A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints

Applied Soft Computing 34 (2015) 838–850

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

A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints Ravi Shankar Kumar, A. Goswami ∗ Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

a r t i c l e

i n f o

Article history: Received 21 September 2012 Received in revised form 4 April 2015 Accepted 14 May 2015 Available online 16 June 2015 Keywords: Inventory Random demand Process shifting Fuzzy mathematical programming Possibility/necessity constraints Particle swarm optimization

a b s t r a c t This paper investigates a single-period production-inventory model in linguistic ‘imprecise’ and statistical ‘uncertain’ environment, where fuzziness and randomness appear simultaneously. In order to capture the stochastic variability of demand uncertainty, we consider it as a random variable. We also model the hazards of imperfect quality production due to the production process shifting from “in-control” state to “out-of-control” state. The process shifting time and fraction of imperfect quality items are characterized as fuzzy random variables (FRVs). The model restricts the budget and allowable shortages. To mitigate the effect of uncertain demand, fluctuating cost parameters, etc., the imposed constraints are considered in fuzzy sense. To model the above scenarios, we formulate a fuzzy mathematical programming problem. By employing the fuzzy expectation, signed distance, and possibility/necessity measure, the fuzzy model is transformed into an equivalent deterministic non-linear programming problem. A fuzzy simulation based particle swarm optimization (PSO) algorithm is also employed to the model. The effectiveness of the model and solution methodologies are demonstrated with the help of numerical illustrations. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In the present day economy, most of the people are conscious about the expenditure on their daily requirements. Every body tries to apply science in their life to get a financial benefit. Though the Inventory Management had come to meet up the special needs of the Second World War, but now a days, its necessity is felt in all the branches of science and technology involved in solving the real-life problems, especially related to economics. Using Inventory Management, one can run a business in a more profitable manner. With the traditional concepts characterizing different scenarios, many economic production quantity (EPQ) models have been developed in deterministic framework. But, today’s business scenarios are floating over time that perturb the key parameters of inventory system. The nature of perturbation may be random, fuzzy or both. Fuzzy random variable (FRV) is a hybrid type of uncertainty that capture the stochastic variability as well as impreciseness (called fuzziness) of the parameters. This paper intends to model a production-inventory in fuzzy random framework.

∗ Corresponding author. Tel.: +91 3222 283650; fax: +91 3222 255303. E-mail addresses: [email protected] (R.S. Kumar), [email protected] (A. Goswami). http://dx.doi.org/10.1016/j.asoc.2015.05.024 1568-4946/© 2015 Elsevier B.V. All rights reserved.

In most of the real-life situations, the production system may undergo imperfect production process due to machine failure. A production system usually deteriorate continuously due to over usage or age factors such as corrosion, fatigue, etc. When a production system produces 100% good-quality items, then we say that process is in “in-control” state, whereas, when the system produces some imperfect-quality items, then the process is in “out-of-control” state. Generally, at the beginning of the period, the process is “in-control” state. After a period of time, owing to the continuous deterioration, the process may shift from this “incontrol” state to an “out-of-control” state. The non-conforming produced items must be rejected, repaired or reworked. In deterministic environment, remarkable study was made by [1–5] in this direction. In the initial phase, Rosenblatt and Lee [1] studied the effects of process deterioration on the traditional EPQ model, wherein the random deterioration from “in-control” state to “outof-control” state is exponentially distributed. Hyun Kim and Hong [5] extended the Rosenblatt and Lee’s [1] model by considering elapsed time until the process shift as arbitrary random variable. Chung and Hou [2] further extended Hyun Kim and Hong’s [5] model by allowing shortages that are backlogged. Liao et al. [6] presented an integrated EPQ model that incorporates the maintenance learning effect, the impact of restoration action such as imperfect repair, rework and preventive maintenance on the damage of a deteriorating production system. Sana [7] presents an EPQ model

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

wherein production rate is taken as a decision variable, and defective item is proportional to production rate and machine shifting time. The statistical ‘uncertainty’ and linguistic ‘impreciseness’ are the natural phenomena, which always occur in each and every real-life business transaction. Dynamic change in demand, money value, man power, etc. are the components that engender the perturbation in the key parameters on inventory system. With the development of fuzzy set theory [8–10], fuzzy random variable (FRV) [11,12] and fuzzy mathematical programming [13–17], inventory modeling problem in fuzzy framework received attention of researchers. Hu et al. [18] pointed out, the elapsed time until the production process shifts may be a FRV. In general, estimation of machine shifting time depends upon previous data records and experts’ experiments, which may engender fuzziness as well as randomness in final estimation of shifting time. However, Hu et al. [18] enhanced the Chung and Hou [2] model by considering elapsed time as a FRV. Wang and Tang [19] assumed it as a fuzzy variable, while Zhang et al. [20] assumed it as a random fuzzy variable. Authors’ like [21–24] extended the classical continuous review and periodic review inventory models in fuzzy random environment. Dey and Chakraborty [21] and Dutta et al. [22] considered lead-time demand as fuzzy variables and annual demand as discrete FRVs. Chang et al. [23] and Lin [24] considered lead-time demand as FRVs, while annual demand as fuzzy numbers. Recently, Jana et al. [25] developed a fuzzy inventory model for deteriorating items, wherein the scheduling period is taken as a random variable while budget and space constraints are taken in fuzzy and random fuzzy sense. Guchhait et al. [26] addressed a two-warehouse inventory model for variable demand wherein cost coefficient are considered as fuzzy numbers, and employed a hybrid meta-heuristic based on particle swarm optimization (PSO) and genetic algorithm (GA) to find the solution. Ye and Li [27] developed a single-period production-inventory model for singlemanufacturer and single-buyer by considering fuzzy demand rate, and employed GA to obtain the optimal solution. In recent trend of inventory modeling, fuzzy mathematical programming via soft constraint are being widely used. Maity and Maiti [28] developed a multi-item production-inventory model under fuzzy restricted capital and storage space. Chou et al. [29] incorporated key parameters as a trapezoidal fuzzy numbers in inventory management system for the empty container allocation problem, and optimized the model using the fuzzy mathematical programming technique. Panda et al. [30] developed a multi-item production-inventory model, wherein capital and shortage cost constraints were taken in fuzzy sense. Inuiguchi et al. [15] applied the possibilistic programming based on possibility and necessity measure to solve the production planning problem. Wang and Shu [31] developed a supply chain inventory model by incorporating the possibility theory. Maiti and Maiti [32] developed a multiitem inventory model with two warehouse storing facility under possibility constraints. Maity [33] extended Maity and Maiti [28] multi-item production-inventory model by incorporating the credibility measure to represent the imprecise constraints. Vasant [34] has formulated a fuzzy mathematical programming to maximize the profit of production planning problem, wherein fuzzy parameters are characterized with S-curve membership function. Jimenez et al. [35] and Diaz-Madronero et al. [36] developed multi-objective fuzzy mathematical models for production planning and supply chain, respectively. Recently, Vasant [37] provided hybrid methods to solve the fuzzy non-linear programming problem such as line search, simulated annealing and pattern search. In real-life business transaction, it often happens that the management takes the decisions under some restrictions. Investment in the form of stock of items turn out to be locking of money that can be used in other places (Dey and Chakraborty [21]). Hence, the

