Optimal production inventory policy for defective items with fuzzy time period

Applied Mathematical Modelling 34 (2010) 810–822

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Optimal production inventory policy for defective items with fuzzy time period S. Mandal b, K. Maity a,*, S. Mondal b, M. Maiti b a b

Department of Mathematics, Mugberia Gangadhar Mahavidyalaya, Purba Medinipur, Mugberia 721 425, India Department of Mathematics, Vidyasagar University, Paschim Medinipur, Midnapore 721 102, India

a r t i c l e

i n f o

Article history: Received 9 November 2007 Received in revised form 15 May 2009 Accepted 3 June 2009 Available online 18 July 2009 Keywords: Fuzzy time period Dynamic production Defective items Credibility

a b s t r a c t In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Since the development of EOQ model by Harris [1], a lot of research works has been reported in the literature (cf. Naddor [2], Hadly and Whitin [3] and others). In classical inventory models, normally static lot size models are formulated. But, because of the dynamic manufacturing environment, the static models are not adequate in analyzing the behavior of such systems and in designing the optimal policies for their control. For this reason, dynamic models of production inventory systems have been considered by some researchers (cf. Hu and Loulon [4], Kirk [5], Hu and Dong [6], Minner and Kleber [7], Feng and Yan [8], Zhang et al. [9] and others). In these models, demand and/or production are assumed to be continuous functions of time. During the last two decades, many researchers (cf. Aliyu and Andhani [10], Bhunia and Maiti [11], Giri and Chaudhuri [12], Andijani and Aldajani [13] and others) have given considerable attention to the area of inventory of deteriorating/defective/perishable items, since the life time of an item is not infinite while it is in storage and/or all units can’t be produced exactly as per the prescribed specifications. Recently, Goyal and Giri [14] have presented a review article on the recent trends in modelling with deteriorating items listing all important publications in this area up to 2001.The application of control theory in production inventory control analysis is now-a-days gradually increasing due to its dynamic behavior. Many research papers (cf. Athans and Falb [15], Axsater and Rosling [16], Bellman and Kalaba [17], Blanchini et al. [18], Bounkel et al. [19], * Corresponding author. Tel.: +91 9434611354. E-mail addresses: [email protected] (K. Maity), [email protected] (M. Maiti). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.06.031

S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

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Branicky et al. [20], Kiesmuller et al. [21], Kleber et al. [22], Kuo [23], etc.) have been published in this regard. Bertrand [24] presented a list of earlier workers who published several papers in application of state space techniques and control theory to production inventory system. Later Axsater [25], Andijani and Aldajani [13] utilized optimal control theory to obtain optimal production policy for production inventory systems where items are deteriorating at a constant rate. Optimal control problems usually deal with both objective function and the constraints. The objective function is the cost function that need to be minimized with respect to time, fuel, energy, etc. The constraints are usually the system dynamics, the limits of the system states and the control effort. Now-a-days, with the advent of multi-nationals in the developing countries, there is a stiff competition amongst the multi-nationals to capture the market. Thus in the recent competitive market, the inventory/stock is decoratively exhibited and colorably displayed through electronic media to attract the customers and thus to boost the sale. For this reason, Datta and Pal [26], Mandal and Phaujder [27], Maity and Maiti [28] and others considered linear form of stock dependent demand. The marketing system has recognized this relationship and incorporated it into shelf-space allocation models. These models assumed the demand rate as a function of the shelf-space allocated to the product and sometimes to the shelf-space allocated to the substitute and /or complementary products. Also as the demand is stock dependent, a gorgeous decoration is needed and this is why the total space varies. This is a cause of fuzzyness of the storage space. But, the above mentioned models are developed for the fixed (deterministic) time horizon. In the case of multi-item inventory models, till now, it is assumed that each is exhausted after a fixed time period and this time period is same for all items. This is a hard assumption on the real world situations. Actually, every year a seasonable product does not end at a particular time. In a year, several season products do have also different time periods. There is an inherent uncertainty in these time horizons. This uncertainty can be represented by fuzzy number. Thus the time periods of the items made out of seasonable products are fuzzy. This is also true in the case of fast moving items also. In this paper, for the first time, we have introduced this concept. We have used optimal control formulation with imprecise time horizons for production inventory problems of multi-items. Till now, in the optimal control literature, none has considered a model with imprecise time period. Here a multi-item production inventory control dynamic problem with fuzzy time horizon has been formulated and solved under storage capacity constraint. Shortages are not allowed. Production is time dependent and demand is stock dependent. The relevant inventory costs of the system are production and holding costs which are fuzzy. The whole system is under a single management house and the business of each item is considered separately. Here the cost of each item is expressed in the form of an integral as an optimal control problem and is optimized where the items compete each other in the limited warehouse space which is fuzzy in nature.Thus it is a constrained multi-objective problem. As the system is under a single management house, total cost is taken into consideration and there are desired inventory level and the satisfactory production level on the total stock and the total production levels, respectively. The constrained multi-objective problem is reduced to a single objective by Global Criteria Method (GCM). The fuzzy space constraint is converted to a crisp one using possibility measure. The fuzzy integral is transformed to crisp integrals by the help of Fuzzy Riemann Integral (FRI) method. The crisp Optimal control problems are optimized using the Kuhn–Tucker conditions and a non-linear optimization technique, Generalised Reduced Gradient (GRG) method. The optimum production and stock levels are determined for different demand functions. Numerical experiments are performed to illustrate the models. 2. Preliminaries 2.1. Fuzzy set ~ in a universe of discourse X is defined as the following set of pairs: A fuzzy set a

~ ¼ fðx; la~ ðxÞÞ : x 2 Xg; a ~ and la~ ðxÞ is called the membership vawhere la~ : X ! ½0; 1 is a mapping called the membership function of the fuzzy set a ~. ~. The larger la~ ðxÞ is the stronger grade of membership form in a lue or degree of membership of x 2 X in the fuzzy set a 2.1.1. Convex fuzzy set ~ of the universe of discourse X is convex iff 8x1 ; x2 2 X, A fuzzy set a

la~ ðkx1 þ ð1  kÞx2 Þ P minðla~ ðx1 Þ; la~ ðx2 ÞÞ when 0 6 k 6 1: 2.1.2. Normal fuzzy set ~ of the universe of discourse X is called a normal fuzzy set implying that there exists at least one x 2 X such A fuzzy set a that la~ ðxÞ ¼ 1. 2.1.3. Fuzzy number Fuzzy numbers are assumed to be convex, normal, upper semicontinuous and compactly supported fuzzy subsets of the ~ : R ! ½0; 1. real line, a

