi (t) 8 Xi (t) 2 P (t). 5: while (t < maxit) do 6: for i=1 to n do 7: Generate r1 and r2 randomly in the range (0,1). 8: Vi (t + 1) = ! Vi (t) + 1 r1 fXbesti (t) Xi (t)g + 2 r2 fXgbest (t) Xi (t)g. 9: if (Vi (t + 1) < svv) then 10: Vi (t + 1) = svv 11: end if 12: if (Vi (t + 1) > evv) then 13: Vi (t + 1) = evv 14: end if 15: Xi (t + 1)=Xi (t)+Vi (t + 1). 16: if Xi (t + 1) < spv then 17: Xi (t + 1) = spv 18: end if 19: if Xi (t + 1) > epv then 20: Xi (t + 1) = epv 21: end if 22: if Check constraint(Xi (t + 1)) = 0 then 23: Xi (t + 1) = Xi (t) 24: end if 25: if (f (Xi (t + 1)) > f (Xbesti (t))) then 26: Xbesti (t + 1)=Xi (t + 1) 27: end if 28: if (f (Xi (t + 1)) > f (Xgbest (t))) then 29: Xgbest (t + 1)=Xi (t + 1) 2:

30: 31: 32: 33:

end if end for

Calculate avgfit. if jf (Xgbest (t + 1))

avgfitj   then

16

Supply chain model under three level trade credit policy

break t=t+1 end while Output: Xgbest(t). End Algorithm. This algorithm requires comparison of objective values due to a solution(Xi) and its best position (Xbesti), shown in the line number 22 of the PSO algorithm. Also requires another comparison of objective values due to a solution(Xi) and global best position(Xgbest), shown in the line number 25 of the PSO algorithm. These comparisons are required in each iteration. To solve dierent models this comparison is made in dierent approaches, which are discussed in x5.3. 5.2 Implementation and testing : Incorporating the above functions and values, the algorithm is implemented using C-programming language. The algorithm is tested against a list of crisp valued standard benchmark non-linear test functions. This list of test functions along with their global optima are listed in Appendix A. Results of these test functions are obtained for number of swarm(n)=10, lower bound of searching velocity vector(svv)=-0.1, upper bound of searching velocity vector(evv)=0.1 and =0.00001, presented in Table-7. From this table, it is clear that the algorithm is good enough to draw the decisions for any crisp valued single objective optimization problem. 5.3 Solution of Models : The decision variables Tr , Mr , Mc , m and  for the models given by (57), (59) and (61) are obtained using the PSO mentioned above for some particular values of  and the optimum value of  is determined from the parametric study on . Crisp Model : For crisp model, objective values due to two solutions Xi and Xj are valid real numbers, so they are compared using law of trichotomy. Fuzzy Model : If Z~1 and Z~2 be the objective values due to solution X1 and X2 respectively, then X1 is better than X2 if Cr(Z~1 > Z~2)  0:5. This is a valid comparison as Cr(Z~1 > Z~2 )+ Cr(Z~1  Z~2 )=1. So in the fuzzy model, the objective value Z~a =(Za ; Za ; Za ) given by (59), can be directly maximized following PSO by comparing two fuzzy objectives using credibility measure. Rough Model : If Z1 and Z2 be the objective values due to solution X1 and X2 respectively, then X1 is better than X2 if T r(Z1 > Z2)  0:5. This is a valid comparison as T r(Z1 > Z2 ) + T r(Z1  Z2 )=1. So in the rough model, the objective value Za =f[Za ; Za ][Za ; Za ]g given by (61), can be directly maximized following PSO by comparing two rough objectives using trust measure. 34: 35: 36: 37: 38: 39:

end if

1

1

6. Numerical Illustrations

2

2

3

3

4

The Crisp models are solved for dierent scenarios using dierent numerical values. To illustrate the dierent scenarios, 3 set of parametric values are taken here. For each set of parametric values, three scenarios (scenario-1,2,3) are optimized. For a particular set of 6.1. Crisp Model :