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management may restrict the maximum investment. Furthermore, shortages are not good, especially for long term businesses. Usually, some impatient customers are not willing to wait for the product, and they may leave the system. This leads to loss of reliability of the organization. Thus, the management may allow restricted shortage in the scheduling period. Moreover, it is rare to seem that budget and shortage constraints are precise (Maiti and Maiti [32]). Let us consider a situation in which at the beginning of the business, the organization start with some capital, but during the business period due to the recission or unexpected increment in demand, the decision maker may later decreases or increases the capital. In such a situation, a flexibility in the capital constraint is desirable. Accordingly, maximum allowable shortage constraint must be flexible. In this paper we formulate a fuzzy mathematical programming problem for a production-inventory under fuzzy budget and shortage constraints. The purpose of this study is twofold: (1) To model a fuzzy stochastic production-inventory for single-period by integrating the facts of process shifting and rework of imperfect quality items. In this process, we consider that demand during the scheduling period is a real valued random variable. Process shifting is modeled by considering elapsed time until machine shifts from “in-control” state to “out-of-control” state, and fraction of imperfect quality items are FRVs. The cost coefficients are fuzzy numbers. Furthermore, we consider the maximum budget and maximum allowable shortages are trapezoidal fuzzy numbers. (2) To develop a rigorous methodology to convert the possibility and necessity constraints equivalent to deterministic (hard) constraints. In this process, the imprecise possibility and necessity constraints are represented with pre-defined confidence, and we propose and prove two lemmas in the next section to proceed the same. As it is evident from the literature review and the best of our knowledge, no such inventory model have developed till now in such scenarios. The rest of the paper is organized as follows: In Section 2, we discuss basic definitions and properties of FRV and fuzzy mathematical programming, and propose two lammas that derive the formulae to find the equivalent deterministic constraint from the possibility and necessity constraints. Section 3 provides the notations and assumptions which are used throughout the article. In Section 4, the model is mathematically derived in random framework. In Section 5, the model is extended in fuzzy random environment, and is converted to deterministic environment by using fuzzy expectation and signed distance method. Then, next two sections provide the solution procedure and a numerical example to illustrate the developed model. In Section 8, we have discussed the use of the model and gave some suggestion to the decision makers or management. Finally, some concluding remarks with future scope of the research have been made in Section 9. 2. Preliminaries Before presenting the proposed production-inventory model in fuzzy random framework, we need to introduce with some basic definitions and properties of FRV and fuzzy mathematical programming. 2.1. Fuzzy random variable Definition 2.1. [Kwakernaak [11]] Let (˝, B, P) be the probability space, wherein ˝ is sample space, B is the -algebra of the subsets of ˝ and P is the probability measure. Let us assume that X be a random variable on (˝, B, P) with probability density function f(x). Again we assume that F be the family of fuzzy numbers in real line ˜ induced by R. Then a fuzzy random variable (FRV), denoted by X, the real valued random variable X, is a mapping from ˝ to F.

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For any ˛ ∈ [0, 1] and all ω ∈ ˝, the real valued mappings X˛− : + + ˝ → R and X˛+ : ˝ → R satisfy X˛− (ω) = X(ω)− ˛ and X˛ (ω) = X(ω)˛ , respectively, and are real-valued random variables on (˝, B, P). The + ˛-cut of a FRV is usually denoted as X˛ (ω) = [X(ω)− ˛ , X(ω)˛ ], and is called a random interval on (˝, B, P). Remark 2.1. Let X˜ be a FRV induced by random variable X, and X˛ = [X˛− , X˛+ ], where X˛− and X˛+ are real valued random variables. Further, let us denote (X) as the sigma algebra of subsets of ˝ generated by the random variables X˛− and X˛+ , ˛ ∈ [0, 1]. Then, in ˜ brief, we can say that (X) is the sigma algebra generated by X. Definition 2.2. [Kwakernaak [11]] A FRV X˜ is called nonnegative (or positive) if X˜ (y) = 0 for y < 0 (or y  0), where X˜ is the ˜ ω ∈ ˝. membership function of the fuzzy number X(ω),

Definition 2.6. A fuzzy number a˜ = (a1 , a2 , a3 , a4 ) on real line R is called trapezoidal fuzzy number if its membership function a˜ is defined as:

a˜ (x) =

+ − Property 2.2 (Kwakernaak [11]). [EX]− ˛ = E[X˛ ] and [EX]˛ = + E[X˛ ].

Definition 2.3 (Yao and Wu [38]). Let a˜ be a fuzzy number with + ˜ is d(˜a, 0) = ˛-cut a˛ = [a− ˛ , a˛ ]. Then the signed distance of a 1 2

1 0

+ (a− ˛ + a˛ )d˛.

According to Definition 2.3, the signed distance of fuzzy expec˜ is tation E(X)

˜ 0) = d(E(X),

1 2

1





E[X˛− ] + E[X˛+ ] d˛,

(2.1)

0



otherwise.

Definition 2.7 (Liu and Iwamura [13]). The mathematical programming (linear or non-linear) with fuzzy parameter is defined as: max (x, a˜ ) ˜) i (x, a

(2.5)

 k˜ i , i = 1, 2, . . ., m,

where x = (x1 , x2 , . . ., xn ) is a decision vector, a˜ = (˜a1 , a˜ 2 , . . ., a˜ n ) is a vector of fuzzy parameters, (x, a˜ ) is the objective function, i (x, a˜ ) are constraint functions and k˜ i are fuzzy numbers, i = 1, 2, . . ., m . The fuzzy mathematical programming (2.5) is not well defined, because the inherent vagueness in the parameters obscure the meaning of max as well as constraints. So, Liu and Iwamura [13] suggested to convert the fuzzy objective as well constraints to the equivalent crisp objective and constraints, respectively. In this paper we employ signed distance (Yao and Wu [38]) and possibility/necessity measure to attain the same. However, the equivalent deterministic mathematical programming of (2.5) become as: max (x)



where E[X˛− ] =

(2.4)

a4 − x ⎪ ⎪ , a3  x  a4 ; ⎪ ⎪ a4 − a3 ⎪ ⎪ ⎩ 0,

s. t. Property 2.1 (Kwakernaak [11]). The fuzzy expectation of FRV X˜ is ˜ in F. For each ˛ ∈ [0, 1], the ˛-cut of E(X) ˜ a unique fuzzy number E(X) + ˜ ˛ = [[EX]− , [EX] ]. is [E(X)] ˛ ˛

⎧ x−a 1 ⎪ , a1  x  a2 ; ⎪ ⎪ a2 − a1 ⎪ ⎪ ⎪ ⎨ 1, a2  x  a3 ;

X − f (x)dx and E[X˛+ ] = ˝ ˛



X + f (x)dx. ˝ ˛

Definition 2.4. [Kwakernaak [11], Hwang and Yao [39]] Two FRVs X˜ and Y˜ are said to be independent if the sigma algebras (X) and (Y) of subsets of R successively generated by X and Y are independent. Theorem 2.1 (Kwakernaak [11]). If two non-negative FRVs X˜ and Y˜ ˜ are independent, then E(X˜ Y˜ ) = E(X)E( Y˜ ).