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2.2. Possibility, necessity and credibility constraints ~ be two fuzzy quantities with membership functions ~ and b Let a and Prade [29,30] and others [18,19,12]

la~ ðxÞ and lb~ ðxÞ, respectively. Then according to Dubois

~ ¼ fsupðminðl ðxÞ; l~ ðyÞÞ; x; yR; x  yg; ~  bÞ Posða ~ a b ~ ¼ finf ðmaxð1  l ðxÞ; l~ ðyÞÞ; x; yR; x  yg; ~  bÞ Nesða ~ a

ð1Þ ð2Þ

b

where the abbreviation Pos represents possibility, Nes represents necessity and  is any of the relations >; <; ¼; 6; P. The relationships for possibility, necessity and credibility constraints: The dual relationship of possibility and necessity requires that

~ ~ ¼ 1  Posða ~  bÞ: ~  bÞ Nesða

ð3Þ

Also necessity measures satisfy the condition

~ ¼ 0: ~ Nesða ~  bÞÞ ~  bÞ; MinðNesða The relationships between possibility and necessity measures satisfy also the following conditions (cf. Dubois and Prade [24]): ~ P Nesða ~ Nesða ~ > 0Þ ) Posða ~ ¼ 1 and Posða ~ < 1 ) Nesða ~ ¼ 0. ~  bÞ ~  bÞ; ~b ~  bÞ ~  bÞ ~  bÞ Posða ~ ~ ~ ~ ~ By Zadeh’s extension principle, if a; bR and c ¼ f ða; bÞ where f : R  R ! R be a binary operation then membership function l~c of ~c is defined as

l~c ðzÞ ¼ supfminðla~ ðxÞ; lb~ ðyÞÞ; x; yR and z ¼ f ðx; yÞ; 8 zRg:

ð4Þ

Recently based on possibility measure and necessity measure, the third set function Cr, called credibility measure, was analyzed by Liu and Liu [31] is as follows:

CrðaÞ ¼

1 ½PosðaÞ þ NecðaÞ for any a 2 2R : 2

ð5Þ

It is easy to check that Cr satisfies the following conditions: (i) Crð/Þ ¼ 0 and CrðRÞ ¼ 1. (ii) CrðaÞ 6 CrðbÞ when ever a; b 2 2R and a  b. Thus, Cr is also a fuzzy measure defined on ðR; 2R Þ. Besides, Cr is self dual, i.e., CrðaÞ ¼ 1  CrðaC Þ for any a 2 2R . In this paper, based on the credibility measure the following form can be defined as:

CrðaÞ ¼ ½qPosðaÞ þ ð1  qÞNecðaÞ

ð6Þ

for any a 2 2R and 0 < q < 1. It also satisfies the above conditions. 2.2.1. Triangular fuzzy number ~Þ (see Fig. 1) is the fuzzy number with the membership function Triangular fuzzy number (TFN) (a mapping: la~ ðxÞ : R ! ½0; 1

8 0; > > > xa1 > > ; > < a2 a1 x la~ ðxÞ ¼ aa33a ; 2 > > > > 0; > > :

la~ ðxÞ, a continuous

for  1 < x < a1 ; for a1 6 x < a2 ; for a2 6 x 6 a3 ; for a3 < x < 1:

~ ¼ ða1 ; a2 ; a3 Þ) be a triangular fuzzy number and r is a crisp number. Then according to Liu and Iwamura [32,33], MaLet a ity and Maiti [34] we define possibility measure and necessity measure as following:

Posða P rÞ ¼

Necða P rÞ ¼

8 > < 1; > :

a3 r a3 a2

0; 8 1; > > > < a2 r

a2 a1

> 0; > > :

if r 6 a2 ; ; if a2 6 r 6 a3 ; if r P a3 ; if r 6 a1 ; ; if a1 6 r 6 a2 ; if r P a2 :

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S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

~. Fig. 1. Membership function of TFN a

The credibility measure for TFN can be define as

Crða P rÞ ¼

Crða 6 rÞ ¼

8 1; if r 6 a1 ; > > > qÞr > < aa2 aqr  ð1 ; if a1 6 r 6 a2 ; a2 a1 2 1 qða3 rÞ > ; > a a > > : 3 2 0; 8 0; > > > > q ra1 ; < a2 a1

ð7Þ

if a2 6 r 6 a3 ; if r P a3 ; if r 6 a1 ; if a1 6 r 6 a2 ;

ð8Þ

a3 qa2 ð1qÞr > ; if a2 6 r 6 a3 ; > a3 a2 > > : 1; if r P a3 :

Based on the credibility measure, Liu and Liu [31] presented the expected value operator of a fuzzy variable as follows: Definition 1. Let X be a normalized fuzzy variable the expected value of the fuzzy variable X is defined by

E½X ¼

Z

1

CrðX P rÞdr 

Z

0

0

CrðX 6 rÞdr:

ð9Þ

1

When the right hand side of (9) is of form 1  1, the expected value is not defined. Also, the expected value operation has been proved to be linear for bounded fuzzy variables, i.e., for any two bounded fuzzy variables X and Y, we have E½aX þ bY ¼ aE½X þ bE½Y for any real numbers a and b are of same sign. ~ is ~ ¼ ða1 ; a2 ; a3 Þ) is a triangular fuzzy number and r is a crisp number. The expected value of a Lemma 1. a

~ ¼ E½a

1 ½ð1  qÞa1 þ a2 þ qa3  where 0 < q < 1: 2

Proof. From (9), by using (7) and (8) we have

~ ¼ E½a

Z

1

Crða P rÞdr 

0

¼

Z

0

a1

dr þ

Z

a2

a1



Z

0

Crða 6 rÞdr ¼

1

Z 0

a1

Crða P rÞdr þ

Z

a2

Crða P rÞdr þ a1

Z

a3

Crða P rÞdr þ 0

a2

  Z a3  a2  qa1 ð1  qÞr qa3 qr 1   dr þ dr ¼ ½a1 ð1  qÞ þ a2 þ qa3 : a2  a1 2 a2  a1 a3  a2 a3  a2 a2

ð10Þ

3. Evaluation of integrals with fuzzy limits 3.1. Fuzzy Riemann Integral (FRI) method ~ ¼ ½b ~L ; b ~U ~Ua  and b ~L ~L ~U ~ U ~ ~ ¼ ½a ~La ; a Let a a a , where aa ; ba are the lower bounds and aa ; ba are the upper bounds of the fuzzy numbers a ~ respectively. Let F R be a set of all fuzzy real numbers induced by the fuzzy real number system R. We define the relaand b, x1  ~ x2 iff ~ x1 and ~ x2 are induced by the same real number x. Then  is an equivalence relation inducing the tion  on F R as ~ ~ ~ ~ ~ equivalence classes ½x ¼ a=a  x. The quotient set F R =  is the set of all equivalence classes. Then the cardinality of F R =  is

