An integrated production inventory model under interactive fuzzy credit period for deteriorating item with several markets

Applied Soft Computing 28 (2015) 453–465

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

An integrated production inventory model under interactive fuzzy credit period for deteriorating item with several markets Bibhas Chandra Das a,∗ , Barun Das b,1 , Shyamal Kumar Mondal a,2 a b

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal 721102, India Department of Mathematics, Sidho Kanho Birsha University, Purulia, West Bengal 723101, India

a r t i c l e

i n f o

Article history: Received 21 March 2013 Received in revised form 18 May 2014 Accepted 30 November 2014 Available online 18 December 2014 Keywords: Integrated production inventory Multiple markets Instantaneous replenishment Part-payment Fuzzy credit period

a b s t r a c t The presence of multiple markets create profitable opportunities to the supply chain system. In this regard, this paper consists of the joint relationship between a manufacturer and multiple markets in which manufacturer offers part-payment to the markets due to their collection of finished products during the production run time. Here it is also considered that manufacturer is facilitated with credit period by raw material supplier where credit period has been presented as an interactive fuzzy fashion. In this paper, two types of deterioration have been assumed such as one for finished products and the other for raw materials. A solution algorithm is presented to get fuzzy optimal profit for the proposed integrated production inventory system optimizing production run time. A numerical example is used to illustrate the proposed model. Finally, sensitivity analysis has been carried out with respect to the major parameters to demonstrate the feasibility of the proposed model. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In real business world, sometimes a manufacturer, supplier and markets/retailer would like to make a long-term cooperative relationship as an integrated system to get a tensionless stable sources of supply and demand of items as well as reliability to gain optimum profit from each other. Globally, the industrial environment gradually becomes more and more competitive and much effort has been made towards the efficiency and effectiveness. So in this connection, the supply chain management plays an important role in the present situation. Pan and Yang [24] worked on integrated inventory system with controllable lead time. Chang et al. [2] addressed the optimal pricing and ordering policy for an integrated inventory model when trade credit is linked to order quantity. Then Chen and Kang [3] developed their integrated inventory model with permissible delay in payments. Ho et al. [13] enriched their integrated inventory model with price-and-credit-linked demand under twolevel trade credit. There are many research works on integrated technique such as Goyal [10], Jaber and Osman [15], Huang [14] and Taleizadeh et al. [30].

∗ Corresponding author. Tel.: +91 9474447356. E-mail addresses: [email protected] (B.C. Das), [email protected] (B. Das), shyamal [email protected] (S.K. Mondal). 1 Tel.: +91 9434994401. 2 Tel.: +91 9433336686. http://dx.doi.org/10.1016/j.asoc.2014.11.057 1568-4946/© 2014 Elsevier B.V. All rights reserved.

In the inventory literature, there are many research works without consideration of production process. But sometimes, consideration of production process becomes necessary to describe a manufacturer’s or supply chain model. Yang and Wee [36] developed an integrated production inventory model for deteriorating items. Law and Wee [18] enriched an integrated productioninventory model for ameliorating and deteriorating items taking account of time discounting. Ouyang et al. [22] took an optimal strategy for an integrated system with variable production rate and trade credit financing. Then Das et al. [8] improves a production policy for a deteriorating item under permissible delay in payments with stock-dependent demand rate. Soni and Patel [25] developed an integrated inventory system with variable production under partial trade credit. In an EPQ model, an optimal lot-sizing policy with production cost effects has been described by Teng et al. [33]. A manufacturer produces an item and it is sold at different markets. Again sometimes, it is seen that the markets have different selling seasons. So, manufacturer/supplier should have to adopt appropriate management policies/strategies in the business with the different markets. In a production inventory model with deteriorating items, He et al. [12] also considered multiplemarket demands. Krichen et al. [17] described a single supplier and multiple cooperative retailers inventory model under permissible delay in payments. Then Pal et al. [23] researched on multi-echelon supply chain model in multiple markets with supply disruption.

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Table 1 Comparison of articles. Author(s) & published (year)

Integrated or not

Deterioration rate

Credit period

Market(s)

With production

Ghare and Schrader [9] Covert and Philip [6] Goyal [10] Goyal [11] Chu et al. [4] Pan and Yang [24] Abad and Jaggi [1] Yang and Wee [36] Teng et al. [32] Jaber and Osman [15] Law and Wee [18] Ouyang et al. [22] Chang et al. [2] Chen and Kang [3] Das et al. [8] Huang [14] He et al. [12] Chung and Liao [5] Ho [13] Krichen et al. [17] Liang and Zhou [19] Mishra and Mishra [21] Yan et al. [35] Mahata [20] Pal et al. [23] Soni and Patel [25] Su [26] Das et al. [7] Taleizadeh et al. [30] Taleizadeh and Jolai [31] Teng et al. [33] Wu et al. [34] Present paper

Not Not Integrated Integrated Not Integrated Integrated Integrated Not Integrated Integrated Integrated Integrated Integrated Not Integrated Not Integrated Integrated Integrated Not Not Integrated Not Not Integrated Integrated Integrated Not Not Not Not Integrated

Exponential Weibull distribution No No Constant No No Yes No No Weibull distribution No No No Constant No Constant No No No Constant Fuzzy Yes Exponential distribution Constant Random Constant Constant Constant Constant No Constant Constant

No No No Crisp Crisp No Crisp No Crisp Crisp No Crisp Crisp Crisp Crisp Crisp No Crisp Crisp Crisp Crisp Crisp No Crisp No Crisp Crisp Crisp No No Crisp Crisp Crisp and fuzzy

Single Single Single Single Single Single Single Single Single Single Single Single Single Single Single Single Multiple Single Single Multiple Single Single Single Single Multiple Single Single Single Single Single Single Single Multiple

No No No No No Yes No Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes No Yes Yes No Yes No Yes

In the business world, it is seen that a manufacturer produces the items which deteriorates as the time goes on. For example, some of the deteriorating items are - volatile liquids, medicine, fruit juice, cold-drinks etc. Ghare and Schrader [9] first derived an economic order quantity (EOQ) model by assuming exponential decay with constant demand. Next, Covert and Philip [6] extended Ghare and Schrader’s constant deterioration rate to a two-parameter Weibull distribution. An EOQ model of deteriorating items was developed by Chu et al. [4]. Mishra and Mishra [21] worked on fuzzified deterioration under cobweb phenomenon and permissible delay in payments. After that, Taleizadeh et al. [28] enriched their inventory model for a deteriorating item with back-ordering and temporary price discount. A lot size model for deteriorating item with expiration dates under two-level trade credit financing was produced by Wu et al. [34]. Now-a-days in most of business, offering of a credit period to settle the account takes an important role. In the literature reviews, it is seen that raw materials supplier offers to manufacturer, manufacturer offers to wholesaler, wholesaler offers to retailers, manufacturer offers to markets a delay period which is crisp in nature to settle the account to entice the other. Goyal [11] derived a single item inventory model under the conditions of permissible delay in payments. Abad et al. [1] developed a seller–buyer model under permissible delay in payments by game theory to determine the optimal unit price with trade credit period, considering that the demand rate is a function of the retail price. Teng et al. [32] developed the optimal pricing and lot-sizing model under permissible delay in payments by considering the difference between selling price and purchase cost when demand rate is a function of the selling price. In supply chain system, Chung et al. [5] developed a simplified algorithm with two part trade credit. Liang and Zhou [19] described a two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment. An

EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain was described by Mahata [20]. Su [26] worked on integrated inventory system with defective items allowing shortages under trade credit. Then Das et al. [7] woked on integrated supply chain model for a deteriorating item with procurement cost dependent credit period. Real business world is full of uncertainties in non-stochastic sense, which leads to estimation of different inventory parameters as fuzzy numbers. Introduction to fuzzy set theory and basic ideas of fuzziness are described by Zimmermann [37]. Ko et al. [16] made a review on soft computing application in supply chain management system. Yan et al. [35] developed their integrated production distribution model with deteriorating item. Taleizadeh et al. [29] described a joint-replenishment inventory control problem using uncertain programming. For solving a fuzzy single-period problem, Taleizadeh et al. [27] developed meta-heuristic algorithms. Then Taleizadeh et al. [31] worked on fuzzy rough EOQ model considering quantity discount and repayment. From Table 1, it is seen that till now no researcher has worked on the integrated EPQ model for deteriorating item with fuzzy credit period in multiple sessional markets. In this paper, a manufacturer receives the deteriorating raw materials from a supplier who offers a credit period (which is fuzzy in nature) to settle his/her account. We have also considered the fuzziness of the credit period in different ways. The manufacturer produces the item which deteriorates over time and supplies to the markets which have different selling seasons. The markets receive the required item as a lot at the beginning of its business. Those markets start their businesses before the production ends, they get an opportunity of part payment at the beginning and the remaining amount should be paid at the end of their own business. Here, the manufacturer and the markets maximize their profits as an integrated form to get rid of tension in the business.

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The organization in this paper is in the following way: Section 2 describes the formulation of the production inventory model with deteriorating item. In Section 3, formulations for non-deteriorating item are derived. Model with fuzzy credit period are developed in Section 4 and in Section 5 numerical examples, sensitivity analysis and some managerial insights have been performed. Finally in Section 6, a conclusion is made. 2. Formulation of a production inventory model for a manufacturer and multiple sessional markets In this paper, it has been considered that a manufacturer produces a single item using raw materials which are purchased from a raw material supplier and the finished products are sold to different markets which have different constant customers’ demands and different sessions of the business. Here, the raw material supplier offers a credit period to the manufacturer, which may be fuzzy in nature. Every market collects all its required quantity of item as per its demand at the starting time of its business period. Here manufacturer also gives an opportunity of initial part-payment to those markets who receive the items during the production run time and the remaining amount will be paid to the manufacturer after the completion of their business, otherwise full payment will be made at the initial time of their business. The manufacturer earns interest by the collected sales revenue from the markets. 2.1. Notations and assumptions 2.1.1. Notations To illustrate the proposed model, the following notations have been used. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi) (xvii) (xviii) (xix) (xx) (xxi) (xxii)

(xiii)

Fig. 1. Manufacturer’s raw material’s inventory for deteriorating item.

1. A single manufacturer and multiple markets have been assumed for a flow of single product. 2. Production rate (p) and production run time (T) both are deterministic variables. 3. Time horizon is finite. 4. Shortages are not allowed. 5. Deterioration rates of raw materials ( r ) and finished products () are deterministic and constant. 6. In practical business world, sometimes it is seen that the credit period offered by the supplier has a little disturbance i.e., it changes due to various factors according to his/her business policy. So in nature, it is vague and imprecise. For this reason here, it has been considered that raw material supplier offers a fuzzy credit period to the manufacturer. Now in fuzzy set theory [37], there are some standard fuzzy numbers such as triangular, trapezoidal, Gaussian etc among which the triangular and trapezoidal fuzzy numbers have been considered for illustration. 7. Since the maintenance cost for production is increased with the increase of production run time, hence in this paper it has been considered that the manufacturer’s selling price (cp ) of a finished product to the markets to be an increasing function of production run time (T) which is defined by cp = ae˛T , where a > 0 and ˛ ≥ 0. 8. The manufacturer offers an opportunity of initial part-payment to those markets who receive the item before the end of production run and the remaining part should be paid at the end of their individual business session. But the markets who receive their required item after the production run time (T), do’nt get this opportunity. Every market receives all items at the starting time of their business period to fulfil their fixed customers’ demand.

C1 : manufacturer’s ordering cost per order. C2 : manufacturer’s setup cost. C3 : all markets’ set-up cost. hr : raw material’s holding cost rate. hp : finished product’s holding cost rate. f: unit usage of raw materials per finished product. p: manufacturer’s production rate. di : customers’ demand rate for i-th market. Qi : quantity received by i-th market from the manufacturer. Ti : starting time of the business of i-th market.  r : constant deterioration rate of raw materials. : constant deterioration rate of finished products. T: time where production is stopped. 2.2. Raw material’s inventory for manufacturer M: credit period offered by the raw material supplier to the manufacturer. The manufacturer receives all of the required quantity of raw Tie : time at which the selling season ends for i-th market. materials instantaneously from the raw material supplier for a procr : manufacturer’s raw material’s purchase cost per unit. duction run when he/she is going to start his/her production (Fig. 1). cp : unit purchase cost by markets. Then as the time goes on, the inventory of raw materials depletes s: unit selling price of markets. gradually due to production and deterioration and vanishes at time n: number of markets where the item is sold. T. The raw material’s inventory Ir (t) at time t is given by the differId : rate of interest earned per year. ential equation I c : rate of interest payable per year.  MTP : total markets profit for deteriorating item. : total manufacturer’s profit WTP : total profit for the integrated system ITP   : total markets profit MTP 2 for non-deteriorating item. WTP 2 : total manufacturer’s profit ITP 2 : total profit for the integrated system

2.1.2. Assumptions The following assumptions have been used to generate this model.

dI r (t) + r Ir (t) = −fp dt

when 0 ≤ t ≤ T

(1)