2.2. Fuzzy mathematical programming

s. t.Pos{ i (x, a˜ )  k˜ i }  i or/and Nes{ i = 1, 2, . . ., m,

˜) i (x, a

 k˜ i }  i ,

(2.6)

where i is predefined confidence level usually interpreted as constraint reliability. Pos{ . }  i interprets x is feasible if and only if the possibility measure of the set { i  k˜ i } is possibly at least i , and Nes{ . }  i interprets x is feasible if and only if the necessity measure of the set { i  k˜ i } is necessarily greater than or equal to i . In next two lemmas, we propose method to convert the fuzzy constraints equivalent to hard constraints. Lemma 2.1. Let a˜ = (a1 , a2 , a3 , a4 ) and b˜ = (b1 , b2 , b3 , b4 ) be two trapezoidal fuzzy numbers. Then for any given confidence level  ˜   if and only if (0 <   1), Pos(˜a  b) (1 − )a4 + a3  (1 − )b1 + b2 .

(2.7)

Proof. According to definition of possibility and Fig. 1, we have Definition 2.5 (Possibility/Necessity [10,13,14]). Let a˜ and b˜ be two fuzzy numbers with membership functions a˜ and b˜ , respectively, then ˜ = sup{min(a˜ (x),  ˜ (y)) : Pos(˜a ∗ b) b

x, y ∈ R, x ∗ y},

(2.2)

⎧ 1, ⎪ ⎪ ⎨

a4 − b1 ˜ = Pos(˜a  b) , 1 = b − b1 + a4 − a3 ⎪ 2 ⎪

⎩

0,

if a3  b2 , a4  b1 ; if a3  b2 , a4  b1 ; if a4  b1 . (2.8)

where the abbreviation Pos represents possibility of occurrence of ˜ and * is any one of the relations <, > , = , , . a˜ ∗ b, The dual relationship of possibility is necessity, and is defined as: ˜ = 1 − Pos(˜a ∗ b), ˜ Nes(˜a ∗ b)

(2.3)

˜  . Then from Eq. (2.8), we First, let us assume that Pos(˜a  b) have either a3  b2 , a4  b1 or a3  b2 , a4  b1 . If a3  b2 , a4  b1 , then a3  b2 ⇒ a3  b2 and a4  b1 ⇒(1 − )a4  (1 − )b1 , and both of these imply (1 − )a4 + a3  (1 − )b1 + b2 . If a3  b2 , a4 −b1 ˜ ⇒  a4  b1 . Then from Eq. (2.8), Pos(˜a  b) b −b +a −a 2

˜ where a˜ ∗ b˜ is complement of a˜ ∗ b. ˜ means the relation a˜ ∗ b˜ is possibly true, Remark 2.2. Pos(˜a ∗ b) ˜ means the relation a˜ ∗ b˜ is necessarily true. and Nes(˜a ∗ b)

1

4

3

⇒(1 − )a4 + a3  (1 − )b1 + b2 . Hence, one part of the lemma is proved. let us assume that inequality (2.7) Conversely, holds. If we consider a4 < b1 , then this ⇒a3  a4 < b1  b2 ⇒(1 − )a4 + a3 < (1 − )b1 + b2 . This is against of our assumption.

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

841

˜ and Pos(˜a  b): ˜ (a) when a3  b2 and (b) when a3 < b2 . Fig. 1. Membership functions of fuzzy numbers a˜ and b,

So, it must be a4  b1 . a4  b1 and inequality (2.7) imply either ˜ = 1  . a3  b2 or a3  b2 . If a3  b2 and a4  b1 , then Pos(˜a  b) the assumption (1 − )a4 + a3  (1 − )b1 + b2 Otherwise, a −b ˜  . The lemma is proved. 䊐 ⇒ b −b4 +a1 −a  . Hence, Pos(˜a  b) 2

1

4

3

Lemma 2.2. Let a˜ = (a1 , a2 , a3 , a4 ) and b˜ = (b1 , b2 , b3 , b4 ) be two trapezoidal fuzzy numbers. Then for any given confidence level  ˜   if and only if (0 <   1), Nes(˜a  b) a1 + (1 − )a2 > b4 + (1 − )b3 .

(2.9)

Proof. According to the definition of necessity and Fig. 2, we have ˜ Nes(˜a  b)

=

˜ 1 − Pos(˜a < b)

= 1−

⎧ 1, ⎪ ⎪ ⎪ ⎨

a2 < b3 , a1 < b4 ; b −a

4 1 , a2 > b3 , a1 < b4 ; 2 = ⎪ b − b + a ⎪ 4 3 2 − a1 ⎪ ⎩

0,

=

⎧ 0, ⎪ ⎪ ⎨

a1  b4 .

a2 < b3 , a1 < b4 ; a2 − b3

b −b +a −a ⎪ ⎪ ⎩ 4 3 2 1 1,

(2.10)

, a2 > b3 , a1 < b4 ; a1  b4 .

˜  . Then we have We first consider that Nes(˜a  b) either a1  b4 or a2 > b3 , a1 < b4 . If a1  b4 , then it must be a2 > b3 . These imply a1  b4 and (1 − )a2 > (1 − )b3 . a1 + (1 − )a2 > b4 + (1 − )b3 . If a2 > b2 , a1 < b4 , Hence, a2 −b3 ˜ ⇒ from Eq. (2.10), Nes(˜a  b)  then b −b +a −a 4

3

2

1

⇒a1 + (1 − )a2  b4 + (1 − )b3 . Hence first part of the lemma is proved. Conversely, let us assume that the inequality (2.9) holds. If we consider a2  b3 , then a1  a2  b3  b4 this implies a1 + (1 − )a2  b4 + (1 − )b3 . This is against of our assumption. So, it must be a2 > b3 . If a2 > b3 , then we have from inequality (2.9), ˜ = 1  . Otherwise, the a1  b4 or a1 < b4 . If a1  b4 , then Nes(˜a  b) a −b assumption a1 + (1 − )a2  b4 + (1 − )b3 leads a −a2 +b3 −b  . 2 1 4 3 ˜  . Hence, the lemma is proved. 䊐 Hence, Nes(˜a > b)

3. Notations and assumptions The following notations are used throughout the paper. 3.1. Notations A B

set-up cost maximum available budget (total production cost)

P production rate S maximum allowable shortages cost scheduling period which is fixed T X random demand during the scheduling period T with uni1 form distribution function f (x) = b−a ,a  x  b Y random elapsed time until the production process shifts “in-control” state to “out-of-control” state with probability density function g(y) holding cost per unit per unit time c1 c2 shortage cost (lost sales) per unit production cost cp cr rework cost of per unit defective item per unit selling price cs cv salvage value of per unit unsold item just after the end of scheduling period T ˇ percentage of defective items produced once the system is in the “out-of-control”, which is random variable with probability density function h(ˇ).  production run-time in a production cycle, which a decision variable expected value of a random variable. E[·] Fuzzy parameters: c˜1 = (c11 , c12 , c13 , c14 ), fuzzy holding cost. c˜2 = (c21 , c22 , c23 , c24 ), fuzzy backordering cost. c˜p = (cp1 , cp2 , cp3 , cp4 ), fuzzy production cost. Y˜ (ω) = (y − 1 , y − 2 , y + 3 , y + 4 ) for ω ∈ ˝, the fuzzy random elapsed time until the production process shifts, where 0  2  1  E[y], 0  3  4 . ˜ = (ˇ − 5 , ˇ − 6 , ˇ + 7 , ˇ + 8 ), fuzzy ranˇ(ω) dom fraction of defective items, where 0  6  5  E[ˇ], 0  7  8  1 − E[ˇ] . B˜ = (B1 , B2 , B3 , B4 ), maximum available budget in fuzzy sense. S˜ = (S1 , S2 , S3 , S4 ), maximum allowable shortages in fuzzy sense.