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equal to the cardinality of the real number system R, since the map R ! F R =  by x ! ½~ x is a bijection. We call F R =  as fuzzy real number system. In practice (for convenience), we take only one element ~ x from each equivalence class ½~ x to form the fuzzy real number system ðF R =  ÞR (roughly speaking), i.e. x=~ x 2 ½~ x; ~ x is the only element from ½~ xÞ. ðF R =  ÞR ¼ ð~ If the fuzzy real number system ðF R =  ÞR consists of closed (bounded) fuzzy real number system. x is a Let ~f : ðF R =  ÞR ! F be a fuzzy-valued function defined on the fuzzy real number system ðF R =  ÞR . Suppose that ~ fuzzy real number induced by a real number x. then we can induce a new fuzzy-valued function ~f : R ! F by ~f ðxÞ ¼ ~f ð~ xÞ. sup~Pa ~ (denoted as ½a ~ ~, we shall discuss the fuzzy Riemann integrals on the fuzzy interval formed by a ~ and b ~; b), pose that b ~ ~ where a and b are two fuzzy real numbers induced by two real numbers a and b, respectively. In order to define the fuzzy ~ for b ~>a ~Pa ~6a ~L P a ~Ua for all a). Now ~ and b ~ (b ~ means b ~ and b Riemann integral, we need to consider the ‘‘length” between a a L U L U U L U L ~ ~ ~ ~ ~ ~a and ðb  a ~Þa ¼ ba  a ~a . We shall consider the interval ½a ~a ; ba  for the lower bound case and the interval ~Þa ¼ ba  a ðb  a ~U  for the upper bound case. ~La ; b ½a a Definition 2. Let ~f ð~ xÞ be a bounded- and closed-fuzzy-valued function defined on the closed-fuzzy real number system ~Pa ~. ðF R =  ÞR , and ~f ðxÞ be induced by ~f ð~ xÞ. Suppose that b ~L  and ½a ~U , respectively, for all a then we ~Ua ; b ~La ; b (i) If ~f ðxÞ is non-negative and ~f La ðxÞ and ~f Ua ðxÞ are Riemann-integrable on ½a a a let

8 hR ~ L i R ~U > ~Ua ; < a~bUa ~f La ðxÞdx; a~bLa ~f Ua ðxÞdx ; if b~a L > a a a Aa ¼ h R ~ U i > ~L 6 a : 0; bLa ~f U ðxÞdx ; ~Ua : if b a a ~ a a

~U  and ½a ~L , respectively, for all a then we ~La ; b ~Ua ; b (ii) If ~f ðxÞ is non-positive and ~f La ðxÞ and ~f Ua ðxÞ are Riemann-integrable on ½a a a let

8 hR ~ U i R b~L ba ~L > ~L > a ~Ua ; > f ðxÞdx; a~Ua ~f Ua ðxÞdx ; if b > a < a~La a a hR ~ U i Aa ¼ ba ~U ~L 6 a ~Ua : if b > a ~L f a ðxÞdx; 0 ; > a > : a ~ and the membership func~; b Under the above conditions, we say that ~f ð~ xÞ is Riemann-integrable on the fuzzy interval ½a R b~ xÞd~ x is defined by, for r 2 A0 , lR b~ ~ ðrÞ ¼ supða=Aa ðrÞ; 0 6 a 6 1Þ. tion of a~ ~f ð~ ~ a

Lemma 2. If f ðxÞ is non-negative then

Z

f ð~ xÞd~ x

~ b

f ðxÞdx ¼ ½gðaÞ; hðaÞ;

ð11Þ

~ a

where

gðaÞ ¼

Z

~L b a

f ðxÞdx;

~U a a

and hðaÞ ¼

Z

~U b a ~La a

~L P a ~Ua ¼ 0; b a

~L 6 a ~Ua ; b a

~L : ~Ua ; b f ðxÞdx for all a a

ð12Þ ð13Þ

Proof. Lemma 2 is already proved in Wu [35]. Method (gðaÞ and hðaÞ can be found analytically). Since gðaÞ is increasing and hðaÞ is decreasing, we have: (i) if gð1Þ 6 r 6 hð1Þ then lðrÞ ¼ 1, (ii) if r < gð1Þ then lðrÞ = max a : 0 6 a 6 1; a is the root of gðaÞ ¼ r, (iii) if r > hð1Þ then lðrÞ ¼ max a : 0 6 a 6 1; a is the root of hðaÞ ¼ r. Since gðaÞ and hðaÞ can be found analytically, we can find the roots of gðaÞ ¼ r or hðaÞ ¼ r using the commercial software LINGO. In practice, we always consider the membership function of the fuzzy number as having bell shape. Thus, we suggest the fuzzy real number ~t having membership function (with spread r)

lðxÞ ¼



jðx  tÞ=rj þ 1; for  r þ t 6 x < r þ t; 0;

otherwise;

which is induced by a real number t. Then real number.

l~t is continuous with compact support. That is, ~t is a closed and bounded fuzzy

S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

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4. Assumptions and notations Production inventory model is developed under the following assumptions: 4.1. Assumptions For ith ði ¼ 1; 2; 3 . . . ; n item, it is assumed that, (i) (ii) (iii) (iv) (v) (vi)

demand rate is stock dependent, defective rate is known and stochastic, shortages are not allowed, unit production cost and holding cost are fuzzy in nature, storage capacity is fuzzy in nature, this is a single period inventory model for each item with fuzzy time period.