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B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

with boundary conditions Ik (Tk−1 ) = Ik−1 (Tk−1 ) − Qk−1

with the boundary condition Ir (T ) = 0

(2)

Using the boundary condition (2), the solution of the Eq. (1) is obtained as fp r (T −t) Ir (t) = [e − 1] r

(3)

The quantity of raw materials received by the manufacturer is fp r T [e − 1] r

Qr = Ir (0) =

(4)



0

= hr .

fp r

fphr

=

r

2

T



T

M 2

= cp .

 p Qi e(Ti −t) ; (1 − e−t ) −  i=0

where Tk−1 ≤ t ≤ Tk (1 ≤ k ≤ m − 1) Ij (t) =

N 

[e

Qi e(Ti −t) ;

i=j

where Tj−1 ≤ t ≤ Tj

Im− (t) =

fp r (T −t) [e − 1]dt r

r (T −M)

Im+ (t) =

− 1 − r (T − M)]

 p (1 − e−t ) − Qi e(Ti −t) ;  n 

(11)

Qi e(Ti −t) ;

(12)

Now at, t = T, Im− (t) = Im+ (t)

which gives

1  log {1 + Qi eTi } p  n

T=

(13)

i=1

Manufacturer’s raw material’s purchase cost

which is the relation between the two variables T and p. Manufacturer’s finished products holding cost HP is given by

= cr .Qr fp r T (e − 1) r

m−1 

HP

(5)

with boundary condition I1 (0) = 0 Tk−1 ≤ t ≤ Tk

n 

Ij ] j=m+1

Ik (t)dt Tk−1

=

 Ij

k−1 

1 1 p {(Tk − Tk−1 ) + (e−Tk − e−Tk−1 )} + (e−Tk − e−Tk−1 ).    Ij (t)dt Tj−1

N 

1 = − (e−Tj − e−Tj−1 ). 



Im−

Qi eTi

i=j T

=

Im− (t)dt Tm−1

=

m−1 

1 1 p {(T − Tm−1 ) + (e−T − e−Tm−1 )} + (e−T − e−Tm−1 ).   

 Im+

Qi eTi

i=0

Tj

=

=

i=1

Tm

Ij (t)dt T

[k = 2, 3, ..., (m − 1)] (6)

Ik + Im− + Im+ +

k=1 Tk

where, Ik =

The manufacturer starts production of the deteriorating item at time t = 0 and then the inventory level increases over time. After time T1 , the first market receives all of his required quantity Q1 instantaneously, then at time T2 the second market receives his required quantity Q2 instantaneously and so on. At time T, the production is stopped, then inventory level decreases due to deterioration of the item and the remaining markets’ instantaneous quantity replenishment and the inventory is vanished for receiving the quantity Qn by the last market. Let the manufacturer’s inventory level in the interval [Ti−1 , Ti ] (i = 1, 2, . . . , n) be Ii (t) and Im− (t), Im+ (t) be the inventory levels in time intervals [Tm−1 , T], [T, Tm ] respectively. Then the differential equations of the finished products’ inventory levels at time t in [0,Tn ] are as follows (Fig. 2): 0 ≤ t ≤ T1

= hp [



2.3. Manufacturer’s finished products’ inventory

dI k (t) + Ik (t) = p, dt

(10)

i=m

Qi

dI 1 (t) + I1 (t) = p, dt

(j = m + 1, m + 2, ..., n)

i=1

i=1

= cr .

(9)

m−1

Ir (t)dt

Manufacturer’s sales revenue n 

(8)

k−1

Ik (t) =

{er (T −t) − 1}dt

T

MT

cr fpI c r

dIm+ (t) + Im+ (t) = 0, T ≤ t ≤ Tm dt dIj (t) + Ij (t) = 0, Tj−1 ≤ t ≤ Tj (j = m + 1, m + 2, . . ., n) dt

0

= cr Ic . =

with boundary condition Im− (Tm−1 ) = Im−1 (Tm−1 ) − Qm−1

[er T − r T − 1]



(7)

Ir (t)dt

The manufacturer pays the full payment to the raw material supplier at the end of the credit period M. For this purpose, the manufacturer takes a loan to cover the payment due to the remaining quantity in the period M to T at the rate of interest Ic . So, interest payable by the manufacturer is cr Ic .

Tm−1 ≤ t ≤ T

with boundary conditions Im+ (Tm ) = Jm+1 (Tm ) + Qm , Ij (Tj−1 ) = Ij−1 (Tj−1 ) − Qj−1 (j = m + 1, m + 2, . . . , n) and In (Tn ) = Qn Solving these equations we get,

Raw material’s holding cost is hr .

dI m− (t) + Im− (t) = p, dt

n 

1 = − (e−T − e−Tm ). 

i=m

Qi eTi

Qi eTi

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

457

Fig. 2. Manufacturer’s finished products inventory.

Therefore, =

HP

hp [

 p pT 1  −T + 2 (e−T − 1) + (e k − e−Tk−1 ). Qi eTi    m−1

k−1

k=2

i=1

  1 1 + (e−T − e−Tm−1 ). Qi eTi + (e−T − e−Tm ). Qi eTi   m−1

i=1

i=m

 1  −T − (e − e−Tm ). Qi eTi ]  n

j=m

n

n

i=j

Manufacturer’s interest earned is Fig. 3. i-th market’s inventory for deteriorating item.





n

n

Id Qi cp r(Tn − Ti ) +

Ti ≤T,i=1

Qi cp (1 − r)Id (Tn − Tie )

Ti ≤T,i=1 n 

+

Qi cp Id (Tn − Ti )

Ti >T,i=1

The manufacturer’s profit function: The manufacturer’s total profit WTP is given by WTP(p, T) = sales revenue − raw material’s purchase cost − raw material’s holding cost − finished products holding cost − interest payable + interest earned − ordering cost − set-up cost n 

= cp .