Supply chain model under three level trade credit policy

17

parametric values, among these three scenarios which one gives the best result, the marketing decision is taken for that scenario for that particular set of parametric values. To illustrate the Scenario-1 numerically, one hypothetical example is used whose parametric values are presented below : Experiment-1 : a = 350, b = 100, c = 40, Cw = 0:75, Cr = 1:7  Cw , Ar = 20, Aw = 170, hr =0:07  Cr , hw = 0:04  Cw , Mw = 3, Ip = :10, Ie = :08, = 1:4. For these assumed parametric values, results are obtained due to dierent values of  and presented in Table-1. It is found from Table-1 that pro t Za is maximum for  = 2. Thus, best marketing decision for Scenario-1 under the above consideration is Max Za=122.7290 for Tr =3.1552, Mr =2.9985, Mc=0.4594, m=2.5217, =0.6862 & =2. Please insert Table-1 here The demand, D depends on the parametric values of a, b, c, Sr (= m  Cr ), , and Mc. Here parametric values of a, b, c, Cr and are chosen, so for the best t of demand(D), the decision variables Mc, m and  are responsible. From the Table-1, it is clearly seen that, if Mc is increases/decreases then  also increases/decreases, but not proportionally, as m is not a xed value. This result leads to a realistic manner for a business transaction. Experiment-2 : a = 500, b = 100, c = 70, Cw = 0:5, Cr = 1:7  Cw , Ar = 15, Aw = 100, hr =0:1  Cr , hw = 0:05  Cw , Mw = 3, Ip = :10, Ie = :08, = 1:3. For these assumed parametric values, results are obtained for scenario-2 due to dierent values of  and it is found that pro t Za is maximum for  = 3. Thus, best marketing decision for Scenario-2 under the above consideration is Max Za=267.8744 for Tr =1.3206, Mr =2.9942, Mc=0.9036, m=2.5532, =0.9079 & =3. Experiment-3 : a = 550, b = 200, c = 50, Cw = 0:5, Cr = 1:7  Cw , Ar = 10, Aw = 100, hr =0:1  Cr , hw = 0:04  Cw , Mw = 3, Ip = :10, Ie = :08, = 1:35. For these assumed parametric values, results are obtained for scenario-3 due to dierent values of  and it is found that pro t Za is maximum for  = 3. Thus, best marketing decision for Scenario-2 under the above consideration is Max Za=292.9692 for Tr =1.0000, Mr =2.9999, Mc=2.0071, m=2.0279, =0.9419 & =3. The optimum results for experiments-2 and 3 are presented in Table-2. Please insert Table-2 here 6.2. Fuzzy Model (FM) : In the fuzzy models, some parameters like ordering cost, interest rate etc. considered as triangular fuzzy numbers. And hence, the total pro t Za also becomes a TFN Z~a. The fuzzy model for all scenarios, the objective value Z~a is directly maximized using credibility measure. The fuzzy model for all scenarios are illustrated numerically with dierent examples. Experiment-4 : A~r = (19; 20; 21); A~w = (168; 170; 172); I~p = (0:09; 0:10; 0:105); I~e = (0:07; 0:08; 0:09), other parametric values are same as Experiment-1. For these assumed parametric values, results are obtained for scenario-1 due to dierent values of , and it is seen that pro t is maximum for  = 2. Experiment-5 : A~r = (14; 15; 17); A~w = (99; 100; 101); I~p = (0:09; 0:10; 0:11); I~e = (0:075; 0:08; 0:085), other parametric values are same as Experiment-2. For these assumed parametric values, results are obtained for scenario-2 due to dierent values of , and it is found that pro t is maximum for  = 3. Experiment-6 : A~r = (9:5; 10; 10:5); A~w = (99:5; 100; 100:5); I~p = (0:095; 0:10; 0:105); I~e =

18

Supply chain model under three level trade credit policy

(0:075; 0:08; 0:085), other parametric values are same as Experiment-3. For these assumed parametric values, results are obtained for scenario-3 due to dierent values of , and it is seen that pro t is maximum for  = 3. The optimum results of fuzzy model for all three scenarios are presented in Table-3. Please insert Table-3 here 6.3. Rough Model (RM) : In the rough models, some parameters like ordering cost, interest rate etc. considered as rough variables. Hence, the total pro t Za also becomes a rough variable Za=f[Za1; Za2][Za3; Za4]g. The rough model for all scenarios, the objective value Za is directly maximized by PSO using trust measure. To illustrate the rough model for dierent scenarios, dierent examples are considered. Experiment-7 : Ar =f[19:5; 20:5][19; 21]g; Aw =f[170; 170:5][169; 171:5]g, Ip = f[0:095; 0:105] [0:09; 0:11]g; Ie=f[0:075; 0:085][0:07; 0:09]g, other parametric values are same as Experiment-1. For these assumed parametric values results, are obtained for scenario-1 due to dierent values of  and it is found that pro t is maximum for  = 2. Experiment-8 : Ar =f[14:5; 15:5][14; 16]g; Aw =f[99:5; 100:5][99; 101]g; Ip = f[0:095; 0:105] [0:09; 0:11]g; Ie=f[0:075; 0:085][0:07; 0:09]g, other parametric values are same as Experiment-2. For these assumed parametric values, results are obtained for scenario-2 due to dierent values of  and it is seen that pro t is maximum for  = 3. Experiment-9 : Ar =f[9:5; 10:5][9; 11]g, Aw =f[99:5; 100:5][99; 101]g; Ip = f[0:095; 0:105][0:09; 0:11]g, Ie=f[0:075; 0:085][0:07; 0:09]g, other parametric values are same as Experiment-3. For these assumed parametric values, results are obtained for scenario-3 due to dierent values of  and it is found that pro t is maximum for  = 3. The best marketing decisions of rough model for all scenarios are presented in Table-4. Please insert Table-4 here 7. Sensitivity Analysis

Sensitivity analyses for the models in three environments under three scenarios with respect to are performed. It is observed that the behaviour of the objective functions with respect to the above parameters are same. For this reason, the results of the crisp model under scenario-1 only are presented in Table-5. From Table-5, it is observed that as the value of increases, the time period for each replenishment of retailer decreases and accordingly total pro t decreases. But it is interesting to note that as decreases, the initial payment of customers uctuates and accordingly demand of customers uctuates neglecting the in uence of on demand, which demands that demand should decreases steadily. However, the relation between  and Mc i.e., as the initial payment is more, the delay in payment for customers is large is well established here. Please insert Table-5 here 8. Some Important Particular Scenarios 8.1 Particular Scenario-1 (Mr = 0) : For some business enterprises speci cally for some

products, it is shown that a supplier oers a delay period to his/her wholesaler's but in return wholesaler does not oer any delay period to his/her retailers, i.e. retailer's credit period, Mr = 0. But to maintain the base demand, retailer oers a delay period to his/her customers. e.g. fruit, bricks, sands etc.

