A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach

Engineering Applications of Artificial Intelligence ] (]]]]) ]]]–]]]

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach Partha Guchhait a,n, Manas Kumar Maiti b, Manoranjan Maiti a a b

Department of Applied Mathematics, Vidyasagar University, Midnapore, Paschim-Medinipur, West Bengal 721102, India Department of Mathematics, Mahishadal Raj College, Mahishadal, Purba-Medinipur, West Bengal 721628, India

a r t i c l e i n f o

abstract

Article history: Received 31 August 2011 Received in revised form 13 July 2012 Accepted 19 October 2012

In this paper, a production inventory model, specially for a newly launched product, is developed incorporating fuzzy production rate in an imperfect production process. Produced defective units are repaired and are sold as fresh units. It is assumed that demand coefficients and lifetime of the product are also fuzzy in nature. To boost the demand, manufacturer offers a fixed price discount period at the beginning of each cycle. Demand also depends on unit selling price. As production rate and demand are fuzzy, the model is formulated using fuzzy differential equation and the corresponding inventory costs and components are calculated using fuzzy Riemann-integration. a-cut of total profit from the planning horizon is obtained. A modified Genetic Algorithm (GA) with varying population size is used to optimize the profit function. Fuzzy preference ordering (FPO) on intervals is used to compare the intervals in determining fitness of a solution. This algorithm is named as Interval Compared Genetic Algorithm (ICGA). The present model is also solved using real coded GA (RCGA) and Multi-objective GA (MOGA). Another approach of interval comparison–order relations of intervals (ORI) for maximization problems is also used with all the above heuristics to solve the model and results are compared with those are obtained using FPO on intervals. Numerical examples are used to illustrate the model as well as to compare the efficiency of different approaches for solving the model. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Fuzzy production Fuzzy demand Fuzzy differential equation Fuzzy Riemann-integration Fuzzy preference ordering Interval Compared Genetic Algorithm

1. Introduction Several researchers of inventory control problem developed their production inventory models with fixed production rate. But production of an item in any manufacturing organization deeply depends on efficiency, effectiveness of the system, i.e., quality of the process output, inventory turnover ratio and so many factors related to the production process, which leads to uncertainty/ impreciseness in any production process. Again due to globalization of market and introduction of multinationals in different developing countries, there is a stiff competition among different companies over the globe for marketing their product. As a result, very frequently they change their product specifications with new features and names. So for these types of products, sufficient past data are not available for the estimation of important inventory parameters like demand, production rate, etc. It is very difficult to estimate these parameters as random numbers because

n

Corresponding author. Tel.: þ91 9434385976. E-mail addresses: [email protected] (P. Guchhait), [email protected] (M. Kumar Maiti), [email protected] (M. Maiti).

estimation of a random parameter requires sufficient amount of past data. As fuzzy estimations are made using experts’ opinion, it is better to estimate parameters like production and demand coefficients using fuzzy numbers to reduce the error. Although a considerable number of research papers have already been published incorporating imprecise inventory parameters (Wee ¨ et al., 2009; Ryu and Yucesan, 2010; Maiti, 2011), none has considered fuzzy production rate in any production inventory model. But it is more appropriate to represent a manufacturing system. Keeping in mind the above-mentioned factor, here attention has been paid to develop an economic production quantity (EPQ) model incorporating fuzzy production rate. Demand has been always one of the most effective factors in the decisions relating to economic ordered quantity (EOQ) model as well as EPQ model. Due to this reason, various formations of consumption tendency have been studied by inventory control practitioners, such as constant demand (Wee et al., 2009), selling price dependent demand (Ouyang et al., 2009), advertisement dependent demand (Maiti and Maiti, 2006), customer credit period dependent demand (Jaggi et al., 2008; Maiti, 2011), seasonal demand (Banerjee and Sharma, 2010), etc. Recently, You et al. (2010) developed an inventory model incorporating trial period

0952-1976/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engappai.2012.10.017