Qi − cr .

i=1

−

cr fpI c r

1 + 

2

fp r T fphr r T (e − 1) − [e − r T − 1] 2 r r

[er (T −M) − 1 − r (T − M)] − hp [

m−1 

(e k=2

−Tk

−e

−Tk−1

+



Qi e i=1

Qi e i=m

Id Qi cp r(Tn − Ti ) +

Ti ≤T,i=1 n

+

).

n 

1 + (e−T − e−Tm ).  n 

k−1 

Ti

1 − 

n 

Ti

(e

−e

−Tm

n 

).

j=m

Qi e

di (T −t) [e ie − 1] 

Qi = Ji (0) =

Ti

(14) Ti

(15)

with boundary condition J(Tie ) = 0 Solving we get,

di T [e ie − 1] 

Holding cost (MHC) for all markets is given by

i=1

Qi e

(i = 1, 2, ...., n)

(16)

The ith market receives the quantity

m−1 

1 + (e−T − e−Tm−1 ).  −T

dJ i (t) = −.Ji (t) − di dt

Ji (t) =

p pT + 2 (e−T − e−T1 )  

n 

portion pay at the end of his/her business period. But those markets arrive after the production run time T, pay the total amount at their business starting time. They pay the initial amount by getting loan from a bank at the rate of interest of Ic per year. Every market earns interest at the rate of Id by depositing sales revenue continuously. The inventory level Ji (t) for the i-th market is governed by the following differential equation (Fig. 3):

]

i=j

Qi cp (1 − r)Id (Tn − Tie )

Ti ≤T,i=1

MHC =

n  i=1 n

=

Ti

i

2

[e(Tie −Ti ) − (Tie − Ti ) − 1]

All markets’ total sales revenue (MPS) is given by

Ti >T,i=1

2.4. The markets’ inventory

Tie

Ji (t)dt

 hm d i=1

Qi cp Id (Tn − Ti ) − C1 − C2

 hm

MPS =

n 

sdi (Tie − Ti )

i=1

The i-th market receives his/her total required quantity of deteriorative item from the manufacturer at the beginning of its selling season to fulfil the customers’ demand rate di . Those markets start their business on or before the production run time T, pay r portion of the price amount payable initially and the remaining (1 − r)

All markets’ total purchase cost (MPC) is given by MPC =

n  i=1

cp .Qi

(17)

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B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

All markets’ total interest earned (MIE) is obtained as





n

MIE =

cp di Id

=

3. Model without deterioration There are also some raw materials and items which deteriorate after a long time beyond of company’s consideration or do’nt deteriorate over time. So in this section, we consider the model for the case of non-deteriorating item i.e, when  = 0 and  r = 0 (Fig. 4). To formulate the model in this case, the calculations are as follows: Total produced quantity is

(Tie − t)dt

Ti

i=1

n 

Tie

1 sdi Id . (Tie − Ti )2 2

i=1

All markets’ total interest payable (MIP) is expressed as n 

MIP =

Qi rsId (Tie − Ti ) +

i=1,Ti ≤T

n 

Qi cp Ic (Tie − Ti )

i=1,Ti ≥T

n 

n 

sdi (Tie − Ti ) −

i=1

−

cp Qi −

i=1

n 

rQi cp Ic (Tie − Ti ) −

i=1,Ti ≤T

Qi cp Ic (Tie − Ti ) +

n 

i=1,Ti ≥T

=

1 2 sdi Id . (Tie − Ti ) 2

fphr .

i=1

m−1

m−1 

1 + (e−T − e−Tm−1 ).  n

i=1

j=m

+

Qi eTi +



+

i=1 n

−

n

n  hm d i=1



i=1,Ti ≤T n



+

i=m

Id Qi cp r(Tn − Ti )

Ti ≤T,i=1

(19)

Qi cp Id (Tn − Ti ) − C1 − C2 − C3

Qi cp Ic (Tie − Ti )

i=1,Ti ≥T

1 sdi Id . (Tie − Ti )2 2

fpc r Ic .

i=1

Now, our objective is to find out optimal values of T, p and ITP such that the integrated total profit function ITP(p, T) be maximum. Lemma 1: Manufacturer’s production rate (p) must satisfy the condition

k

i=1

Qi eTi

[e(Tie −Ti ) − (Tie − Ti ) − 1]

n 

Qi rcp Ic (Tie − Ti ) −

i=1

Ti >T,i=1

i

2

n 

n 

Qi cp (1 − r)Id (Tn − Tie ) +

sdi (Tie − Ti ) −

n 

1 −T (e − e−Tm ). 

i=j

n 

Ti ≤T,i=1 n

Qi e(Ti −Tk )

1 − e−Tk

]

when Tk ≤ T

(T − M)2 2

Quantity of raw material’s received by manufacturer is Qr = fpT Manufacturer’s purchase cost is cr Qr = fpc r .T

Proof:

Manufacturer’s finished products holding cost (HP) is given by

p 

Interest payable by manufacturer is

k−1

k=2

 1  −T − (e − e−Tm ). Qi eTi ] + 

Ik (t) =

T2 2

 pT 1  −T p (e k − e−Tk−1 ). Qi eTi + 2 (e−T − e−T1 ) +   

−hp [

p ≥ Max[

n  

1 − e−Tk − Tk ≤ T,

  pT 2  − (Tk − Tk−1 ). Qi − (T − Tm−1 ). Qi 2 m−1

Qi · e(Ti −Tk )

=

HP

i=1

Now, when

k=2

Ik (Tk )≥Qk

  p 1 − e−Tk − Qi · e(Ti −Tk ) ≥Qk  

i.e,

p≥

k 

i=1

Qi · e

(Ti −Tk )

i=1

1 − e−Tk

k−1

n 

−(T − Tm ).

n

i.e,

n

WTP(p, T ) + MTP(p, T ) fp fphr r T cr fpI c r (T −M) −cr . (er T − 1) − [e − r T − 1] − [e − 1 − r (T − M)] 2 2 r r r

=



1 Qi p

Raw material’s holding cost is

(18)

So, the total profit (ITP) for the integrated system is written as ITP(p, T )

Qi

i=1

m di [e(Tie −Ti ) − (Tie − Ti ) − 1] − C3 2

i=1

T =

i.e,

n  h

n 

n  i=1

Therefore, all markets’ total profit (MTP) is given by MTP(p,T) = MPS − MPC − MHC − MIP + MIE − set-up cost =

=

pT

(20)

i=m

i=1

Qi +

n 

i

2 i=1

.(Tie − Ti )2

n 

(Tj − Tj−1 ).

j=m+1

Holding cost for all of the markets is n  hm d

m−1

i=j

i=1

Qi

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

459

Fig. 4. Manufacturer’s finished products inventory for non-deteriorating item.

The i-th market’s received quantity Qi is given by

Then maximizing ITP2 (p, T), the optimum values of the required variables and objective function will be obtained for this case.