Supply chain model under three level trade credit policy

19

Under the above consideration, the best marketing decision is to be made by maximize the joint pro t of the wholesaler and the retailer with Retailer's cycle length(Tr ), Customer's credit period(Mc), a fraction of the total amount purchased to be paid initially by the customer(), mark-up of the selling price(m) and number of retailer's cycle() as the decision variables. The joint pro t of the wholesaler and the retailer consists the sales revenue, purchasing cost, ordering cost, holding cost and the amount of earned interest and the payable interest. Here, the computation is performed to calculate the amount of the payable and earned interest only, and all other costs are equal to the proposed general model. Retailer's Pro t : Interest have to be paid for the amount, IPr = Cr (IP1 + IP2), where IP1 = Interest to be paid for the sold units during [0; Tr ] Z Tr = (1 ) D Mc dt = D (1 ) Tr Mc (63) 0

Z Tr

= Interest to be paid for the unsold units during [0; Tr ] = q(t) dt = D2 Tr2(64) 0 Earned interest for the amount, IEr = Cr IE , where IE = 0: (65) Hence, the Retailer's pro t per unit time can be written as 1 Tr2 Zra = fQ Sr D Tr Cr hr D 2 Ar Ip IP r + Ie IE r g Tr = D Sr D Cr T2r hr D T1 fAr + Ip IP r Ie IE r g (66) r IP2

Wholesaler's Pro t : Let, [ MTrw ] = n, a positive integer.

Interest to be paid for the amount, IP w = Cw IP , where IP = f Q (n + 1) QgfMw (n + 1) Tr g + f Q (n + 2) Qg Tr + ::: +f Q ( 1) Qg Tr = Q( n 1)[ T2r ( + n) Mw ] (67) Interest earned for the amount, IE w = Cw IE , where nTr IE = Q Mw + Q (Mw Tr ) + ::: + Q fMw n Tr g = Q (n + 1)) [Mw 2 ] (68) Thus, the Wholesaler's pro t per unit time can be written as 1 f Q S  Q C h Q T ( 1) A I IP + I IE g Zwa = w w w r w p w1 e w1  Tr 2 = D Sw D Cw hw D Tr ( 2 1) 1T f Aw + Ip IP w IeIE w g (69) r The joint pro t of the wholesaler and the retailer per unit is given by Za = Zra + Zwa (70) where, Zra and Zwa are given by (66) and (69) respectively. 8.2 Particular Scenario-2 (Mc = 0) : In real practice, often it is seen that a supplier

20

Supply chain model under three level trade credit policy

oers a permissible delay period to wholesaler to boost the demand of the item also in turn wholesaler also oers a delay period to his/her retailers. But retailer does not oer any credit period to his/her customers, e.g., in a shopping complex, a customer does not get any credit period from retailer, also in the dhaba besides the highways, etc. Here, a business strategy has been proposed considering customer's credit period Mc = 0. Under the above consideration, the best marketing decision is to be made by maximize the joint pro t of the wholesaler and the retailer with Retailer's cycle length(Tr ), Retailer's credit period(Mr ), mark-up of the selling price(m) and number of the retailer' cycle() as the decision variables. The joint pro t of the wholesaler and the retailer consists the sales revenue, purchasing cost, ordering cost, holding cost and the amount of earned interest and the payable interest. Here, the computation is performed to calculate the amount of the payable and earned interest only, and all other costs are equal to the proposed model. Here, two cases arises : (i) Mr < Tr and (ii) Mr > Tr Retailer's Pro t : Case-1 : Mr < Tr

Interest to be paid for the amount, IPr = Cr (IP1 + IP2), where IP1 = Interest to be paid for the sold units during [Mr ; Tr ] Z Tr D = D (t Mr ) dt = (Tr Mr )2 2 1

(71)

Mr

and

IP2

= Interest to be paid for the unsold units during [Mr ; Tr ] Z Tr D = q(t) dt = (Tr Mr )2 2

(72)

Mr

Earned interest for the amount, IEr = Cr IE , where 1

IE

= Interest earned for the sold units during [0; Mr ] =

Z Mr 0

D(Mr

t) dt =

Thus, the retailer's pro t per unit time is given by 1 fQ S D T C h D Tr2 A I IP + I IE g Zra = r r r r r p r e r Tr 2 = D Sr D Cr T2r hr D T1 fAr + Ip IP r Ie IE r g 1