4.2. Notations n number of items e Mð¼ ðM1 ; M2 ; M 3 ÞÞ storage space available which is imprecise in nature. For the ith ði ¼ 1; . . . ; nÞ item time length of the cycle which is fuzzy in nature Tei Di ðtÞð¼ di0 þ di1 xi ðtÞÞ demand rate at time t where di0 ; di1 are known U i ðtÞð¼ ui0 þ ui1 t þ þ uim t m Þ production rate at time t where ui0 ; ui1 ; . . . ; uim are unknown the inventory level at time t xi ðtÞ per unit area ai desired inventory level Ii0 the satisfactory production level pi0 rate of defectiveness which is stochastic d~i production cost per unit item which is fuzzy Cf ui hei holding cost per unit item per unit time which is fuzzy

5. Model formulation 5.1. Multi-Objective Dynamic Model (MODM) with fuzzy integral and fuzzy space constraint Let there be n-defective items under a production inventory system. Here, though the system is under a single management, the production inventory process for each item is considered separately to minimize the corresponding cost. The items are stored together in a single go-down of finite dimension. These are produced at a variable rate U i ðtÞ and defective at a stochastic rate, dbi . Demand of the items are stock dependent and the stock level at time, t decreases due to consumption. Shortages are not allowed. The differential equation for ith item representing the above system during a fixed time horizon, T i is

X_ i ðtÞ ¼ ð1  dbi ÞU i ðtÞ  Di ðtÞ

ð14Þ

and

0 6 U i ðtÞ 6 ui ;

Di ðtÞ P 0;

0 6 t 6 Ti:

Assuming the warehouse of finite capacity, minimization of total costs consisting of holding and production costs for each item leads to

Minimize eJ ¼

n Z Tei X i¼1

subject to

n X

ð hei X i ðtÞ þ Cfui U i ðtÞÞdt;

i ¼ 1; 2; . . . ; n;

ð15Þ

0

e X i ðtÞai 6 M;

i ¼ 1; 2; . . . ; n:

ð16Þ

i¼1

5.2. Conversion of MODM to single objective problem Here, for ith item, the desired inventory level Ii0 , and satisfactory production rate pi0 are assumed to be the reference points for state ðX i Þ and control ðU i Þ variables, respectively. With these values, we apply GCM for optimization of present optimal control problem. In this case, as the system is under a single management the corresponding problem is

816

S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

Minimize eJ ¼

n Z Tei X

ð hei ðX i ðtÞ  Ii0 Þ2 þ Cfui ðU i ðtÞ  pi0 Þ2 Þdt;

ð17Þ

0

i¼1

subject to the constraint (16), i ¼ 1; 2; . . . ; n. 5.3. Conversion of fuzzy constraint to a crisp one The above fuzzy budget constraint is converted into crisp constraint with the help of the possibility function (cf. In this problem, space constraint is considered to be of the form as n X

H

2.2).

e X i ðtÞai 6 M;

i¼1

e are fuzzy storage capacity. where M

now let

n X

ai xi ðtÞ ¼ n:

ð18Þ

i¼1

Using possibility distribution we have

e  n P 0Þ > g: Posð M

ð19Þ

With the help of Maity and Maiti [34] the problem becomes

2 3 Z Tei Z Tei n X 2 2 4 Minimize eJ ¼ ð hei ðX i ðtÞ  Ii0 Þ þ Cfui ðU i ðtÞ  pi0 Þ Þdt þ kðM3 ð1  gÞ þ gM2  nÞdt 5: 0

i¼1

ð20Þ

0

6. Equivalent representation of the proposed model Taking hei and cf ui are TFN and using credibility theory (cf. Lemma 1), we have the problem as

2 Z Tei n X 4 Minimize eJ ¼ ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt 0

i¼1

þ

Z Tei 0

3 kðM 3 ð1  gÞ þ gM2  nÞdt 5:

ð21Þ

7. Solution of Multi-Objective Non-Linear Programming (MONLP) problem using FRI method

Here let U i ðtÞ ¼ ui0 þ ui1 t þ þ uim tm : From Eq. (14), we have

xi ðtÞ ¼ ðbi0 þ bi1 t þ bi2 t2 þ bi3 t3 þ þ bim t m Þ  Di0 þ ðDi0  bi0 Þedi1 t ;

ð22Þ

where

bir ¼ ð1  Eð dbi ÞÞ

m   X jðr1Þ ð1Þj uij ðj!Þ= di1 ðr!Þ ;

r ¼ 0; 1; 2; . . . ; m:

ð23Þ

j¼r

Here let the fuzzy real number ~t have the membership function (with spread

l~t ðxÞ ¼



r)

jðx  tÞ=rj þ 1; for  r þ t 6 x < r þ t; 0; otherwise;

which is induced by a real number T i . We have that a ¼ l~t ðxÞ ¼ jðx  tÞ=rj þ 1 implies x ¼ t þ rð1  aÞ or t  rð1  aÞ. Thus tea L ¼ t  rð1  aÞ and tea U ¼ t þ rð1  aÞ Then the fuzzy integral is

Minimize eJ ¼

2 Z Tei n X 4 ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt 0

i¼1

þ

Z Tei 0

3 kðM 3 ð1  gÞ þ gM2  nÞ5dt:

ð24Þ

817

S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

Using Wu [35]. The problem can be formed as

h i Minimize eJ La ; eJ Ua

ð25Þ

for all a, where

2 Z eT L n X ia 4 Ja ¼ ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt

eL

0

i¼1

þ

3

Z Teia L

kðM3 ð1  gÞ þ gM2  nÞdt5

0 n Z X

¼

0

i¼1

þ

T i rð1aÞ

Z

T i rð1aÞ

ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt

 kðM 3 ð1  gÞ þ gM 2  nÞdt ;

ð26Þ

0

and

2 Z Teia U n X eJ U ¼ 4 ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt a 0

i¼1

þ

Z Teia U 0

¼

n Z X

Z

kðM 3 ð1  gÞ þ gM 2  nÞdt 5

T i þrð1aÞ

0

i¼1

þ

3

T i þrð1aÞ

ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt

kðM 3 ð1  gÞ þ gM 2  nÞdt

 ð27Þ

0

5.1. Particular cases 5.1.1. Model 1: quadratic production function For m ¼ 2, i.e., production function is quadratic,

let U i ðtÞ ¼ ui0 þ ui1 t þ ui2 t2 :

ð28Þ

From Eqs. (22) and (23), the optimal stock,

where

xi ðtÞ ¼ ðbi0 þ bi1 t þ bi2 t 2 Þ  Di0 þ ðDi0  bi0 Þedi1 t ;

ð29Þ

, 2 X jðr1Þ j b bir ¼ ð1  Eð di ÞÞ ð1Þ uij ðj!Þ ðdi1 ðr!ÞÞ;

ð30Þ

r ¼ 0; 1; 2

j¼r

and the problem becomes

0 Ja ¼ @ L

n Z X 0

i¼1

e

T Lia

½ðð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 ÞðX i ðtÞ  Ii0 Þ2 

þ½ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 ÞðU i ðtÞ  pi0 Þ2 Þdt þ ¼

n Z X i¼1

T i 2ð1aÞ

"