3.2. Assumptions The following assumptions are made while developing the model. 1. The cycle consists of only one period. 2. The scheduling period starts with production and demand which occur simultaneously. Production starts at the beginning of the scheduling period and during the production run-time, the machine may shift from an “in-control” state to an “out-ofcontrol” state. 3. Once the machine shifts from “in-control” to “out-of-control” state, some defective items will be produced. The number of defective items produced are not depend on the process shifting

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R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

˜ or equivalently Nes(˜a  b): ˜ (a) when a2 < b3 and (b) when a2 > b3 . Fig. 2. Pos(˜a < b)

4. 5. 6.

7. 8.

time Y. Thus elapsed time Y and percentage of defective items ˇ are independent random variables. The production rate is constant. Total demand over the scheduling period is stochastic and uniform. Shortage should not occur during the production run-time . Hence we assume that P > Tb , where b is the maximum demand which may occur during the scheduling period T. All defective items produced during [0, ] are detected just after , and rework cost for defective items will be incurred. The maximum investment and maximum allowable shortages are limited.

incurs for this case. The inventory level for this case is governed by the following differential equations:

⎧ x ⎪ P − , if 0 < t  ; ⎪ ⎪ T ⎪ ⎨

dI(t) x = − , dt T ⎪ ⎪ ⎪ ⎪ ⎩−x, T

In this section we mathematically formulate the model under the assumptions discussed in previous section. The presence of randomness in demand parameter engender two situations, either shortages do not occur or shortages occur (Sarkar and Moon [40]). Case I: Shortages do not occur When produced quantity P is sufficient to fulfill the customers demand during the period T, then shortages do not occur in the system. In such a situation, system earns salvage value for the remaining items after meeting up the demand. The inventory level for this case is governed by the following differential equations:

dI(t) = dt ⎩

T

x − , T

(4.1) if   t  T,

T

x

P − t, ⎪ T ⎪ ⎪ ⎪ ⎩ x

− (t − T1 ), T

I(T1 ) = 0 ⇒

(4.4)

if   t  T1 ;

T1 =

if T1  t  T.

PT . x

Since, in this case shortages are occurred, so it must be I(T) < 0. This implies x > P. The total revenue is sum of the revenue earned by the sale of products within period T and total salvage earned by the sale of product after T. Hence, the expected total revenue is





P

xf (x)dx +

cs = b−a



b

P 2  2 + a2 Pb − 2



P

+ cv

Pf (x)dx P



(P − x)f (x)dx a

(4.5)

(P − a)2 + cv . 2(b − a)

The production cost is

⎧

 x ⎪ ⎨ P − T t, if 0 < t  ; ⎪ ⎩ P − x t,

if 0 < t  ;

a

PC = cp P.

with initial condition I(0) = 0. Solutions of (4.1) are

I(t) =

if T1  t  T,

 ⎧

x P− t, ⎪ ⎪ T ⎪ ⎪ ⎨

ER = cs

⎧ x ⎨ P − , if 0 < t  ;

(4.3)

with conditions I(0) = 0, I(T1 ) = 0. Solutions of (4.3) are

I(t) = 4. Mathematical formulation of the model

if   t  T1 ;

(4.6)

The expected holding cost is



P

EHC = c1 a

(4.2)

if   t  T.

 

+ c1

0

 b  

P

Since shortages do not occur, thus we must have I(T)  0 ⇒ P  x. Case II: Shortages occur When produced items P is not sufficient to fulfill the demand during the period T, then system undergoes to stock out situation. The shortage quantity is not backlogged. Hence, lost sales cost

=−

P−

c1 b−a

P−

0



P22T + log 2

x T

PT −

b P





T

tdt +



P −



 x T P 2 2

 tdt +



T1



P −



(P − a) −

 

x t dt f (x)dx T

 

x t dt f (x)dx T

T 2 2 (P  − a2 ) 4



P 2 (b − P) . − b

(4.7)

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

The expected shortage cost is





b

ESC = c2

b

x T

[−I(T )]f (x)dx = c2 P

P



T−

PT x

843

˜ and E[( − Y˜ )+ ] are the fuzzy expectations of FRVs ˇ ˜ where E[ˇ] + ˜ and ( − Y ) , respectively.

 f (x)dx (4.8)

5.2. Equivalent deterministic cost function

2

c2 (b − P) . = 2(b − a) The expected rework cost is ERC = cr PE[ˇ( − Y )+ ],

(4.9)

In this section, we employ signed distance method (Yao and Wu [38]) to find the equivalent deterministic cost function from the fuzzy expected cost function (5.1). For this, we first have to find the   ˛-cut of EP(). The left and the right end points of ˛-cut of the EP() are

− Y)+

where ( = max { − y, 0}. The expected total profit EP() is

EP − ˛ () =

EP() = ER − PC − EHC − ESC − ERC

=

cs b−a



P22

Pb −

− cp P −



 PT −

b P

P 2



(P − a) 2(b − a)

(P − a) −

2

P 2 (b − P) − b



 Pb −

P 2  2 + a2 2

c2 (b − P) − 2(b − a)

× [c14 − (c14 − c13 )˛] −

c2 (b − P)  S. 2(b − a)



P 2 PT − 2

P22T + log 2

b P





+ cv

+

(5.2)

EP + ˛ () =

cs b−a

 Pb −

P 2  2 + a2 2

 + cv

(P − a)2 2(b − a)

1 − P[cp1 + (cp2 − cp1 )˛] − b−a T P22T − (P 2  2 − a2 ) + log 4 2





P 2 PT − 2

b P

−

(P − a)

P 2 (b − P) b



× [c11 + (c12 − c11 )˛] −

(b − P) [c21 + (c22 − c21 )˛] 2(b − a)

+ −

− ]E[[( − Y˜ ) ]˛ ]. − cr PE[ˇ˛

(5.3)

˜ 5.2.1. Determination of the ˛-cut of E[ˇ] The fuzzy expectation of a FRV is a fuzzy number, and its ˛-cut is a closed interval. According to Property 2.2, the ˛-cut of fuzzy expectation of a FRV is equal to the expectation of ˛-cut of FRV. We ˜ and E[( − Y˜ )+ ]. The here use that lemma to find the ˛-cut of E[ˇ] − + ˜ is ˇ˛ = [ˇ , ˇ ] = [ˇ − 5 + ( 5 − 6 )˛, ˇ + ˛-cut of the FRV ˇ ˛ ˛ 8 − ( 8 − 7 )˛], 0  ˛  1. So, the expectation of the random − + , ˇ˛ ] is interval [ˇ˛ − + [E[ˇ˛ ], E[ˇ˛ ]] =



(ˇ − 5 + ( 5 − 6 )˛)h(ˇ)dˇ, ˝

(ˇ + 8 ˝



−( 8 − 7 )˛)h(ˇ)dˇ .