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

2

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dependent demand. All of them developed their models in crisp environment, i.e., demand coefficients are considered as crisp number. But, as discussed earlier, it is better to estimate demand coefficients with fuzzy numbers. In the present market situation, it is observed that some manufacturers offer price discount, specially for newly launched products for a certain time period at the beginning of each cycle. As a result, demand increases automatically due to the low unit price. After that specified period, the manufacturer withdraws the additional discount and thus unit price increases. By this process, demand increases due to the fact that some customers have already accustomed with the product during the price discount period and do not switch over to other products though price discount is withdrawn. This process of boosting a product is commonly practiced by different manufacturers specially when a product is newly launched in the market. Again, though offering of price discount boost the demand of an item, nature of demand is always fuzzy in nature. In the literature only, Pal et al. (2009) addressed a price discount inventory model. Till now none has considered price discounted fuzzy demand in an EPQ model. The presence of fuzzy demand as well as fuzzy production rate leads to fuzzy differential equation of instantaneous state of inventory level. Till now fuzzy differential equation is little used to solve fuzzy inventory models though the topics on fuzzy differential equations have been rapidly growing in the recent years. The first impetus on solving fuzzy differential equation was made by Kandel and Byatt (1978). An extended version of their work had been published after 2 years (Kandel and Byatt 1978). After that different approaches have been made by several authors to solve fuzzy differential equations (Kaleva, 1987; Buckley and Feuring, 2000; Vorobiev and Seikkala, 2002; Chalco-Cano and Roman-Flores, 2009). Again inventory models are normally developed with infinite lifetime for products. In reality lifetime of a product (i.e., duration of time for which demand of the item exits compare to other competitive items) rapidly changes due to several factors- innovation of new technology, introduction of new features to the item, environmental effect, etc., so planning horizon of the EPQ model of an item is finite and fuzzy/random in nature. Few research papers have already been published incorporating this assumption (cf. Pal et al., 2009; Roy et al., 2009). In this paper, an EPQ model is presented with fuzzy production rate and fuzzy demand in an imperfect production process, i.e., not all produced units are of perfect quality. In each cycle, after the end of production process, defective units are repaired and are sold as fresh units. Demand depends on unit selling price and price discount period offered by the manufacturer cum retailer. After the discount period demand depends only on unit selling price. Also it is assumed that the planning horizon of the model is imprecise in nature, which leads to the imprecise constraint— sum of all cycle lengths is less than the length of imprecise planning horizon. For any feasible solution the constraint should hold well with at least some possibility/necessity b (Zadeh, 1978; Dubois and Prade, 1980; Liu and Iwamura, 1998). Two models are developed depending on the possibility or necessity measure of the fuzzy constraint. Fuzzy differential equation (Buckley and Feuring, 2000) and fuzzy Riemann integration (Wu, 2000) are used to develop the mathematical formulation of the model and a-cut of the total profit is derived, which is an interval. Since there is no exact method for solving an optimization problem with interval objective function, here a-cut of total profit is optimized using a heuristic. For different values of a, results are obtained and tabulated/plotted to find the nature (membership function) of the profit. Here time duration of production ðt 1 Þ, discount period (t0) and mark up of unit selling price ðm1 Þ are decision variables. For illustration, two numerical examples are used and