1

r

1

D 2 Mr(73)

2

1

1

(74)

Case-2 : Mr > Tr

Interest to be paid for the amount, IPr = Cr IP , where IP = 0 Earned interest for the amount, IEr = Cr IE , where IE = Interest earned for the sold units during [0; Tr ] Z Tr D = D (Mr t) dt = [Mr2 (Mr Tr )2 ] 2 2

(75)

2

0

(76)

Supply chain model under three level trade credit policy

21

Thus, the retailer's pro t per unit time is given by 1 fQ S D T C h D Tr2 A I IP + I IE g Zra = r r r r r p r e r Tr 2 = D Sr D Cr T2r hr D T1 fAr + Ip IP r Ie IE r g r 2

2

2

2

2

(77)

Wholesaler's Pro t : Case-1 : Mr < Tr Let, [ MTrw ] = n, where n is a positive integer.

Here two cases arise: Sub-case(i) : n Tr + Mr  Mw Interest to be paid for the amount, IP w = Cw IP , where IP = f Q (n + 1) Qgf(n + 1) Tr + Mr Mw g + f Q (n + 2) Qg Tr + ::: +f Q ( 1) Qg Tr = Q( n 1)[ T2r ( + n) + Mr Mw ] (78) Interest earned for the amount, IE w = Cw IE , where IE = Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw n Tr Mr ) = Q (n + 1)) [Mw Mr nT2 r ] (79) Sub-case(ii) : n Tr + Mr > Mw Interest to be paid for the amount, IP w = Cw IP , where IP = f Q n Qgfn Tr + Mr Mw g + f Q (n + 1) Qg Tr + ::: + f Q ( 1) Qg Tr = Q( n)[ T2r ( + n 1) + Mr Mw ] (80) Interest earned for the amount, IE w = Cw IE , where IE = Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw (n 1) Tr Mr ) = n Q [Mw Mr (n 21) Tr ] (81) 11

11

12

12

Wholesaler's pro t per unit time for Mr < Tr is given by ( D Sw D Cw hw D Tr ( 2 1)  1Tr f Aw + Ip IP w Zwa = D Sw D Cw hw D Tr ( 2 1)  1Tr f Aw + Ip IP w

11

1

12

Case-2 : Mr > Tr Let, [ MTrw ] = n, where n is a positive integer.

Ie IE w Ie IE w

11 12

g if n Tr + Mr  M(82) w g otherwise

Here two cases arise: Sub-case(i) : (n 1) Tr + Mr  Mw Interest to be paid for the amount, IPw = Cw IP , where IP = f Q n Qgfn Tr + Mr Mw g + f Q (n + 1) Qg Tr + ::: + f Q ( 1) Qg Tr = Q( n)[ T2r ( + n 1) + Mr Mw ] (83) 21

22

Supply chain model under three level trade credit policy

Interest earned for the amount, IE w = Cw IE , where IE = Q fMw Mr g + Q fMw Tr Mr g + ::: + Q fMw (n 1) Tr Mr g = n Q [Mw Mr (n 21) Tr ] (84) Sub-case(ii) : (n 1) Tr + Mr > Mw Interest to be paid for the amount, IP w = Cw IP , where IP = f Q (n 1) Qgf(n 1) Tr + Mr Mw g + f Q n Qg Tr + ::: +f Q ( 1) Qg = Q( n + 1)[ ( + n2 2) Tr + Mr Mw ] (85) Interest earned for the amount, IE w = Cw IE , where IE = Q (Mw Mr ) + Q (Mw Tr Mr ) + ::: + Q (Mw (n 2) Tr Mr ) = (n 1) Q [Mw Mr (n 22) Tr ] (86) 21

22

22

Wholesaler's pro t per unit time for Mr > Tr is given by ( D Sw D Cw hw D Tr ( 2 1)  1Tr f Aw + Ip IP w Ie IE w g if (n 1) Tr + Mr  Mw Zwa = (87 (  1) 1 D Sw D Cw hw D Tr 2 Ie IE w g otherwise  Tr f Aw + Ip IP w Hence the joint pro t of the wholesaler and the retailer per unit time for Mr  Tr is given by  Zra + Zwa if Mr < Tr Za = (88) Zra + Zwa if Mr > Tr where Zra , Zra , Zwa and Zwa are given by (74), (77), (82) and (87) respectively. 8.3 Particular Scenario-3 (Mr = Mc = 0) : In spite of huge competitive market, there are some products in the market for which retailer and the base customer does not be obtained any credit period from the wholesaler and the retailer respectively, but wholesaler obtains a credit period from the supplier, e.g., various types of intoxicated products, the seasonal products, etc. Here, a business strategy has been proposed considering retailer's credit period Mr = 0 and customer's credit period Mc = 0. Under the above consideration, the best marketing decision is to be made by maximize the joint pro t of the wholesaler and the retailer with Retailer's cycle length(Tr ), mark-up of the selling price(m) and number of the retailer' cycle() as the decision variables. The joint pro t of the wholesaler and the retailer consists the sales revenue, purchasing cost, ordering cost, holding cost and the amount of earned interest and the payable interest. Here, the computation is performed to calculate the amount of the payable and earned interest only, and all other costs are equal to the proposed model. Retailer's Pro t : Interest to be paid for the amount, IPr = Cr IP , where Z Tr D IP = Interest to be paid for the unsold units during [0; Tr ] = q(t) dt = Tr2 (89) 2 0 Earned interest for the amount IEr = Cr IE , where IE = 0: (90) 21