(P − a)2 − c˜p P 2(b − a)

(5.4) +

5.2.2. Determination of the ˛-cut E[( − Y˜ ) ] For the fixed production run-time , the fuzzy random time period for producing the imperfect quality item is,  − Y˜ (ω) = ( − y − 4 ,  − y − 3 ,  − y + 2 ,  − y + 1 ), where y > 0, + 0  2  1  E[y], 0  3  4 . Let us denote that Z˜ = ( − Y˜ ) . − + + + This is a positive FRV. E[(( − Y˜ ) )˛ ] and E[(( − Y˜ ) )˛ ] repre-

T (P − a) − (P 2  2 − a2 ) 4

P 2 − (b − P) b

˜ − cr PE[ˇ]E[( − Y˜ ) ],

(b − P) [c24 − (c24 2(b − a)



 The fuzzy expected total profit EP(), from Eq. (4.10) can be written as:

c˜1 − b−a



and

5.1. Fuzzy expected cost function

P 2  2 + a2 2

P

P 2 (b − P) − b

˜ The above equations include the ˛-cut of fuzzy expectations of E[ˇ] + and E[( − Y˜ ) ], which are evaluated in next two subsections.

As we discussed in introduction section, we now assume that the elapsed time until the production process shifts from “in-control” state to “out-of-control” state and fraction of imperfect quality ˜ respectively. Since, the fraction of items are FRVs, and are Y˜ and ˇ, imperfect quality items does not depend on machine shifting time, ˜ and Y˜ are independent FRVs. Furthermore, unit production so ˇ cost, holding cost and backordering cost are fuzzy numbers.

Pb −

b

(P − a)

2

(4.11)

5. Mathematical model in fuzzy random environment

cs b−a



P 2 PT − 2

+ +

Hence our aim is to maximize the expected total profit given is Eq. (4.10) subject to the constraints (4.11).

 EP() =



+ − c23 )˛] − cr PE[ˇ˛ ]E[[( − Y˜ ) ]˛ ],

(4.10)

2



(P − a)2 2(b − a)

2

2

As we discussed in the introduction section, investment and shortages are two important factors of inventory system. Whereas one indicates the locking of money, other engenders goodwill loss. Thus, in order to detract the effects of these inadequacies, we here impose constraints on budget and shortages. Usually, the unit production cost and maximum budget are estimated from the past data record or experts’ opinion. Before deciding the budget and shortages constraints, the decision maker must keep in mind the maximum demand. The maximum budget and maximum allowable shortages can be restricted as: and

+ cv

T P22T − (P 2  2 − a2 ) + log 4 2

T 2 2 (P  − a2 ) 4

− cr PE[ˇ( − Y )+ ].

cp P  B



1 − P[cp4 − (cp4 − cp3 )˛] − b−a

2

+ cv

2

c1 b−a

P22T + log 2

+ a2

cs b−a

 −

2 c˜2 (b − P) 2(b − a)

(5.1)

+ −

sent the expectations of the random variables (( − Y˜ ) )˛ and

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R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850 + +

(( − Y˜ ) )˛ , respectively. Thus, for a fixed production run-time  and fixed values of 1 , 2 , 3 and 4 , the variable y falls in any one of the following intervals (i) y ∈ [0,  − 4 ], (ii) y ∈ [ − 4 ,  − 3 ], (iii) y ∈ [ − 3 ,  + 2 ], (iv) y ∈ [ + 2 ,  + 2 ] or (v) y ∈ [ + 1 , ∞) (see Fig. 3). Case i. When y ∈ [0,  − 4 ], then the fuzzy time period ˜ Z(ω), ω ∈ ˝, for producing the imperfect quality item is shown in Fig. 3a. The ˛-cut of FRV Z˜ is

The expectation of the above random interval is





E[Z˛− ], E[Z˛+ ]



− 4

=

( − y − 4 + ( 4 − 3 )˛)g(y)dy,



0 − 4

 ( − y + 1 − ( 1 − 2 )˛)g(y)dy .

0

(5.5)

[Z˛− , Z˛+ ] = [ − y − 4 + ( 4 − 3 )˛,  − y + 1 − ( 1 − 2 )˛].

Fig. 3. Fuzzy time period for producing imperfect items: (a) when 0  y   − 4 , (b) when  − 4  y   − 3 , (c) when  − 3  y   + 2 , (d) when  + 2  y   + 1 and (e) when + 1  y < ∞.

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

Case ii. When y ∈ [ − 4 ,  − 3 ], then the fuzzy time period ˜ Z(ω), ω ∈ ˝, for producing the imperfect items is shown in Fig. 3b. ˜ for this case is The ˛-cut of Z,



 +

Z− ˛ , Z˛

⎧ ⎪ ⎨ [0,  − y + 1 − ( 1 − 2 )˛] ,

=

845

− + It is obvious from Eqs. (5.4) to (5.9), E[ˇ˛ ], E[ˇ˛ ], E[Z˛− ] and + E[Z˛ ] are positive. Hence, from Eqs. (5.2), (5.3), (5.4) and (5.9), the  is signed distance of EP()

y + 4 −   1; 4 − 3

0˛

⎪ ⎩ [ − y − + ( − )˛,  − y + − ( − )˛] , 0  y + 4 −   ˛  1. 4 4 3 1 1 2 4 − 3

The expectation of the above random interval is





E[Z˛− ], E[Z˛+ ] =

⎧  − 3 ⎪ y + 4 −  ⎪ 0, [ − y + 1 − ( 1 − 2 )˛]g(y)dy , 0  ˛   1; ⎪ ⎪ 4 − 3 ⎪ ⎪ − 4 +( 4 − 3 )˛ ⎪ ⎪ ⎪  − 4 +( 4 − 3 )˛ ⎨ ( − y − 4 + ( 4 − 3 )˛)g(y)dy,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

− 4

(5.6)

− 4 +( 4 − 3 )˛

( − y + 1 − ( 1 − 2 )˛)g(y)dy ,

0

− 4

y + 4 −   ˛  1. 4 − 3

Case iii. When y ∈ [ − 3 ,  + 2 ], then the fuzzy time period ˜ for producing the imperfect quality item is shown in Fig. 3c. The Z, ˛-cut of Z˜ is [Z˛− , Z˛+ ]

 d(EP(), 0) =

= [0,  − y + 1 − ( 1 − 2 )˛].