Qi = di (Tie − Ti )

4. Basic concepts on fuzzy sets (cf. Zimmermann [37])

Therefore, the manufacturer’s total profit (WTP2 ) is given by WTP 2 (p, T )

= Cp

n 

i=1

−hp [

pT 2 − 2

m−1 

−(T − Tm ) n 

i=m

(T − M) T2 − fphr . 2 2 k−1 

(Tk − Tk−1 ).

k=2

n 

+

Qi +

n 

 (x) is called the membership function or grade of membership

A

Qi ]

A which maps X to the membership space M. (When M conof x in A is called non-fuzzy and  (x) tains only the two points 0 and 1, A is identical to the characteristic function of a non-fuzzy set.) The range of the membership function is a subset of the non-negative real numbers whose supremum is finite. A = “real numbers considerably larger than 8” Example:

(21)

i=j

n 

Id Qi cp r(Tn − Ti ) +

Qi i=1

n 

(Tj − Tj−1 )

j=m+1

A = {(x,  (x))|x ∈ X} A

m−1 

Qi − (T − Tm−1 )

i=1

Ti ≤T,i=1 n

+

DEFINITION 1 [Fuzzy set]: If X is a collection of objects denoted generically by x, then a fuzzy set A in X is a set of ordered pairs:

2

Qi − cr pT − fpcr Ic .

Qi cp (1 − r)Id (Tn − Tie )

Ti ≤T,i=1



Qi cp Id (Tn − Ti ) − C1 − C2

A = {(x,  (x))|x ∈ X} A

Ti >T,i=1

All markets’ total profit (MTP2 ) is obtained as (Fig. 5) MTP 2 (p, T )

=

n 

sdi (Tie − Ti ) −

i=1

−

n  i=1

n 

n  hm d

i

cp Qi −

2

Qi cp Ic (Tie − Ti ) +

i=1,Ti ≥T

=

i=1

n 

Qi − cr pT − fpcr Ic .

i=1

−hp [ +

A

n 

k=2

n 

(Tj − Tj−1 )







i=1,Ti ≥T

Qi ] +

k−1

m−1

n

i=1 n

i=1

i=m



(1 + (x − 8)−2 )

Id Qi cp r(Tn − Ti )

Ti ≤T,i=1

i=j

Qi cp (1 − r)Id (Tn − Tie ) +

sdi (Tie − Ti ) −

i=1 n

−

⎪ ⎩

; when

(T − M)2 T2 − fphr . 2 2

m−1

Ti ≤T,i=1 n

+

⎧ 0 ; when x ≤ 8 ⎪ ⎨ −1

   pT 2  (Tk − Tk−1 ). Qi − (T − Tm−1 ) Qi − (T − Tm ) Qi − 2

j=m+1 n

+

where

 (x) =

WTP 2 (p, T ) + MTP 2 (p, T )

= Cp

(22)

1 sdi Id . (Tie − Ti )2 − C3 2

Therefore, the total profit (ITP2 (p, T)) for the integrated system is given by ITP 2 (p, T )

Qi rcp Ic (Tie − Ti )

i=1,Ti ≤T

i=1

n 

n 

.(Tie − Ti )2 −

n 

n 

Qi cp Id (Tn − Ti ) − C1 − C2 − C3

Ti >T,i=1

cp Qi −

i=1

Qi cp Ic (Tie − Ti ) +

n  i=1

n  hm d

i

2

(23)

.(Tie − Ti )2 −

i=1

1 sdi Id . (Tie − Ti )2 2

n  i=1,Ti ≤T

Qi rcp Ic (Tie − Ti )

x>8

460

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

Fig. 5. (a) Raw-materials inventory and (b) ith market’s inventory for the case of non-deterioration.

and (b) Trapezoidal fuzzy number M . Fig. 6. (a) Triangular fuzzy number M DEFINITION 2 [Fuzzy Function]: Let X and Y be the universes f: X → P (Y ) be the set of all fuzzy sets in Y (power set), P (Y ) is and f is a fuzzy function iff  (y) =  (x, y), ∀(x,y)∈ a mapping. Then

sets in X1 , X2 , . . . . . , Xr respectively. Assume that f is a mapping from X to a universe Y, y = f(x1 , x2 , . . . , xr ). Then the extension principle allows us to define a fuzzy set B in Y by

X× Y where  (x, y) is the membership function of the fuzzy relation.

B = (y,  (y)) : y = f (x1 , x2 , ...., xr ), (x1 , ...., xr ) ∈ X, B

f (x)

R

R

f the daily Example: Let X be the set of all workers of a plant, output and y the number of processed work pieces. A fuzzy function f (x) = y. could then be DEFINITION 3 [Fuzzy extension principle]: Let X be a cartesian product of universes X = X1 , X2 , . . . . . , Xr and A 1 , A 2 , .....A r be fuzzy

 sup where  (y) =

B

x∈f −1 (y)

0

 (x)

A

if

f −1 (y) = / ∅

otherwise

Table 2 Results for different values of ˛ and M in Example 1. ˛

0.1

0.2

0.3

0.4

0.5

M

T

p

cp

MTP

WTP

ITP

0.12 0.14 0.16 0.18 0.20 0.12 0.14 0.16 0.18 0.20 0.12 0.14 0.16 0.18 0.20 0.12 0.14 0.16 0.18 0.20 0.12 0.14 0.16 0.18 0.20

0.1911467 0.2251947 0.2594562 0.2943525 0.3299329 0.1863276 0.2197315 0.2529290 0.2866632 0.3209705 0.1817281 0.2144760 0.2466392 0.2792437 0.3123141 0.1773408 0.2094279 0.2404914 0.2721046 0.3039828 0.1731571 0.2045847 0.2347849 0.2652507 0.2959863

35492.64 30049.14 26013.78 22869.43 20348.36 36423.83 30808.95 26698.26 23496.51 20930.73 37358.65 31567.41 27392.11 24134.34 21524.94 38295.53 32349.83 28093.50 24780.89 22128.81 39233.17 33127.75 28800.76 25434.37 22740.38

14.27018 14.31885 14.36799 14.41822 14.46961 14.53156 14.62897 14.72642 14.82611 14.92819 14.78445 14.93041 15.07517 15.22335 15.37513 15.02918 14.22332 15.41427 15.60980 15.81012 15.26612 15.50790 15.74386 15.98551 16.23307

29901.17 29748.84 29419.17 29082.22 28737.46 28144.50 27688.41 27014.64 26345.85 25661.06 28644.91 25646.16 24675.03 23680.97 22662.72 24800.13 23681.17 22400.16 21088.43 19744.58 23207.73 21772.05 20189.13 18568.00 16907.24

15270.94 15540.24 15929.48 16326.68 16732.40 17013.40 17605.15 18316.04 19042.57 16785.73 18698.83 17611.92 20637.70 21686.95 22760.94 20329.68 21561.51 22894.67 24259.00 22655.90 21908.20 23455.29 25087.84 26758.96 28470.08

45172 45289.08 45348.65 45408.90 45469.86 45157.90 45273.56 45330.68 45388.42 45446.79 45143.80 45258.08 45312.73 45367.92 45423.66 45129.81 45242.67 45294.82 45347.43 45400.49 45115.93 45227.34 45276.97 45326.96 45377.31