21

22

22

2

1

2

1

2

1

1

2

2

Supply chain model under three level trade credit policy

Hence, the Retailer's pro t per unit time can be written as 1 fQ S D T C h D Tr2 A I IP + I IE g Zra = r r r r r p r e r Tr 2 = D Sr D Cr T2r hr D T1 fAr + Ip IP r Ie IE r g r

Wholesaler's Pro t : Let, [ MTrw ] = n, where n is a positive integer. Interest to be paid for the amount, IP w = Cw IP , where IP = f Q (n + 1) QgfMw (n + 1) Tr g + f Q (n + 2) Qg Tr + ::: +f Q ( 1) Qg Tr = Q( n 1)[ Tr ( + n) Mw ]

2

23

(91)

(92)

Interest earned for the amount, IE w = Cw IE , where nTr IE = Q Mw + Q (Mw Tr ) + ::: + Q fMw (n 1) Tr g = Q (n + 1) [Mw 2 ] (93) Thus, the Wholesaler's pro t per unit time can be written as 1 f Q S  Q C A h Q T ( 1) I IP + I IE g Zwa = w w w w r p w e w  Tr 2 = D Sw D Cw hw D Tr ( 2 1) 1T f Aw + Ip IP w IeIE w g (94) r The joint pro t of the wholesaler and the retailer per unit is given by Za = Zra + Zwa (95) where, Zra and Zwa are given by (91) and (94) respectively. 8.4 Results of Particular Scenarios : Some results are obtained for the crisp model under the particular scenarios due to dierent . The best marketing decisions for the dierent scenarios are obtained and (for  = 3) presented in Table-6. Following parametric values are assumed to illustrate the crisp model for particular scenarios numerically: a = 350, b = 100, c = 40, Cw = 0:75, Cr = 1:7  Cw , Ar = 20, Aw = 170, hr =0:07  Cr , hw = 0:04  Cw , Mw = 3, Ip = 0:10, Ie = 0:08, = 1:45. Please insert Table-6 here 9. Conclusion

Due to the today's competitive business market, the business enterprises are tied up within a supply chain system consisting production house to base customers. Thus, it is too much important to develop the models under dierent levels. In this paper, for the rst time a supply chain model is developed under three level trade credit policy incorporating customer's credit period and credit amount dependent demand. Here, customer's credit period is oered by the retailer only on a fractional amount of the total purchased amount, whereas retailer and the wholesaler enjoy the full credit on the whole amount. The main aim of this paper is to optimize the total pro t of retailer and the wholesaler considering the retailer's cycle length, retailer's credit period, customer's credit period, fraction of the total amount purchased to be paid initially by the customer and number of business cycles of the retailer as the decision

24

Supply chain model under three level trade credit policy

variables. To optimize the pro t function, a PSO technique is implemented. Due to the rapid changes of cost for various purposes the proposed model is also developed in imprecise environment. Solution methodology for crisp, fuzzy and rough cost parameters of the proposed supply chain model is presented. Also, dierent solutions are given for crisp, fuzzy and rough models under various cases. In this paper, a general supply chain model under three level trade credit policy is developed, which has more practical uses. It is a real life practice in a four-player SC. It can be extended for higher level (i.e. with more players) SC. In the development of this model, it is assumed that all end customers follow the business ethics honestly, they cleared their dues just after the allowed delay period. This is a limitation of the model. Due to the general nature of the proposed model, the model can be extended under various circumstances. For instance, we may extend this model for the deteriorating items, stock-out, quantity discount, discount and in ation rates, and others. Acknowledgements

The authors are heartily thankful to the Honourable Reviewers and the Editor for their contractive comments to improve the quality of the paper. This research work is supported by University Grant Commission of India for Minor Research Project entitle \A study on the in uence of promotional cost sharing, price discount and trade credit on channel pro t/cost in multilevel supply chain under imprecise environment". Appendix A

Following list of test functions are used to test the eciency of the algorithm. From the Table-7, it is observed that the results for dierent test functions are equal to the global optimum in most of the cases and for others results are tends to the global results. Please insert Table-7 here For TF-3, 3 global minima are x1 =[(3.1416,2.2511),(-3.1416,12.2470),(9.4248,2.2470)]. For  2 TF-9, 18 global minima are x = [( 1:4251; 7:0835); ( 7:0835; 1:4252); ( 0:8003; 7:70 83),(-7.7083,-0.8003),(-7.0835,4.8580),(4.8581,-7.0835),(-1.4251,-.08003),(-0.8003,-1.4251),( 7: 0835; 7:7084); ( 7:7083; 7:0835); (4:8581; 5:4829); (5:4829; 4:8580); (5:4829; 7:7083); ( 7:7 084 ; 5:4829); ( 0:8003; 4:8580); (4:8580; 0:8003); (5:4829; 1:4251); ( 1:4251; 5:4829)]. For TF-10, it is shown that the problem is required the maximum number of iteration 940 to converge to its global optima. In this problem search space is taken [400,500], but in the implementation of the algorithm the searching interval for the velocity vector is taken [-0.1,0.1]. For that reason this problem required a large number of iteration for convergence. If the velocity vector searching interval is taken as [-1,1], it is veri ed that the problem required only 126 iteration to converge to its optima. Thus it can be conclude that the algorithm is good enough for this test function also. TF-1 : (Taken from Michalewicz, 1992[24]) F (x1 ; x2 ) = 100(x2 x21 )2 + (x1 1)2 ; x1 + x2 2  0; x1 2 + x2  0; 0:5  x1  0:5; 1:0  x2  1:0:

Supply chain model under three level trade credit policy

25

It has one global minima at (x1; x2) = (0:5; 0:25) and F (0:5; 0:25) = 0:25. TF-2 : (Taken from Yang, 2010[33]) v u n n u1 X 1 1X AC (x1 ; x2 ; :::; xn ) = 20exp[ t xi 2 ] exp[ cos(2xi )] + 20 + e; where 5 n i=i n i=1 n = 1; 2; :::; and 5  xi  5 for i = 1; 2; :::; n. This function has the global minimum AC (x1 ; x2 ; :::; xn ) = 0 at (x1 ; x2 ; :::; xn ) = (0; 0; :::; 0). TF-3 : (Taken from Bessaou & Siary, 2001[2]) RC (x1 ; x2 ) = fx2 [5=(4  2 )]x1 2 +(5=)  x1 6g2 +10 f1 [1=(8)]g cos(x1 )+10; where 5  x1  10, 0  x2  15. This problem has a global minima at three dierent points (x1; x2) = ( ; 12:275); (; 2:275); (9:4278; 2:475) and RC (x1; x2) = 0:397887. TF-4 : (Taken from Yang, 2010[33]) DJ (x1 ; x2 ; :::; xn ) =

n X i=1

xi 2 ;

5:12  xi  5:12 for i = 1; 2; :::; n:

This function has a global minima DJ (x1; x2; :::; xn) = 0 at (x1; x2; :::; xn) = (0; 0; :::; 0). TF-5 : (Taken from Bessaou & Siary, 2001[2]) GP (x1 ; x2 ) = [1 + (x1 + x2 + 1)2  (19 14 x1 + 13 x1 2 14 x2 + 6 x1 x2 + 3 x2 2 )]  [30 + (2 x1 3 x2)2  (18 32 x1 + 12 x12 48 x2 36 x1 x2 + 27 x22)]; 2  xi  2 for i = 1; 2. This problem has 4 local minima and one global minima GP (x1 ; x2 ) = 3 at (x1 ; x2 ) = (0; 1). TF-6 : (Taken from Bessaou & Siary, 2001[2]) F 2(x1 ; x2 ) = 100  (x2 2 x1 ) + (1 x1 ); 2:048  x1 ; x2  2:048: It has one global minima at (x1; x2) = (2:048; 0) and F 2(x1; x2) = 0. TF-7 : (Taken from Bessaou & Siary, 2001[2]) n n n X 1X 1X ixi )2 + ( ixi )4 ; 5  x1 ; x2 ; :::; xn  5: Zn (x1 ; x2 ; :::; xn ) = ( xi 2 ) + ( 2 2 i=1 i=1 i=1 It has one global minima at (x1; x2; :::; xn) = (0; 0; :::; 0) and Zn(0; 0; :::; 0) = 0. TF-8 : (Taken from Bessaou & Siary, 2001[2]) H3;4 (x) =

4 X i=1

ci exp[

3 X j =1

aij (xj

pij )2 ]; 0  xi  1 for i = 1; 2; 3:

(Parameter values are given in Table-8) It has 4 local minima and one global minima H3;4(x) = 3:86278 at x = (0:11; 0:555; 0:855).

26

TF-9 :

Supply chain model under three level trade credit policy

Please insert Table-8 here (Taken from Bessaou & Siary, 2001[2]) SH (x1 ; x2 ) =

5 X i=1

i  cos[(i +1)  x1 + i] 

5 X i=1

i  cos[(i +1)  x2 + i];

10  x1; x2  10:

It has 760 local minima and 18 global minima SH (x1; x2) = 186:7309. TF-10 : (Taken from Yang, 2010[33]) n X p Sn (x1 ; x2 ; :::; xn ) = xi sin( kxi k); 400  xi  500 for i = 1; 2; :::; n: i=1

It has one global minima at (x1; x2; :::; xn) = (420:9687; 420:9687; :::; 420:9687) and Sn(420:96 87; 420:9687; :::; 420:9687) = 418:9829n. Appendix B