The expectation of the above random interval is

  [E[Z˛− ], E[Z˛+ ]] = 0,



+ 2

(5.7)

˜ Case iv. When y ∈ [ + 2 ,  + 1 ], then the fuzzy time period Z, for producing the imperfect items is shown in Fig. 3d. The ˛-cut of Z˜ for this case is

⎧ ⎨ [0,  − y + 1 − ( 1 − 2 )˛] , ⎩

0˛ 0

[0, 0] ,

 − y + 1  1; 1 − 2

 − y + 1  1; 1 − 2 0



E[Z˛− ], E[Z˛+ ]



+ 1 −( 1 − 2 )˛



P 2  2 + a2 2

Pb −

 + cv

4

c + c + c + c  11 12 13 14 4

cr P 2

 1 

−

P

(5.9)

P 2 2

P 2 (b − P) − b

(b − P) 2(b − a)

 (P − a)



c + c + c + c  21 22 23 24 4

˝

+ 1 −( 1 − 2 )˛

( − y + 1 − ( 1 − 2 )˛)g(y)dy (ˇ − 5 + ( 5 − 6 )˛)h(ˇ)dˇ



− 4 +( 4 − 3 )˛

( − y − 4 + ( 4 − 3 )˛)g(y)dy d˛. (5.10)

5.3. Fuzzy constraints As discussed earlier, in various circumstances the investment and allowable shortages may not be expressed as a hard constraint (as it is considered in Eq. (4.11)). Hence, budgetary and shortage constraints which are defined in relation (4.11) can be re-defined in fuzzy sense as: c˜p P  B˜

( − y + 1 − ( 1 − 2 )˛)g(y)dy .

PT −

2

−

0

0



(ˇ + 8 − ( 8 − 7 )˛)h(ˇ)dˇ 0

×



1 b−a

0



(5.8)

(P − a)2 2(b − a)

b

˝

( − y − 4 + ( 4 − 3 )˛)g(y)dy,



×



− 4 +( 4 − 3 )˛

0

+ [EP − ˛ () + EP ˛ ()]d˛ 0

c + c + c + c  p1 p2 p3 p4

+

 − y + 1  ˛  1. 1 − 2

=

1

T P22T − (P 2  2 − a2 ) + log 4 2



˜ for Case v. When y ∈ [ + 1 , ∞), then the fuzzy time period Z, producing the imperfect items is shown in Fig. 3e. The ˛-cut of Z˜ for this case is [0, 0]. Hence, [E[Z˛− ], E[Z˛+ ]] = [0, 0]. After substituting the specific expression of E[Z˛− ] and E[Z˛+ ] for the intervals [0,  − 4 ], [ − 4 ,  − 3 ], [ − 3 ,  + 2 ], [ + 2 ,  + 1 ] and [ + 1 , ∞) from Eqs. (5.5) to (5.8), we have





×

⎧   + −( − )˛  1 1 2 ⎪ ⎪ 0, ( − y + 1 − ( 1 − 2 )˛)g(y)dy , ⎪ ⎪ ⎪ ⎨ − 2 0˛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [0, 0] ,

− P

−

 − y + 1  ˛  1. 1 − 2

The expectation of the above random interval is

[E[Z˛− ], E[Z˛+ ]] =

cs b−a

( − y + 1 − ( 1 − 2 )˛)g(y)dy . − 3

[Z˛− , Z˛+ ] =

=

1 2

and

2 c˜2 (b − P) ˜  S. 2(b − a)

(5.11)

But, these soft constraints are not well defined, because meaning of the inequalities are not clear due to the involvement of fuzziness in the parameters. Therefore, we have to convert these fuzzy

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R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

constraint equivalent to crisp. For this purpose, we express these as possibility and necessity constraints with predefined confidence level as suggested by Liu and Iwamura [13,14], Inuiguchi et al. [15], and Maity and Maiti [28].



and Pos

˜ and Pos where Pos(˜cp P  B)



2 c˜ 2 (b − P)  S˜ 2(b − a) c˜ 2 (b−P)2 2(b−a)

 S˜

 2 ,



(5.19)

6. Solution procedure estimate that the

(1 − 1 )cp1 P + 1 cp2 P  (1 − 1 )B4 + 1 B3 2

2

(5.12)

2

2 c22 (b − P) (1 − 2 )c21 (b − P) + 2(b − a) 2(b − a)

2 c22 (b − P) (1 − 2 )c21 (b − P) + 2(b − a) 2(b − a)



c (b−P) relations cp P  B˜ and 22(b−a)  S˜ will be implemented with maximum chance at least 1 and 2 , respectively. Then, we use Lemma 2.1, to find the crisp constraints, which are equivalent to the possibility constraints (5.12). The crisp constraints are

and

2

and

 (1 − 2 )S4 + 2 S3 .

5.3.1. Possibility constraints If both constraints are possibility type, as ˜  1 Pos(˜cp P  B)

(1 − 1 )cp3 P + 1 cp4 P  (1 − 1 )B2 + 1 B1

The foregoing discussion provides a methodology to convert the fuzzy mathematical programming equivalent to deterministic mathematical programming using signed distance method and possibility and necessity measures, which will give exact solution to the problem. In order to find the competency of the proposed method and also for comparative studies, we employ fuzzy simulation based PSO algorithm. In this section we now discuss both solution approaches in brief.

2

6.1. Signed distance approach

≤ (1 − 2 )S4 + 2 S3 .

(5.13)

5.3.2. Necessity constraints If both constraints are necessity type, as



˜  1 and Nes Nes(˜cp P  B)

2 c˜ 2 (b − P)  S˜ 2(b − a)

  2 .

(5.14)

Then, we can find the crisp constraints equivalent to necessity constraints (5.14) by using the Lemma 2.2, as

The defuzzified cost Eq. (5.10) comprises with total revenue and total costs including production, holding, shortages and rework cost of defective items. This is a profit function, and our aim is to maximize this, subject to budgetary and shortage constraints. The budgetary and shortage constraints are either possibility, necessity or one is necessity and other is possibility type as discussed in Sections 5.3.1–5.3.3, respectively. The mathematical formulation finally generates a non-linear programming problem (NLPP) as:

 max d(EP(), 0) subject, to, the constraints one of the equation (5.13),

(1 − 1 )cp3 P + 1 cp4 P  (1 − 1 )B2 + 1 B1 2

and

2 c24 (b − P) (1 − 2 )c23 (b − P) + 2(b − a) 2(b − a)

(5.15), (5.17) or (5.19) and 0 <  < T.

2

 (1 − 2 )S2 + 2 S1 .

(5.15)

5.3.3. Mixture of possibility and necessity constraints If budgetary constraint is possibility type and shortage constraint is necessity type, that is,



˜  1 and Nes Pos(˜cp P  B)

2

c˜ 2 (b − P)  S˜ 2(b − a)



 2 .

(5.16)

The crisp constraints which are equivalent to imprecise constraints (5.16) are

and

2 c24 (b − P) (1 − 2 )c23 (b − P) + 2(b − a) 2(b − a)

6.2. Fuzzy simulation approach According to Maiti and Maiti [32], the fuzzy simulation approach (FSA) first approximate the nearest interval to the fuzzy objective function, and then formulate a multi-objective problem. In this paper we use the population based random search multi-objective particle swarm optimization (MOPSO) algorithm to the proposed model. + ˜ If A˛ = [A− ˛ , A˛ ] is ˛-cut of fuzzy number A, then, according ˜ is to Grzegorzewski [41], the nearest interval approximation of A defined as, [LA, RA] = [



(5.17)

If budgetary constraint is necessity type and shortage constraint is possibility type, that is,



2 c˜ 2 (b − P)  S˜ 2(b − a)

0

A− ˛ d˛,

1 0

A+ ˛ d˛]. Hence, the nearest inter-

 of Eq. (5.1) val approximation of the fuzzy expected profit EP() is

2

 (1 − 2 )S2 + 2 S1 .