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

A = {(−1, 0.5), (0, 0.8), (1, 1), (2, 0.4)} and f(x) = x2 . Example: Let Then by applying the extension principle, we get that

is given by The centroid value of M M ∗ = M0 +

B = f ( A) = {(0, 0.8), (1, 1), (4, 0.4)}

1 (2 − 1 ) 3

M

follows:



 sup

M

mean value of M). (ii)  (x) is piece wise continuous. M Example: The following fuzzy set is fuzzy number: approximately 5 = {(3, . 2), (4, . 6), (5, 1), (6, . 7), (7, . 1)} But {(3, . 8), (4, 1), (5, 1), (6, . 7)} is not a fuzzy number because (4) and (5) = 1.

x∈ITP −1 (y)

ITP (y)=

0 if

i.e., ITP (y) =

M

According to assumption-6 in section §2.1.2, in this paper it is considered that the raw material supplier gives an opportunity to Here, the the manufacturer by offering a fuzzy credit period (M). is represented in two different forms such as triancredit period M gular fuzzy number and trapezoidal fuzzy number. So due to fuzzy the optimum value of integrated profit funccredit period (M), tion ITP(p, T) in Eq. (19) will be different for various values of M with some degree of belongingness. Therefore in such situation, the profit function will be fuzzy in nature and is denoted by ITP (p,T ) (M), where −cr .

y = ITP (p,T ) (x) and ITP −1 (y) = /  (p,T )

 (x) if

(p,T )

M

4.1. Model with fuzzy credit period

=

(25)

Using the extension principle in fuzzy set theory, the member can be obtained as ship function denoted by ITP (y) of ITP (p,T ) (M)

is a convex DEFINITION 4 [Fuzzy number]: A fuzzy number M of the real line R such that normalized fuzzy set M (i) It exists exactly one x0 ∈ R with  (x0 ) = 1 (x0 is called the

ITP (p,T ) (M)

461



M

y = ITP (p,T ) (x) and ITP −1 (y) =  (p,T )

⎧ −1 ITP (p,T ) (y) − M0 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

M0 + 2 − ITP −1 (y) (p,T ) 2

0

if

y1 ≤ y ≤ y2

if

y2 ≤ y ≤ y3

otherwise

where y1 = ITP(p,T) (M0 − 1 ), y2 = ITP(p,T) (M0 ), y3 = ITP(p,T) (M0 + 2 ).

can be determined in the Now, the centroid value of ITP (p,T ) (M) following way (Fig. 6):

fp r T fphr r T cr fpI c r (T −M ) − 1 − r (T − M)] (e − 1) − [e − r T − 1] − [e 2 2 r r r

 pT 1  −T p (e k − e−Tk−1 ). Qi eTi + 2 (e−T − e−T1 ) +    m−1

−hp [

k−1

k=2

i=1

  1 1 + (e−T − e−Tm−1 ). Qi eTi + (e−T − e−Tm ). Qi eTi   m−1

i=1

 1  −T − (e − e−Tm ). Qi eTi ] +  n

j=m n

+





+

sdi (Tie − Ti ) −

i=1 n

−



+

n 

i=j

Ti ≤T,i=1 n

n  hm d



i=1

2

Qi rcp Ic (Tie − Ti ) −

i=m

Id Qi cp r(Tn − Ti )

[e

(Tie −Ti )

n 

Qi cp Ic (Tie − Ti )



⎧ x−M + 0 1 ⎪ if M0 − 1 ≤ x ≤ M0 ⎪ ⎪ 1 ⎨ ⎪ ⎪ ⎪ ⎩

M0 + 2 − x 2

0 otherwise

−∞∞

if M0 ≤ x ≤ M0 + 2

M

(26)

ITP (y)dy −∞

1 sdi Id . (Tie − Ti )2 2

as the triangular fuzzy numLet us consider the credit period M = (M0 − 1 , M0 , M0 + 2 ), where 0 < 1 < M0 and ber such as M 0 < 2 ; 1 , 2 are determined by the decision makers. Now, the is defined as follows: membership function of M

∞

yITP (y)dy CI(p, T ) =

i=1,Ti ≥T

triangular fuzzy number (TFN) 4.2. M:

M



− (Tie − Ti ) − 1]

i=1

 (x) =

(24)

Qi cp Id (Tn − Ti ) − C1 − C2 − C3

Ti >T,i=1

i

i=1,Ti ≤T n



n

Qi cp (1 − r)Id (Tn − Tie ) +

Ti ≤T,i=1 n

n

=

M

1 (y1 + y2 + y3 ) 3

trapezoidal fuzzy number (TPFN) 4.3. M: as a trapezoidal fuzzy number Considering the credit period M = (M0 − 1 , M0 − 2 , M0 + 3 , M0 + 4 ), where 0 < 2 < 1 < M0 M and 0 < 3 < 4 ; 1 , 2 , 3 , 4 are determined by the decision is defined as follows: makers. Now, the membership function of M

 (x) =



M

⎧ M0 − 2 − x ⎪ 1− if M0 − 1 ≤ x ≤ M0 − 2 ⎪ ⎪ 1 − 2 ⎪ ⎪ ⎪ ⎨1 if M0 − 2 ≤ x ≤ M0 + 3 ⎪ x − M0 − 3 ⎪ 1− ⎪ ⎪ 4 − 3 ⎪ ⎪ ⎩ 0

otherwise

if

M0 + 3 ≤ x ≤ M0 + 4

462

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

Then the centroid value of  (x) is



M

M ∗ = M0 +

1 (3 + 4 − 2 − 1 ) 4

(27)

Using the extension principle in fuzzy set theory, the member can be obtained as ship function denoted by ITP (y) of ITP (p,T ) (M) follows:

M

 sup x∈ITP −1

ITP (y) =

(p,T )

M

0

(y)

 (x)

y = ITP (p,T ) (x) and ITP −1 (y) =  (p,T )

if

i.e., ITP (y) =

M

y = ITP (p,T ) (x) and ITP −1 (y) = /  (p,T )

if

M

⎧ y M0 − 2 − ITP −1 ⎪ (p,T ) ⎪ ⎪ 1− ⎪ ⎪ 1 − 2 ⎪ ⎪ ⎪ ⎨ 1

if

y2 ≤ y ≤ y3

⎪ (y) − M0 − 3 ITP −1 ⎪ (p,T ) ⎪ ⎪ 1− ⎪ ⎪  4 − 3 ⎪ ⎪ ⎩ 0

y1 ≤ y ≤ y2

if

if

y3 ≤ y ≤ y4

otherwise

where, y1 = ITP(p,T) (M0 − 1 ), y2 = ITP(p,T) (M0 − 2 ), y3 = ITP(p,T) (M0 + 3 ) and y4 = ITP(p,T) (M0 + 4 )

is obtained as follows: Then the centroid value of ITP (p,T ) (M) 

∞

yITP (y)dy CI(p, T ) =



−∞ ∞

M

(28)

ITP (y)dy −∞

=

M

1 (y1 + y2 + y3 + y4 ) 4

4.4. Defuzzification algorithm to get the optimum value of T To get the optimum value of the production run time (T) in the proposed integrated model with fuzzy credit period, the following steps are necessary. Step 1: At first to get the expression of fuzzy integrated profit

ITP (p,T ) (M). Step 2: Calculate the centroid value (M* ) of the fuzzy credit period M. Step 3: Determine the membership function (ITP (y)) of ITP (p,T ) (M).