0

ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ

ia

0

1 kðM3 ð1  gÞ þ gM 2  nÞdt A

! 2 X 2 X ð2  dkj Þbij bik tjþk þ ðDi0  bi0 Þ2 e2di1 t þ D2i0 þ I2i0 j¼0 k¼j

þ2ðDi0  bi0 Þedi1 t

! ! 2 2 X X j j ðbij t Þ  ðDi0 þ Ii0 Þ  2ðDi0 þ Ii0 Þ bij t þ 2Di0 Ii0 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ j¼0

j¼0

! !# 2 X 2 2 X X 2 k jþk j dt ð2  dj Þuij uik t þ ðpi0 Þ  2pi0 uij t



j¼0 k¼j

þk

Z TeL

Z 0

T i 2ð1aÞ

j¼0

M3 ð1  gÞ þ gM 2 

n X i¼1

! !

ai ððbi0 þ bi1 ðT i  2ð1  aÞÞ þ bi2 ðT i  2ð1  aÞÞ2 Þ þ ðDi0  bi0 Þedi1 ðT i 2ð1aÞÞ Þ dt ;

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S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

0 2 00 11 jþkþ1 n 2 X 2 X X ððT  2ð1  a ÞÞ B 6 BB CC i J La ¼ @ ð2  dkj Þbij bik AA 4ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ@@ ðj þ k þ 1Þ i¼1 j¼0 k¼j ðDi0  bi0 Þ2 e2di1 ðT i 2ð1aÞÞ þ ðD2i0 þ I2i0 ÞðT i  2ð1  aÞÞ þ 2ðDi0  bi0 Þedi1 ðT i 2ð1aÞÞ 2di1 ! ! 2 X bij ðDi0 þ Ii0 Þ j j1 j ððdi1 ðT i  2ð1  aÞÞÞ  jðdi1 ðT i  2ð1  aÞÞÞ þ þ ð1Þ j!Þ   a di1 ðdi1 Þjþ1 j¼0 ! 2 X bij 2ðDi0 þ Ii0 Þ ðT i  2ð1  aÞÞjþ1 þ 2Di0 Ii0 ðT i  2ð1  aÞÞ þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ jþ1 j¼0 ! !# 2 2 2 X XX ðT i  2ð1  aÞÞjþkþ1 ðT i  2ð1  aÞÞjþ1 2 k þ ðpi0 Þ ðT i  2ð1  aÞÞ  2pi0 ð2  dj Þuij uik uij  jþkþ1 jþ1 j¼0 k¼j j¼0



þ k ðM 3 ð1  gÞ þ gM 2 ÞðT i  2ð1  aÞÞ 

n X

ai

i¼1

ðDi0  bi0 Þ di1 ðT i 2ð1aÞÞ Di0 ðT i  2ð1  aÞÞÞ  e di1

bi0 ðT i  2ð1  aÞÞ þ bi1

ðT i  2ð1  aÞÞ2 ðT i  2ð1  aÞÞ3 þ bi2 2 3

 :

ð31Þ

Similarly, J Ua will be obtained by replacing ðT i  2ð1  aÞÞ of J La with ðT i  2ð1  aÞÞ. @JLa @J U a ¼ @u ¼ 0; j ¼ 0; 1; 2. From Kuhn–Tucker conditions, we have, @u ij ij Using above relations, we get,

" ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þð1  di Þ 2bi0 ðT i  2ð1  aÞÞ þ

ðDi0  bi0 Þ 2

di1

e2di1 ðT i 2ð1aÞÞ þ

bi1 ðT i  2ð1  aÞÞ2 di1

2bi2 2edi1 ðT i 2ð1aÞÞ bi0 bi1 bi2 ðT i  2ð1  aÞÞ3 þ þ ðdi1 ðT i  2ð1  aÞÞ þ 1Þ þ 3 ððdi1 ðT i  2ð1  aÞÞÞ2 3di1 di1 di1 d2i1 di1 #  Di0 þ bi0 2ðDi0  bi0 Þ di1 ðT i 2ð1aÞÞ 2ðDi0 þ bi0 Þ e  ðT  2ð1  a ÞÞ þ2di1 ðT i  2ð1  aÞÞ þ 2Þ   i 2 di1 di1 di1  2 þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ 2ui0 ðT i  2ð1  aÞÞ þ ui1 ðT i  2ð1  aÞÞ2 þ ðui2 ðT i  2ð1  aÞÞ3 3 ! ðT i  2ð1  aÞÞ 1 di1 ðT i 2ð1aÞÞ ;  2 e 2pi0 ðT i  2ð1  aÞÞ ¼ kð1  di Þai di1 di1 þ

ð32Þ

" ð1  di Þ 2bi1 ðDi0  bi0 Þ 2di1 ðT i 2ð1aÞÞ ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ 2bi0 ðT i  2ð1  aÞÞ þ e ðT i  2ð1  aÞÞ3  2 di1 3 di1 bi1 2bi2 bi2 2edi1 ðT i 2ð1aÞÞ ðT i  2ð1  aÞÞ2 þ bi0 ðT i  2ð1  aÞÞ2  ðT i  2ð1  aÞÞ3 þ ðT i  2ð1  aÞÞ4  2 di1 3di1 2 di1 ! bi1 bi2 Di0 þ bi0  bi0 þ ðdi1 ðT i  2ð1  aÞÞ þ 1Þ þ 2 ððdi1 ðT i  2ð1  aÞÞÞ2 þ 2di1 ðT i  2ð1  aÞÞ þ 2Þ  di1 di1 di1 !# 2 2ðDi0  bi0 Þ di1 ðT i 2ð1aÞÞ ðT i  2ð1  aÞÞ ðT i  2ð1  aÞÞ  e ðT i  2ð1  aÞÞ  2ðDi0 þ bi0 Þ  di1 di1 2  2 ui2 þð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ ui0 ðT i  2ð1  aÞÞ2 þ ðui1 ðT i  2ð1  aÞÞ3 þ ðT i  2ð1  aÞÞ4 3 3 ! i ðT i  2ð1  aÞÞ ðT i  2ð1  aÞÞ2 1 di1 ðT i 2ð1aÞÞ 2 þ  3 e pi0 ðT i  2ð1  aÞÞ ¼ kð1  di Þai  2 2di1 di1 di1 

and

ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ

ð1  di Þ 3 di1

"

2

4bi0 ðT i  2ð1  aÞÞ þ

4bi1 di1 2bi2 di1 ðT i  2ð1  aÞÞ3 þ ðT i  2ð1  aÞÞ5 3 5

þ

2ðDi0  bi0 Þ 2di1 ðT i 2ð1aÞÞ 4bi2 e þ 2bi1 ðT i  2ð1  aÞÞ2  2bi0 di1 ðT i  2ð1  aÞÞ2 þ ðT i  2ð1  aÞÞ3 di1 3

þ

2bi0 di1 bi1 di1 ðT i  2ð1  aÞÞ3  bi2 ðT i  2ð1  aÞÞ4 þ ðT i  2ð1  aÞÞ4 þ 4edi1 ðT i 2ð1aÞÞ 3 2

2

2

ð33Þ

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S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

bi0 bi1 bi2 Di0 þ bi0 þ ðdi1 ðT i  2ð1  aÞÞ þ 1Þ þ 3 ððdi1 ðT i  2ð1  aÞÞÞ2 þ 2di1 ðT i  2ð1  aÞÞ þ 2Þ  di1 d2i1 di1 di1 4ðDi0  bi0 Þedi1 ðT i 2ð1aÞÞ ððT i  2ð1  aÞÞ 

!