˜  1 and Pos Nes(˜cp P  B)

It is not possible to solve analytically this constrained non-linear programming problem by using any classical method or optimization technique. So, we use Mathematica and MATLAB softwares.

1

(1 − 1 )cp1 P + 1 cp2 P  (1 − 1 )B4 + 1 B3 2

(6.1)



 2 .

(5.18)

Then, the crisp constraints which are equivalent to imprecise constraints (5.18) are,



1

EP − ˛ d˛,

[LEP, REP] = 0

1

 EP + ˛ d˛ ,

(6.2)

0

+ where EP − ˛ and EP ˛ are respectively, obtained in Eqs. (5.2) and (5.3) via (5.4) and (5.9). We now employ Ishibuchi and Tanaka [42] and Maiti and Maiti  [32] method that suggest maximization of fuzzy cost function EP() is equivalent to maximize the functions LEP and CEP = (LEP + REP)/2.

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850 Table 1 The parameters of MOPSO algorithm. pop size max iter w 1 2

847

in an external archive of solutions, called repository. The repository provide an approximation of Pareto-optimal front. The entire schema of MOPSO is shown by a flowchart in Fig. 4.

50 100 0.15 0.5 0.5

7. Numerical illustration In support of above developed model and methodologies, we consider a numerical example with the following input data (in appropriate units): a = 1000, b = 1200, P = 4000, c˜1 = (17, 19, 21, 24), c˜2 = (7, 9, 12, 14), cr = 8, c˜p = (98, 102, 106, 110), cs = 130, cv = 90, h[ˇ] = 10, 0 < ˇ < 0.1, g[y] = 6e−6y . 1 = 0.03, 2 = 0, 3 = 0.04, 4 = 0.08, 5 = 0.04, 6 = 0.02, 7 = 0.02, 8 = 0.05, 1 = 0.90, 2 = 0.95, T = 0.4, and fuzzy parameters B˜ and S˜ are given in Table 2. The MOPSO algorithm is coded in the MATLAB R2012b, while NLPP via signed distance approach is solved in Mathematica 8.0 software by using the command NMaximize. Both MATLAB R2012b and Mathematica 8.0 were executed on Intel CoreTM i3 1.80 GHz CPU with 4.00 GB RAM and Windows 8 operating system. The parameters setting of MOPSO algorithm is given in Table 1. We execute the MOPSO algorithm 10 times for each instance and report the best of them. Thus, total 10 × 4 ×4 = 160 runs are performed to obtain the more reliable results. When both budgetary and shortage constraints are possibility type, then the optimal solutions for both approaches are obtained in Table 2. The MOPSO algorithm provides trade-off between LEP and REP. For instance B˜ = (100000, 104000, 107000, 115000) and S = (3000, 3200, 3600, 4000), possibility budgetary constraint with 1 = .90 and possibility shortage constraint with 2 = .95, MOPSO gives a single point (18299, 23198) in the Pareto front as shown in Fig. 5 (Note that we have converted the maximization problem into minimization and then coded. Hence, the actual result is minus times of result shown in the figure). As Eq. (6.2) shows, the maximization of LEP and CEP are non-conflicting objectives. Hence, MOPSO converges to a single point. However, this is a near optimal, and gives a rough idea of profit estimation. This profit estimation can be interpreted as maximum profit is 23198 while minimum is 18299. On the other hand, signed distance approach gives as exact optimal solution as the profit is 25083. Solutions for other data set are presented in remaining rows of Tables 2–5, and can be interpreted in same way. When B˜ increases and S˜ decreases, then total profits for all cases are increased, this favors the real life situation. When both the constraints are necessity type, the solutions are shown in Table 3. Moreover, when one is possibility type and other is necessity type, then results are shown in Tables 4 and 5. The signed distance approach provides better result than fuzzy simulation based MOPSO algorithm for all instances as Tables 2–5 indicate.

Thus, Eq. (5.1) generates the following multi-objective problem with necessity and/ or possibility constraints, as max[LEP, CEP] subject to the constraints one of the equation (5.13), (5.15), (5.17) or (5.19)

(6.3)

and 0 <  < T.

6.2.1. Particle swarm optimization (PSO) The particle swarm optimization (PSO) is inspired by the social behavior of bird flocking or fish swimming, and is proposed by Kennedy and Eberhart [43]. It is a population based random search algorithm that find the near-optimal solution to the single-objective optimization problem. In order to solve the multiobjective problem, Coello et al. [44] extended the original PSO to the multi-objective PSO (MOPSO). MOPSO is a powerful meta-heuristic algorithm, which has been widely employed in inventory modeling problems (for detail, see, Moslemi and Zandieh [45], and references therein). In PSO, a set of potential solutions, called swarm, and an individual potential solution of swarm, called particle execute over the iterations. Every particle of the swarm has five individual characteristics: (i) position, (ii) velocity, (iii) objective function value related to the position, (iv) the best value explored by the individual particle (called pbest) so far, and (v) the best from the swarm (called gbest). The velocity and position updating of particles in any iteration is done according to the following equations:

vˆ ij = wvij + 1 r1 (pbest ij − xij ) + 2 r2 (gbest i − xij ), xˆ ij = xij + vˆ ij , where vij and vˆ ij are the previous and current velocities of ith particle in the jth dimension, w is inertia weight, r1 and r2 are random numbers uniformly distributed in [0, 1], 1 and 2 are personal and social learning coefficients, respectively, pbestij is the jth element of personal best for particle i, and gbesti is the ith element of the global best. In multi-objective optimization problem, the selection of gbest is a key component, because the optimum solutions are a set of Pareto-optimal. The simplest approach to select the gbest might be random selection of non-dominated solutions. Although random selection may increase the diversity of the solution set by exploring the search space, it may decrease the convergence rate by destroying the memory at the time of execution. So, we use crowding distance (CD) criterion in the paper to overcome this problem. MOPSO stores the non-dominated solutions found so far

8. Discussion Based on the solutions obtained in Tables 2–5, we can say that the management must keep in mind the expected demand when he/she decides the budgetary constraint. The management should

Table 2 Both of the constraints are possibility type with 1 = .90 and 2 = .95. B˜

S˜

Signed distance approach

MOPSO

*

*

Total profit

Total profit LEP

(100000, 104000, 107000, 115000) (105000, 108000, 112000, 118000) (110000, 112000, 116000, 120000) (115000, 120000, 122000, 125000)

(3000, 3200, 3600, 4000) (2500, 2700, 3200, 3500) (2000, 2400, 2700, 3000) (1500, 1800, 2200, 2500)

0.264 0.276 0.284 0.284

25,083 25,743 25,847 25,847

0.243 0.244 0.265 0.280

18,299 19,357 19,872 20,123

CEP 23,198 24,338 24,928 25,311

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R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

Fig. 4. Flowchart of MOPSO.