M

Step 4: Calculate the centroid value CI(p, T) of ITP (p,T ) (M). Step 5: To get the optimal value of T and p, maximize the crisp integrated total profit CI(p, T) using the standard LINGO software. The above algorithm has been depicted by the flowchart shown in Fig. 7 for the case of triangular fuzzy number.

5. Numerical illustrations To illustrate the proposed models numerically, the following examples have been considered. Example 1: A manufacturing company receives the rawmaterials from a supplier to produce a required item where the raw-materials deteriorate at the rate of 0.10. Here, the manufacturer’s raw-materials holding cost rate is $4 and the supplier offers a crisp credit period to the manufacturer to settle the account. Then using 1.2 units of raw-materials, the manufacturer produces a finished item which also deteriorates at the rate of 0.15. The company sells the finished item to three different markets which have different selling seasons such as [0.09, 0.20], [0.20, 0.32] and [0.40, 0.60] in year with their annual demand rates being 15,000, 14,000 and 16,000 units respectively. The company’s purchase cost of rawmaterials is $8 per unit and selling price parameters are a = 14,

Fig. 7. Flowchart for solution method.

˛ = 0.3 [according to assumption 7]. The holding cost rates for company’s raw-materials and finished products are $4 and $5 respectively. Also here, the ordering cost and set-up cost of the manufacturer as well as all markets’ set-up cost are $2000, $3000 and $2000 respectively. The rates of selling price and holding cost for each market are $20 and $7 respectively. The manufacturer earns the interest at the rate of $0.08 on deposited revenue. At the same time, the company also pays an amount to the supplier at the rate of $0.12. Now, the company is interested to find out a table showing the optimal profits individually and jointly with the markets along with optimal production rate, optimal production run time and the manufacturer’s selling price for different values of ˛ and M. Solution: Since the objective functions of the proposed models are highly non-linear, hence these can not be optimized analytically. So, the standard LINGO software has been used to get the optimal solution of such type of models utilizing the above parameters. It is a powerful general non-linear solver which includes a

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

463

Table 3 Results of Example 2 for different values of M. ˛

M

T

p

cp

MTP

WTP

ITP

0.3

0.12 0.14 0.16 0.18 0.20

0.1275 0.1487 0.1699 0.1911 0.2129

51220 43912 38431 34169 30675

14.55 14.64 14.73 14.83 14.92

29226 28609 27989 27364 26900

19436 20062 20691 21324 21857

48662 448671 48680 48688 48757

Table 4 Results of Example 3 for different values of 1 , 2 . ˛

1

2

* (= 2 − 1 )

M*

T

p

MTP

WTP

ITP

CI

0.3

0.10 0.08 0.06 0.04 0.02

0.02 0.04 0.06 0.08 0.10

−0.08 −0.04 0 0.04 0.08

0.1333 0.1467 0.16 0.1733 0.1867

0.1800 0.2183 0.2586 0.3011 0.3454

37714 31015 26105 22348 19415

26495 25531 24312 23010 21634

18680 19744 20999 22334 23739

45175 45275 45311 45344 45373

45045 45194 45275 45353 45429

number of ways to find optimal solution for non-linear model. It is based upon a Generalized Reduced Gradient (GRG) algorithm. However to get a good feasible solution quickly, it also incorporates successive linear programming. This non-linear solver takes advantage of sparsity for improve speeds and more efficient memory usage. The obtained results have been shown in Table 2. Example 2: The problem in Example 1 for non-deteriorating item when ˛ = 0.3. Solution: The results obtained by LINGO is shown in Table 3. Example 3: The problem in Example 1 when supplier’s offered trade credit period is a triangular fuzzy number with M0 = 0.16 and the raw-materials and finished item are both deteriorating for ˛ = 0.3. Solution: The solutions are followed by the steps below. Step 1: for 1 = 0.10 and 2 = 0.20 get M* = 0.1333 from Eq. (25). Step 2: in Eq. (19), putting M = M0 − 1 = 0.06, M = M0 = 0.16 and M = M0 + 2 = 0.18, find y1 (p, T), y2 (p, T) and y3 (p, T) respectively. Step 3: find the expression CI(p, T ) = 13 [y1 (p, T ) + y2 (p, T ) + y3 (p, T )] Step 4: maximizing CI(p, T) using LINGO get T = 0.18, p = 37714 and for these values of T and p get ITP = 45175, MTP = 26495 and WTP = 18680. Step 5: repeat all the previous steps to find the results for other values of 1 and 2 . The obtained results are shown in Table 4. Example 4: The problem is same as in Example 1 for deteriorating item when the supplier’s offered credit period is a trapezoidal fuzzy number. Solution: The obtained results are shown in Table 5. Example 5: The same problem in Example 1 when supplier’s offered trade credit period is a triangular fuzzy number with M0 = 0.16 and the raw-materials and finished item are both nondeteriorating for ˛ = 0.3. Solution: The obtained results are shown in Table 6. Example 6: The problem is same as Example 1 when supplier’s offered trade credit period is a trapezoidal fuzzy number with M0 = 0.16 and the raw-materials and finished item are both nondeteriorating for ˛ = 0.3.