1 ððdi1 ðT i  2ð1  aÞÞÞ2 þ 2di1 ðT i  2ð1  aÞÞ þ 2ÞÞ di1 2

ðT i  2ð1  aÞÞ3 di1 2ðDi0 þ bi0 Þð2ðT i  2ð1  aÞÞ  2ðT i  2ð1  aÞÞ2 di1 þ Þ þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ 3   2ui0 ðui1 2ui2 2p  ðT i  2ð1  aÞÞ3 þ ðT i  2ð1  aÞÞ4 þ ðT i  2ð1  aÞÞ5  i0 ðT i  2ð1  aÞÞ3 3 2 5 3 ! 2 3 2ðT i  2ð1  aÞÞ ðT i  2ð1  aÞÞ ðT i  2ð1  aÞÞ 1 di1 ðT i 2ð1aÞÞ þ ¼ kð1  di Þai ; ð34Þ  þ e 3 2 4 3di1 3 di1 di1 di1 " ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þð1  di Þ 2bi0 ðT i þ 2ð1  aÞÞ þ

ðDi0  bi0 Þ 2

di1

e2di1 ðT i þ2ð1aÞÞ þ

bi1 ðT i þ 2ð1  aÞÞ2 di1

2bi2 2edi1 ðT i þ2ð1aÞÞ bi0 bi1 bi2 ðT i þ 2ð1  aÞÞ3 þ þ ðdi1 ðT i þ 2ð1  aÞÞ þ 1Þ þ 3 ððdi1 ðT i þ 2ð1  aÞÞÞ2 3di1 di1 di1 d2i1 di1 #  Di0 þ bi0 2ðDi0  bi0 Þ di1 ðT i þ2ð1aÞÞ 2ðDi0 þ bi0 Þ  þ2di1 ðT i þ 2ð1  aÞÞ þ 2Þ  e  ðT i þ 2ð1  aÞÞ þ ð1=2Þðcui1 ð1  q2 Þ 2 di1 di1 di1   2 þ cui2 þ cui3 q2 Þ 2ui0 ðT i þ 2ð1  aÞÞ þ ui1 ðT i þ 2ð1  aÞÞ2 þ ðui2 ðT i þ 2ð1  aÞÞ3  2pi0 ðT i þ 2ð1  aÞÞ 3 ðT i þ 2ð1  aÞÞ 1 di1 ðT i þ2ð1aÞÞ  2 ðe Þ; ð35Þ ¼ kð1  di Þai ð di1 di1 þ

ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ

" ð1  di Þ 2bi1 ðDi0  bi0 Þ 2di1 ðT i þ2ð1aÞÞ 2bi0 ðT i þ 2ð1  aÞÞ þ e ðT i þ 2ð1  aÞÞ3  2 di1 3 di1

bi1 2bi2 bi2 2edi1 ðT i þ2ð1aÞÞ ðT i þ 2ð1  aÞÞ2 þ bi0 ðT i þ 2ð1  aÞÞ2  ðT i þ 2ð1  aÞÞ3 þ ðT i þ 2ð1  aÞÞ4  2 di1 3di1 2 di1 ! bi1 bi2 Di0 þ bi0 2  bi0 þ ðdi1 ðT i þ 2ð1  aÞÞ þ 1Þ þ 2 ððdi1 ðT i þ 2ð1  aÞÞÞ þ 2di1 ðT i þ 2ð1  aÞÞ þ 2Þ  di1 di1 di1 !# 2ðDi0  bi0 Þ di1 ðT i þ2ð1aÞÞ ðT i þ 2ð1  aÞÞ ðT i þ 2ð1  aÞÞ2 þ ð1=2Þðcui1 ð1  q2 Þ  e ðT i þ 2ð1  aÞÞ  2ðDi0 þ bi0 Þ  di1 di1 2   2 ui2 þ cui2 þ cui3 q2 Þ ui0 ðT i þ 2ð1  aÞÞ2 þ ðui1 ðT i þ 2ð1  aÞÞ3 þ ðT i þ 2ð1  aÞÞ4  pi0 ðT i þ 2ð1  aÞÞ2 3 3 ! 2 ðT i þ 2ð1  aÞÞ ðT i þ 2ð1  aÞÞ 1 di1 ðT i þ2ð1aÞÞ ð36Þ þ  3 e ¼ kð1  di Þai  2 2di1 di1 di1



and

ð1=2Þðhi1 ð1  q1 Þ þ hi2 þ hi3 q1 Þ þ

ð1  di Þ 3 di1

"

2

4bi0 ðT i þ 2ð1  aÞÞ þ

4bi1 di1 2bi2 di1 ðT i þ 2ð1  aÞÞ3 þ ðT i þ 2ð1  aÞÞ5 3 5

2ðDi0  bi0 Þ 2di1 ðT i þ2ð1aÞÞ 4bi2 e þ 2bi1 ðT i þ 2ð1  aÞÞ2  2bi0 di1 ðT i þ 2ð1  aÞÞ2 þ ðT i þ 2ð1  aÞÞ3 di1 3 2

2

2bi0 di1 bi1 di1 ðT i þ 2ð1  aÞÞ3  bi2 ðT i þ 2ð1  aÞÞ4 þ ðT i þ 2ð1  aÞÞ4 þ 4edi1 ðT i þ2ð1aÞÞ 3 2 ! bi0 bi1 bi2 Di0 þ bi0 2  þ ðdi1 ðT i þ 2ð1  aÞÞ þ 1Þ þ 3 ððdi1 ðT i þ 2ð1  aÞÞÞ þ 2di1 ðT i þ 2ð1  aÞÞ þ 2Þ  di1 d2i1 di1 di1