At time nTr , retailer has completed his/her n-th cycle and also paid the full amount for the nQ units to the wholesaler. Again wholesaler got the revenue for Q-units of retailer's (n + 1)-th cycle at the time (nTr + Mr ). Again since nTr + Mr  Mw and there is no payment during (nTr + Mr ; Mw ), so wholesaler got the full revenue of the (n + 1)Q units till Mw . Thus at time Mw , wholesaler have to pay interest for the unpaid amount i.e., for the amount of fQ (n + 1)Qg units. At time f(n + 1)Tr + Mr g, wholesaler obtained the payment(amount of Q units) for retailer's (n + 2)-th cycle. Thus, the interest have to pay for the amount of fQ (n + 1)Qg units for the time f(n + 1)Tr + Mr Mw g. Also the next payment (amount of Q units) will occurred at the time f(n +2)Tr + Mr g. Thus the interest have to pay for the amount of [fQ (n + 1)Qg Q] = fQ (n + 2)Qg for the time [f(n + 2)Tr + Mr g f(n + 1)Tr + Mr g] = Tr . Preceding in this manner, for the last cycle i.e., for -th cycle, interest have to pay for the amount of fQ ( 1)Qg for the time [f( 1)Tr + Mr g f( 2)Tr + Mr g] = Tr . Hence the result. Discussion about Eq. (25): Wholesaler obtained a credit period Mw from supplier and oers a credit period Mr to the retailer. In the time Mw , retailer's (n + 1)-th cycle is running and since nTr + Mr  Mw, so the retailer paid for (n +1)Q units and wholesaler earned interest for this (n + 1)Q units. At time Mr , retailer paid the amount of Q units to the wholesaler for retailer's 1st cycle, and hence wholesaler earned interest on this Q units for (Mw Mr ) time. Again at (Tr + Mr ), retailer pays o the amount of Q units of retailer's 2nd cycle and consequently wholesaler earned interest for this Q units for the time (Mw Tr Mr ). In the similar manner, at (nTr + Mr ), retailer paid the amount for Q units of (n + 1)-th cycle and then wholesaler earned interest for this Q units for the time (Mw nTr Mr ). Discussion about Eq. (27): At time nTr , retailer has completed his/her n-th cycle and pays o the amount of Q units for his/her n-th cycle to the wholesaler at the time (n 1)Tr + Mr . Thus wholesaler obtain the amount of nQ units till (n 1)Tr + Mr . Again wholesaler got the revenue for the Q-units of retailer's (n + 1)-th cycle at the time (nTr + Mr ). Since nTr + Mr > Mw and there is no payment during ((n 1)Tr + Mr ; Mw ), so wholesaler got the full revenue of the nQ units till Mw . Thus at time Mw , wholesaler have to pay interest for Discussion about Eq. (23):

Supply chain model under three level trade credit policy

27

the unpaid amount i.e., for the amount of (Q nQ) units. At time (nTr + Mr ), wholesaler obtained the payment(amount of Q units) for the retailer's (n + 1)-th cycle. Thus, the interest have to pay for the amount of (Q nQ) units for the time (nTr + Mr Mw ). Also the next payment (amount of Q units) will occurred at the time [(n + 1)Tr + Mr ]. Thus the interest have to pay for the amount of [(Q nQ) Q] = [Q (n + 1)Q] for the time [f(n + 1)Tr + Mr g fnTr + Mr g] = Tr . Preceding in this manner, in the last cycle i.e., in -th cycle, interest have to pay for the amount of fQ ( 1)Qg for the time [f( 1)Tr + Mr g f( 2)Tr + Mr g] = Tr . Hence the result. Discussion about Eq. (29): At the time Mw , retailer's (n + 1)-th cycle is running and since nTr + Mr > Mw, so the retailer paid for nQ units and wholesaler earned interest for this nQ units. At time Mr , retailer paid the amount of Q units to the wholesaler for retailer's 1st cycle, and hence wholesaler earned interest on this Q units for (Mw Mr ) time. Again at (Tr + Mr ), retailer pays o the amount of Q units of retailer's 2nd cycle and consequently wholesaler earned interest for this Q units for the time (Mw Tr Mr ). In the similar manner, at f(n 1)Tr + Mr g, retailer paid the amount for Q units of n-th cycle and then wholesaler earned interest for this Q units for the time fMw (n 1)Tr Mr g. References [1] Aggarwal, S.P., Jaggi, C.K. (1995). Ordering policies of deteriorating items under permissible delay in payments. Journal of the Operational Research Society, 46, 658-662. [2] Bessaou, M., Siarry, P. (2001). Agenetic algorithm with real-value coding to optimize multimodal continuous function. Structural and Multidisciplinary Optimization, 23, 63-74. [3] Chen, L.H., Kang, F.S. (2010). Integrated inventory models considering the two level trade credit policy and a price-negation scheme. European Journal of Operational Research, 205, 47-58. [4] Chung, K.J. (2008). Comments on the EOQ model under retailer partial trade credit policy in supply chain. International Journal of Production Economics, 114, 308-312. [5] Chung, K.J. (2011). The simpli ed solution procedures for the optimal replenishment decisions under two level trade credit policy depending on the order quantity in a supply chain system. Expert Systems with Application. 38, 13482-13486. [6] Chung, K.J. (2013). The EPQ model under conditions of two levels of trade credit and limited storage capacity in supply chain management. International Journal of Systems Science, 44, 1675-1691. [7] Chung, K.J., Cardenas-Barron, L.E., Ting, P.S. (2014). An inventory model with non-instantaneous receipt and exponentially deteriorating items for an integrated three layer supply chain system under two levels of trade credit. International Journal of Production Economics, 155, 310-317. [8] Chung, K.J., Liao, J.J. (2004). Lot-size decisions under trade credit depending on the ordering quantity. Computers & Operations Research, 31, 909-928. [9] Dubois, D., Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications, Academic Press, NewYork. [10] Goyal, S.K. (1985). Economic order quantity under conditions of permissible delay in payment. Journal of the Operational Research Society, 36, 335-338. [11] Guchhait, P., Maiti, M.K., Maiti, M. (2010). Multi-item inventory model of breakable items with stockdependent demand under stock and time dependent breakability rate. Computers & Industrial Engineering, 59, 911-920. [12] Guchhait, P., Maiti, M.K., Maiti, M. (2014). Inventory model of a deteriorating item with price and credit linked fuzzy demand : A fuzzy dierential equation approach. Operational Research Society of India, 51, 321-353. [13] Ho, C.H. (2011). The optimal integrated inventory policy with price-and-credit linked demand under two-level trade credit. Computers & Industrial Engineering, 60, 117-126. [14] Huang, Y.F. (2003). Optimal retailers ordering policies in the EOQ model under trade credit nancing. Journal of the Operational Research Society, 54, 1011-1015.