Solution: The obtained results are shown in Table 7. 5.1. Sensitivity analysis Effects on the fuzzy model with deteriorating item for change of the values of the parameters di ,  and  r have been shown in Table 8. Now from Table 8, it is observed that p, MTP, WTP and CI increase with the increase of di . When  increases, the values of MTP and CI decrease but WTP at first decreases and then increases. In this case the pattern of p can not be predicted. Again when  r increases, WTP and CI decrease but MTP and p increase. 5.1.1. Comparison between the optimum production run times due to TFN and TPFN by Fisher’s t-test In the fuzzy integrated model, two optimum production run times have been obtained using TFN and TPFN. Now question is that does there exist any significance difference between these two values ? If exists, then how much? To get this answer, it can be tested that the null hypothesis H0 : T TFN (mean of values of T for TFN) = T TPFN (mean of values of T for TPFN) against the alternative hypothesis H1 : T TFN = / T TPFN on the basis of the results presented in Tables 4 and 5. This hypothesis can be tested using t-distribution. The test statistic is t=

T TFN − T TPFN



(1/n1 ) + (1/n2 )

s.

which follows t-distribution with (n1 + n2 − 2) degrees of freedom, where s2 =

n1 s2 TFN + n2 s2 TPFN n1 + n2 − 2

2 Here n1 = 5, n2 = 5, T TFN = 0.26068, T TPFN = 0.2482, sTFN = 2 0.00428, sTPFN = 0.00278. Therefore, degrees of freedom = 8 and value of t = 0.7204. Since the calculated value of t < the tabulated value of t0.05 , we accept the null hypothesis H0 with 95% confidence limit and

Table 5 Results of Example 4 for different values of 1 , 2 , 3 , 4 . ˛

1

2

3

4

0.3

0.10 0.08 0.04 0.04 0.06

0.04 0.04 0.02 0.02 0.04

0.02 0.02 0.02 0.04 0.08

0.04 0.06 0.04 0.06 0.10

* (= 3 + 4 − 2 − 1 ) −0.08 −0.04 0 0.04 0.08

M*

T

p

MTP

WTP

ITP

CI

0.14 0.15 0.16 0.17 0.18

0.1901 0.2100 0.2421 0.2770 0.3218

35692 32260 27920 24337 20873

26195 25780 24814 23751 22367

18997 19502 20499 21588 22992

45192 45282 45313 45339 45359

45077 45196 45288 45336 45356

464

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

Table 6 Results of Example 5 for different values of 1 , 2 ˛

1

2

* (= 2 − 1 )

M*

T

p

MTP

WTP

ITP

CI

0.3

0.10 0.08 0.06 0.04 0.02

0.02 0.04 0.06 0.08 0.10

−0.08 −0.04 0 0.04 0.08

0.1333 0.1467 0.16 0.1733 0.1867

0.1533 0.1867 0.2200 0.2533 0.2867

42587 34982 29682 25776 22779

28474 27495 26688 25692 24685

20191 21161 22014 22991 23976

48665 48656 48702 48683 48661

48554 48581 45661 48675 45686

Table 7 Results of Example 6 for different values of 1 , 2 , 3 , 4 . ˛

1

2

3

4

* (= 3 + 4 − 2 − 1 )

M*

T

p

MTP

WTP

ITP

CI

0.3

0.10 0.08 0.04 0.04 0.06

0.04 0.04 0.02 0.02 0.04

0.02 0.02 0.02 0.04 0.08

0.04 0.06 0.04 0.06 0.10

−0.08 −0.04 0 0.04 0.08

0.14 0.15 0.16 0.17 0.18

0.18 0.21 0.20 0.23 0.28

36276 31094 32649 28390 23321

27692 26985 27281 26390 24887

20961 21709 21443 22319 23768

48653 48694 48724 48709 48655

48546 48612 48704 48697 48635

conclude that there is no significant difference between the mean production run time (T) for TFN and TPFN. 5.1.2. Managerial insights Based on the results obtained in Tables 2–7 and from Fig. 8, the following managerial insights are obtained. (1) From Table 2, it is seen that with the increase of M, the optimal values of T, cp , WTP and ITP increase but p and MTP decrease when ˛ is fixed. Again considering M fixed, the optimal values of T, MTP and ITP decrease but cp , p and WTP increase with the increase of ˛. It implies that for the crisp model with deteriorating item, if the supplier’s offered credit period be longer then production run time, markets’ unit purchase cost, manufacturer’s profit and integrated profit will be more but markets’ total profit’ll be lesser. Again for fixed credit period, when the purchase cost parameter ˛ be more, then production run time, markets’ profit and integrated profit is less but markets’ unit purchase cost, production rate and manufacturer’s profit is more. (2) From Table 3, it is seen that for the crisp model with nondeteriorating item, the optimal values of all variables and objective functions have same characteristics as the case under deteriorating item.

(3) From Fig. 8, it is observed that for a fixed value of credit period (M), the integrated total profit (ITP) initially increases with the increase of production run time (T). Then after getting its maximum value, it decreases. So, the integrated profit is a concave function. (4) Tables 4–7 reveal that as imprecise nature (i.e., difference * ) increases, the centroid value (i.e, M* ) of fuzzy credit period increases and the rate of change of the centroid value (i.e, CI) of fuzzy integrated profit with respect to total profit (ITP) increases. It is also concluded that, the trade credit period (M* ) and the production rate (p) are inversely proportional to each other whereas the production period T is directly proportional to M* . (5) From Table 8, it is concluded that when the markets’ demands increase then production run time, manufacturer’s profit and integrated profit increase. More deterioration of finished products implies less markets’ profit but manufacturer’s profit fluctuates. Again, more raw-materials’ deterioration implies more markets’ profit and production run time but less manufacturer’s profit. (6) From the comparison test between the optimum production run times due to TFN and TPFN in Section 5.1.1, it is inferred that

Table 8 Effect of changes in different parameters. Parameter

% of change

T

p

MTP

WTP

ITP

CI

di

−30 −20 −10 0 +10 +20 +30

0.2586 0.2586 0.2586 0.2586 0.2586 0.2586 0.2586

18273 21136 23494 26105 28715 31326 33936

16419 19350 21681 24312 26944 29575 32206

13199 16133 18400 20999 23599 26199 28799

29618 35483 40081 45311 50543 55774 61005

29593 35454 40048 45275 50503 55731 60958



−30 −20 −10 0 +10 +20 +30

0.2200 0.2200 0.2232 0.2586 0.3258 0.3606 0.3610

30438 30549 30215 26105 20671 18666 18694

25845 25723 24506 24312 22119 20901 20762

20279 20130 20074 20999 22948 23962 23903

46123 45853 45580 45311 45067 44863 44665

46081 45810 45538 45275 45038 44837 44638

r

−30 −20 −10 0 +10 +20 +30

0.3599 0.3599 0.2962 0.2586 0.2333 0.2200 0.2200

18610 18610 22724 26105 28984 30770 30770

21179 21179 23160 24312 25078 25480 25480

24456 24331 22242 20999 20153 19677 19602

45635 45510 45402 45311 45231 45157 45082

45609 45484 45371 45275 45191 45114 45039

B.C. Das et al. / Applied Soft Computing 28 (2015) 453–465

465

Fig. 8. Graph of T Vs ITP for M = 0.16.

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