þ

1 ððdi1 ðT i þ 2ð1  aÞÞÞ2 þ 2di1 ðT i þ 2ð1  aÞÞ þ 2ÞÞ di1 # 2 ðT i þ 2ð1  aÞÞ3 di1 2 2ðDi0 þ bi0 Þð2ðT i þ 2ð1  aÞÞ  2ðT i þ 2ð1  aÞÞ di1 þ Þ 3  4ðDi0  bi0 Þedi1 ðT i þ2ð1aÞÞ ððT i þ 2ð1  aÞÞ 

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S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822

  2ui0 ðui1 2ui2 2p 3 4 5 3 ðT i þ 2ð1  aÞÞ þ ðT i þ 2ð1  aÞÞ þ ðT i þ 2ð1  aÞÞ  i0 ðT i þ 2ð1  aÞÞ 3 2 5 3 ! 2ðT i þ 2ð1  aÞÞ ðT i þ 2ð1  aÞÞ2 ðT i þ 2ð1  aÞÞ3 1 di1 ðT i þ2ð1aÞÞ þ 4e  þ : ð37Þ 3 2 3di1 3 di1 di1 di1

þ ð1=2Þðcui1 ð1  q2 Þ þ cui2 þ cui3 q2 Þ ¼ kð1  di Þai

Solving Eqs. (32)–(37), with the help of GRG technique we get, ui0 ; ui1 and ui2 and then find the optimal rate of production, ðU i ðtÞÞ in terms of a. At first, we obtain the value of ui0 ; ui1 and ui2 and the corresponding objective value for a ¼ 1. Then according to FRI method we obtain the value of a for the corresponding objective value. 5.1.2. Model 2 For m = 1, i.e., linear production function, let U i ðtÞ ¼ ui0 þ ui1 t. Putting ui2 ¼ 0 and solving Eqs. (32)–(37), we get, ui0 ; ui1 and then find the optimal rate of linear production function in terms of a. At first, we obtain the value of ui0 and ui1 and the corresponding objective value for a ¼ 1. Then according to FRI method we obtain the value of a for the corresponding objective value. 5.1.3. Model 3 For m = 0, i.e., constant production, i.e., U i ðtÞ ¼ ui0 . As before, putting ui2 ¼ ui1 ¼ 0, and solving Eqs. (32)–(37), we get, ui0 , objective value and the corresponding a according to the FRI-method. 8. Numerical experiments 8.1. Input data We take two items, i.e., n ¼ 2. 8.1.1. Fuzzy input data f2 ¼ about 0.2, i.e., e ¼ about 100, i.e., ðM 1 ; M 2 ; M 3 Þ ¼ ð90; 100; 110Þ; f h1 ¼ about 0.21, i.e., ðh11 ; h12 ; h13 Þ ¼ ð0:2; 0:21; 0:22Þ; h M C u1 ¼ about 1.6, i.e., ðC u11 ; C u12 ; C u13 Þ ¼ ð1:5; 1:6; 1:8Þ; g C u2 ¼ about 1.4, i.e., ðC u21 ; C u22 ; C u23 Þ ¼ ðh21 ; h22 ; h23 Þ ¼ ð0:18; 0:2; 0:21Þ; g f2 ¼ about 5. f1 ¼ about 6 and T ð1:3; 1:4; 1:6Þ, T 8.1.2. Crisp input data d1 Þ ¼ 0:03; Eðc d2 Þ ¼ d10 ¼ 10; d20 ¼ 9; d11 ¼ 1:1; d21 ¼ 1:2; p10 ¼ 15; p20 ¼ 16; r 1 1 ¼ 0:4; r 12 ¼ 0:41; r21 ¼ 0:41; r 22 ¼ 0:42; Eðc 0:01; a1 ¼ 0:3; a2 ¼ 0:6; I10 ¼ 5; I20 ¼ 6; L ¼ 0:1; g ¼ 0:6. 8.2. Results of Model 1 Optimal productions and a for the corresponding optimum costs for Model 1 (i.e., for quadratic production rate) are presented below: H We have gð1Þ ¼ 318:93 ¼ hð1Þ, therefore lðJÞ (defined in 3.2) = 1. So we expect that cost (i.e., J) will become $318.93 with the membership value 1.0. Now we have the following table lðxÞ is the membership of the Fuzzy integral: 8.3. Results of Model 2 Optimal productions and a for the corresponding optimum costs for Model 2 (i.e., for linear production rate) are presented below: H We have gð1Þ ¼ 264:65 ¼ hð1Þ, therefore lðJÞ (defined in 3.2) = 1. So we expect that cost (i.e., J) will become $264.65 with the membership value 1.0. Now we have the following table lðxÞ, the membership of the Fuzzy integral: 8.4. Results of Model 3 Optimal productions and a for the corresponding optimum costs for Model 3 (i.e., for constant production rate) are presented below: H We have gð1Þ ¼ 227:10 ¼ hð1Þ, therefore lðJÞ (defined in 3.2) = 1. So we expect that cost (i.e., J) will become $227.10 with the membership value 1.0. Now we have the following table lðxÞ is the membership of the Fuzzy integral. 9. Discussion Here a multi-item inventory-production model with stochastic defective rate is formulated and solved by GCM, FRI, Kuhn–Tucker condition and GRG methods. In reality different items are not finished at the same time period in a market

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S. Mandal et al. / Applied Mathematical Modelling 34 (2010) 810–822 Table 1 Optimum cost for Model 1. J