R.S. Kumar, A. Goswami / Applied Soft Computing 34 (2015) 838–850

849

Table 3 Both of the constraints are necessity type with 1 = .90 and 2 = .95. B˜

S˜

Signed distance approach

MOPSO

*

*

Total profit

Total profit LEP

(100000, 104000, 107000, 115000) (105000, 108000, 112000, 118000) (110000, 112000, 116000, 120000) (115000, 120000, 122000, 125000)

(3000, 3200, 3600, 4000) (2500, 2700, 3200, 3500) (2000, 2400, 2700, 3000) (1500, 1800, 2200, 2500)

0.229 0.240 0.251 0.264

19,840 22,059 23,766 25,038

0.228 0.237 0.250 0.260

12,079 14,489 16,457 18,097

CEP 16,828 19,266 21,279 22,985

Table 4 The possibility budgetary constraint with 1 = .90 and necessity shortages constraints with 2 = .95. B˜

S˜

Signed distance approach 

*

Total profit

MOPSO *

Total profit LEP

(100000, 104000, 107000, 115000) (105000, 108000, 112000, 118000) (110000, 112000, 116000, 120000) (115000, 120000, 122000, 125000)

(3000, 3200, 3600, 4000) (2500, 2700, 3200, 3500) (2000, 2400, 2700, 3000) (1500, 1800, 2200, 2500)

0.265 0.277 0.284 0.284

25,177 25,764 25,847 25,847

0.257 0.271 0.275 0.286

18,300 19,357 19,872 20,123

CEP 23,199 24,338 24,928 25,311

Table 5 The necessity budgetary constraint with 1 = .90 and possibility shortages constraint with 2 = .95. B˜

S˜

Signed distance approach 

*

Total profit

MOPSO *

Total profit LEP

(100000, 104000, 107000, 115000) (105000, 108000, 112000, 118000) (110000, 112000, 116000, 120000) (115000, 120000, 122000, 125000)

(3000, 3200, 3600, 4000) (2500, 2700, 3200, 3500) (2000, 2400, 2700, 3000) (1500, 1800, 2200, 2500)

try to invest maximum amount as far as possible in order to ensure the maximum profit, but anyhow it should not exceed the production cost of maximum demand quantity. On the other hand, minimum investment should not be less than the production cost of minimum demand quantity. We suggest the organization to fixed the budgetary constraint equal to about the production cost of the expected demand quantity, otherwise, the NLPP may produce infeasible solution.

0.229 0.240 0.251 0.264

19,840 22,059 23,766 25,038

0.219 0.232 0.248 0.247

12,077 14,490 16,454 18,094

CEP 16,826 19,267 21,275 22,981

The numerical illustrations examine the competency of both our approach based on signed distance and fuzzy simulation based PSO algorithm. When both constraints are possibility type, then both approaches generate more profits compare to the case when both constraints are necessity type. When budgetary constraint is possibility type and shortage constraint is necessity type, then both approaches give the best results. The reason is, the budget is used higher level (i.e., [B3 , B4 ]) and shortage is allowed at lower level (i.e., [S1 , S2 ]). The results with the necessity budgetary constraint and possibility shortage constraint for both approaches almost tie with the results with necessity budgetary and shortage constraints. The trends of solutions of both approaches are same, but signed distance based our approach gives better result than PSO algorithm throughout the investigation. In brief we can say that budgetary constraint play vital role in the optimal decision policy. It may be concluded that if the decision maker is ready to spend higher level budgetary constraint (i.e., [B3 , B4 ]), they should opt for possibility constraint for budgetary constraint. At the same time, he/she should use necessity shortage constraint which in turn will reduce the shortage quantity.

9. Conclusion

Fig. 5. Pareto solution for possibility budgetary constraint B˜ = (100000, 104000, 107000, 115000) and possibility shortage constraint S˜ = (3000, 3200, 3600, 4000).

In this article we have modeled a single-period productioninventory problem in fuzzy random framework with imprecise budgetary and shortage constraints. To capture the real-life business situations, the key parameters, namely, demand and cost-coefficients are taken as random variable and fuzzy numbers, respectively. Moreover, we have assumed that machine shifting from “in-control” state to “out-of-control” state and fraction of defective items as FRVs. In order to deal with uncertainties like unexpected increment in demand during the scheduling

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period, recession, price hike, etc., we have restricted budget and shortage constraints by fuzzy numbers. To model the aforementioned scenario, we have formulated a fuzzy mathematical programming problem. To the best of our knowledge, no such production-inventory model has been developed till now. However, fuzzy expectation and signed distance method are employed to find the equivalent deterministic cost function from the fuzzy random cost function. In order to convert the soft constraint equivalent to the hard constraint, two lemmas are proposed in Section 2. Numerical illustrations are provided to explain the working procedure of the proposed model and methodology. In order to accomplish the comparative analysis of the proposed methodology, we also employed fuzzy simulation based PSO algorithm to the numerical illustrations. It is also observed that the proposed model will produce the best result when decision makers consider budgetary constraint as possibility type and shortage constraint as necessity type. This observation is also supported by PSO algorithm. Throughout the investigation, we found that the methodology based of signed distance perform better than PSO. In general production process depends upon several unreliable factors such as availability of workers, supply of raw material, etc. Hence, fuzzy or random production rate is more suitable than constant one. This model can be extended by considering production rate as fuzzy or random variable. One more possible extension of this model is to incorporate the preventing maintenance technology to prevent the machine shifting probability. Acknowledgements The authors express their sincere thank to the editor and the anonymous referees for their valuable comments and constructive suggestions that improve the quality of the paper. We are also thankful to Mr. K. Karthik, B. Tech student of Industrial & Systems Engineering, IIT Kharagpur for the computational support given while carrying out this research. References [1] M.J. Rosenblatt, H.L. Lee, Economics production cycles with imperfect production processes, IIE Trans. 18 (1986) 48–55. [2] K. Chung, K. Hou, An optimal production runtime with imperfect production processes and allowable shortages, Comput. Oper. Res. 30 (2003) 483–490. [3] M.K. Salameh, M.Y. Jaber, Economics production quantity model for items with imperfect quality, Int. J. Prod. Econ. 64 (2000) 59–64. [4] P.A. Hayek, M.K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Prod. Plan. Control 12 (6) (2001) 584–590. [5] C. Hyun Kim, Y. Hong, An optimal run length in deteriorating production process, Int. J. Prod. Econ. 58 (1999) 183–189. [6] G.L. Liao, Y.H. Chen, S.H. Sheu, Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance, Eur. J. Oper. Res. 195 (2009) 348–357. [7] S.K. Sana, An economic production lot size model in an imperfect production system, Eur. J. Oper. Res. 201 (2010) 158–170. [8] L. Zadeh, Fuzzy sets as a basis for theory of possibility, Fuzzy Sets Syst. 1 (1978) 3–28. [9] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold, New York, 1991. [10] D. Dubois, H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Inf. Sci. 30 (1983) 183–224. [11] H. Kwakernaak, Fuzzy random variables: definition and theorems, Inf. Sci. 15 (1) (1978) 1–29. [12] M.A. Gil, M. López-Dí az, D.A. Raleascu, Overview on the development of fuzzy random variables, Fuzzy Sets Syst. 157 (2006) 2546–2557. [13] B. Liu, K.B. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (1998) 227–237. [14] B. Liu, K.B. Iwamura, A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets Syst. 100 (1998) 229–233.

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