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Supply chain model under three level trade credit policy

[15] Huang, Y.F. (2006). An inventory model under two levels of trade credit and limited storage space derived without derivatives. Applied Mathematical Modelling, 30, 418-436. [16] Huang, Y.F. (2007). Economic order quantity under conditionally permissible delay in payments. European Journal of Operational Research, 176, 911-924. [17] Huang, Y.F., Hsu, K.H. (2008). An EOQ model under retailer partial trade credit policy in supply chain. International Journal of Production Economics, 112, 655-664. [18] Jaggi, C.K., Goyal, S.K., Goel, S.K. (2008). Retailers optimal replenishment decisions with credit-linked demand under permissible delay in payments. European Journal of Operational Research, 190, 130-135. [19] Lee, H.M., Yao, J.S. (1998). Theory and Methodology: Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operational Research, 109, 203-211. [20] Liu, B. (2002). Theory and Practice of Uncertain Programming, Heidelberg: Phsica-Verlag. [21] Mahata, G.C. (2012). An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain. Expert Systems with Applications, 39, 3537-3550. [22] Maiti, M.K. (2011). A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon. European Journal of Operational Research. 213, 96-106. [23] Maiti, M.K., Maiti, M. (2007). Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm. European Journal of Operational Research, 179, 352-371. [24] Michalewicz, Z. (1992). Genetic algorithms + data structures = evolution programs. Berlin: Spinger. Chapter 2 & 3. [25] Min, J., Zhou, Y.W., Zhao, J. (2010). An inventory model for deteriorating items under stock dependent demand and two-level trade credit. Applied Mathematical Modelling, 34, 3273-3285. [26] Ouyang, L.Y., Teng, J.T., Goyal, S.K., Yang, C-T. (2009). An economic order quantity model for deteriorating items with partially permissible delay in payments to order quantity. European Journal of Operational Research, 194, 418-431. [27] Pawlak, Z. (1982). Rough Sets, International Journal of Information and Computer Science, 11, 341-356. [28] Soni, H.N. (2013). Optimal replenishment policies for deteriorating items with stock sensitive demand under two level trade credit and limited capacity. Applied Mathematical Modelling, 37, 5887-5895. [29] Teng, J.T. (2009). Optimal ordering policies for a retailer who oers distinct trade credits to its good and bad customers. International Journal of Production Economics, 119, 415-423. [30] Teng, J.T., Chang, C.T. (2009). Optimal manufacture's replenishment policies in the EPQ model under two level trade credit policy. European Journal of Operational Research, 195, 358-363. [31] Teng, J.T., Cheng, C.T., Chern, M.S., Chan, Y.L. (2007). Retailer's optimal ordering policies with trade credit nancing. International Journal of Systems Science, 38, 269-278. [32] Teng, J.T., Yang, H.L., Chern, M.S. (2013). An inventory model for increasing demand under two levels of trade credit linked to order quantity. Applied Mathematical Modelling, 37, 7624-7632. [33] Yang, X.-S. (2010). Test problems in optimization, in: Engineering Optimization: An Introduction with Metaheuristic Appplications, John Wiley & Sons. [34] Zadeh, L.A. (1965). Fuzzy Sets. Information and Control, 8, 338-356. [35] Zadeh, L.A. (1978). Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28.

Research Highlights

ä Proposed model has been developed incorporating three level trade credit policy. ä Selling price, customers’ credit period and customers’ credit amount induced demand. ä Model is formulated in crisp, fuzzy and rough environments. ä Solutions of imprecise models are found without using crisp equivalent of objectives. ä The marketing decision has made from both wholesaler’s and retailer’s point of view.

1