Membership-value

u10

u11

u12

u20

u21

u22

280.00 (287.00 304.00 313.00 318.93 323.00 328.00 354.00

0.62 0.70 0.85 0.94 1.00 0.95 0.88 0.67

14.12 14.03 13.87 13.78 13.72 13.67 13.62 13.46

0.68 0.66 0.63 0.61 0.60 0.59 0.58 0.54

0.77 0.74 0.67 0.63 0.60 0.58 0.56 0.49

14.98 14.87 14.75 14.53 14.38 14.21 13.84 13.65

0.64 0.62 0.60 0.58 0.52 0.49 0.45 0.42

0.70 0.68 0.64 0.62 0.59 0.56 0.53 0.52

Table 2 Optimum cost for Model 2. J

Membership-value

u10

u11

u20

u21

234.00 246.00 250.00 258.00 264.65 270.00 277.00 291.00 296.00

0.72 0.81 0.85 0.93 1.00 0.94 0.87 0.73 0.67

15.14 15.14 15.14 15.14 15.13 15.13 15.13 15.13 15.13

0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01

16.15 16.14 16.14 16.14 16.13 16.13 16.13 16.13 16.13

0.04 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02

Table 3 Optimum cost for Model 3. J

Membership-value

u10

u20

199.00 212.00 216.00 223.00 227.10 230.00 237.00 246.00 249.00

0.65 0.82 0.86 0.96 1.00 0.95 0.88 0.77 0.73

12.58 12.56 12.56 12.55 12.55 12.54 12.54 12.53 12.52

14.71 14.69 14.68 14.67 14.66 14.66 14.65 14.64 14.63

and these time periods are uncertain in fuzzy sense. For FRI method the objective values decrease as well as increase with the decrement of membership values. The salesman will have to choose which one will be better for him. For multi-objective model formulation, it is under continuous production control and so the objective function takes different values for different values of time t. Three types of production rate are considered here namely, quadratic (i.e., Model 1), linear (i.e., Model 2) and constant (i.e., Model 3) production rate. The objective value among these three models is minimum for Model 3. So a decision-maker will always want to take the constant production. Here, the optimal costs with their corresponding membership functions are presented. Tables 1–3 give the values of total cost for Models 1–3, respectively. As the time period is fuzzy, the values of total cost are also imprecise and are obtained with the corresponding membership functions. 10. Conclusion A multi-objective and multi-item defective dynamic system with a resource constraint with different fuzzy time periods has been solved for the first time via GCM, FRI, Kuhn–Tucker condition and GRG methods. For the first time, a dynamic production inventory model with imprecise time periods under space constraint has been formulated and solved. The formulation and analysis presented here can be extended to other production inventory problems with different types of demand, deterioration, price discount, shortage cost, budget constraint, etc. Acknowledgements The Author thanks UGC for Minor Project (F.No.: PSW-062/07-08, SNO: 86023), Govt. of India.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

F. Harris, Operations and Cost Factory Management Services, A.W. Shaw Co., Chicago, 1913. E. Naddor, Inventory Systems, John Wiley, New York, 1966. G. Hadley, T.M. Whitin, An Optimal Final Inventory Model, Prentice Hall, 1975. J.Q. Hu, R. Loulon, Multi-product production/inventory control under random demands, IEEE Trans. Autom. Contr. 40 (1995) 350–355. D.E. Kirk, Optimal Control Theory. An Introduction, Prentice Hall, New Jersey, 1970. J.Q. Hu, X. Dong, Optimal control for systems with deterministic production cycles, IEEE Trans. Autom. Contr. 40 (1995) 782–787. S. Minner, R. Kleber, Optimal control of production and remanufacturing in a simple recovery model with linear cost functions, Spektrum 23 (2001) 3– 24. Y. Feng, H. Yan, Optimal production control in a discrete manufacturing system with unreliable machines and random demands, IEEE Trans. Autom. Contr. 45 (12) (2000) 2280–2296. Q. Zhang, G. George, E. Boukas, Optimal control of a marketing system, IEEE Trans. Autom. Contr. 46 (3) (2001) 416–427. M.D.S. Aliyu, A.A. Andhani, Multi-item-multi-plant inventory control of production systems with shortages/backorders, Int. J. Syst. Sci. 30 (1999) 533– 539. A.K. Bhunia, M. Maiti, A deterministic inventory replenishment problem for deteriorating items with time-dependent demand and shortages for the finite time horizon, Opsearch 34 (1997) 51–61. B.C. Giri, K.S. Chaudhuri, An economic production lot-size model with shortages and time dependent demand, IMA J. Math. Appl. Business Indust. 10 (1999) 203–211. A. Andijani, M. Aldajani, Analysis of deteriorating inventory/production systems using a linear quadratic regulator, Eur. J. Operat. Res. 106 (1998) 82– 89. S.K. Goyal, B.C. Giri, Recent trends modelling of deteriorating inventory, Eur. J. Operat. Res. 134 (2001) 1–16. M. Athans, P.L. Falb, Optimal Control: An Introduction to the Theory and its Applications, McGraw-Hill, New York, 1966. S. Axsater, K. Rosling, Multi-Production-Inventory Control: Material Requirements Planning or ReorderPoint Policies, Eur. J. Operat. Res. 75 (1994) 405–412. R. Bellman, R. Kalaba, Dynamic Programming and Modern Control Theory, Academy Press, New York, 1965. F. Blanchini, S. Miani, R. Pesenti, P. Rinaldi, Control policies for multi-inventory systems with uncertain demand, in: Proceedings of the IEEE Conference on Decision and Control, Oriando, FL, USA, 2001. M. Bounkel, L. Tadj, Y. Benhadid, Optimal control of a production system with inventory-level-dependent demand, Appl. Math. E – Notes 5 (2004) 36– 43. M.S. Branicky, V.S. Borkar, S.K. Mitter, A unified framework for hybrid control: model and optimal control theory, IEEE Trans. Autom. Contr. 43 (1) (1998) 31–45. G. Kiesmuller, S. Minner, R. Kleber, Optimal control of a one product recovery system with backlogging, IMA J. Math. Appl. Business Indust. 11 (2000) 189–207. R. Kleber, S. Minner, G. Kiesmuller, A continuous time inventory model for a product recovery system with multiple options, Int. J. Product. Econ. 79 (2002) 121–141. Y. Kuo, Optimal adaptive control policy for joint machine maintenance and product quality control, Eur. J. Operat. Res. 171 (2) (2006) 586–597. J. Bertrand, Analysis of production control system for a diffusion department, Int. J. Syst. Sci. 11 (1980) 589–606. S. Axsater, Control theory concepts in production – inventory control, Int. J. Syst. Sci. 16 (2) (1985) 161–169. T.K. Datta, A.K. Pal, Order level inventory system with power demand pattern for items with variable rate of deterioration, Ind. J. Pure Appl. Math. (1988) 1043–1053. B.N. Mandal, S. Phausdar, A note on inventory model with stock dependent consumption rate, Opsearch 26, 43–46. K. Maity, M. Maiti, Production inventory system for deteriorating multi-item with inventory-dependent dynamic demands under inflation and discounting, Tamsui Oxford J. Manage. Sci. 21 (2005) 1–18. D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1988. D. Dubois, H. Prade, System of linear fuzzy constraints, Purdue University Electrical Engineering Technique Report TR-EE 78-13. B. Liu, Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Trans. Fuzzy Syst. 10 (2002) 445–450. B. Liu, K.B. Iwamura, Chance constraint programming with fuzzy parameters, Fuzzy Sets Syst. 94 (1998) 227–237. B. Liu, K.B. Iwamura, A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets Syst. 100 (1998) 229–233. K. Maity, M. Maiti, Possibility and necessity constraints and their defuzzification – a multi-item production-inventory scenario via optimal control theory, Eur. J. Operat. Res. 177 (2007) 882–896. Hsien-Chung Wu, The Fuzzy Riemann Integral and its Numerical Integration, Fuzzy Sets Syst. 110 (2000) 1–25.