e ¼ ða1 ,a2 ,a3 Þ. Fig. 1. Triangular fuzzy number A

results are obtained. Here, although TFN is used for fuzzy parameters, the solution methodology is quite general and can be used for and type of fuzzy number. There are some research works on inventory control problems, where interval valued objective function is optimized. Maiti and Maiti (2006) developed an inventory model, where interval valued objective function was transformed to an equivalent multi-objective problem following an interval comparison approach (depending on left, right values of the interval numbers) proposed by Ishibuchi and Tanaka (1990) and solved using a MOGA. Gupta et al. (2009) use RBS process in a RCGA for solving an inventory model with interval valued inventory costs. They used order relations of intervals (ORI) for maximization problems (proposed by Mahato and Bhunia, 2006) for ranking the chromosomes, where center and width of intervals were used for comparison. Bera et al. (in press) solved a fuzzy inventory model, where fuzzy parameters were replaced by equivalent interval numbers (following Grzegorzewski, 2002) and objective function has been transformed to different equivalent multi-objective problems using different interval comparison approaches and solved using a MOGA. All these research papers used different approaches to compare interval objectives to find optimal decisions. Merits and demerits of different approaches on comparison of interval numbers have recently been discussed by Sengupta and Pal (2009). According to them FPO on intervals is the best approach for comparison of interval numbers. Due to this reason, in this research paper a modified GA with varying population size is used which can deal with interval objective function, where FPO on intervals is used to compare the intervals in determining fitness of a solution. This is named as ICGA and is used to solve the models. The models are also solved following RCGA (Gupta et al., 2009) and MOGA (Bera et al., in press) using FPO on intervals. At the same time ORI is also used with all the approaches- RCGA, MOGA and ICGA and results are compared with those obtained by the three approaches with FPO on intervals.

2. Preliminaries f Let Ff 1 and F 2 be two fuzzy numbers in R with membership functions m e ðxÞ and m e ðxÞ respectively. Then according to Dubois F1

F2

and Prade (1980) and Zadeh (1978)

f posðFf 1 nF 2 Þ ¼ supfminðm e ðxÞ, m e ðyÞÞ,x,y A R,xnyg F1

F2

ð1Þ

where pos represents possibility, n is any one of the relations 4 , o , ¼ , r , Z and R represents set of real numbers f f f nesðFf 1 nF 2 Þ ¼ 1posðF 1 nF 2 Þ

ð2Þ

where nes represents necessity. Similarly possibility and necessity measures of Ff 1 with respect f f to Ff 2 are denoted by P e ðF 1 Þ and N e ðF 1 Þ respectively and are F2

F2

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

P. Guchhait et al. / Engineering Applications of Artificial Intelligence ] (]]]]) ]]]–]]]

defined as

P e ðFf1 Þ ¼ supfminðm e ðxÞ, m e ðxÞÞ,x A Rg

ð3Þ

N e ðFf 1 Þ ¼ minfsupðm e ðxÞ,1m e ðxÞÞ,x A Rg

ð4Þ

F2

F1

F2

F2

F1

f1 , X f2 . . . , X fn D R and Fuzzy Extension Principle (Zadeh, 1978): If X e ¼ f ðX f1 , X f2 , . . . , X fn Þ, where f : R  R    R-R is a binary X e is defined as operation then membership function me of X X

meX ðzÞ ¼ supfminðme ðx1 Þ, me ðx2 Þ, . . . , me ðxn ÞÞ,x1 ,x2 , . . . ,xn A R X X X

For each z A R,

1

2

f : Rn -Rn is a continuous function, then e f : FðRn Þ-FðRn Þ is well defined function and (see Roman-Flores et al., 2001) e f ðuÞ½a ¼ f ð½ua Þ,

F2

n

and z ¼ f ðx1 ,x2 , . . . ,xn Þg

3

8a A ½0,1,8u A FðRn Þ,

ð8Þ

where f ðAÞ ¼ ff ðaÞ=a A Ag. Interval number: An interval number A DR is defined as follows: A ¼ ½AL ,AR  ¼ fx9AL r x r AR g Interval A, alternatively, is represented by its mid-point m(A) and the half-width w(A) as A ¼ /mðAÞ,wðAÞS, where mðAÞ ¼ ðAL þ AR Þ=2 and wðAÞ ¼ ðAR AL Þ=2.

ð5Þ

e ¼ ða1 ,a2 ,a3 Þ (cf. Fig. 1) has Triangular fuzzy number (TFN): A TFN A three parameters a1 ,a2 ,a3 , where a1 o a2 o a3 and is characterized by the membership function me, given by A 8 xa 1 > for a rx r a > 1 2 > a a > < 2 1 ð6Þ meðxÞ ¼ a3 x for a rx r a A 2 3 > > > a3 a2 > : 0 otherwise: e ¼ ðb1 ,b2 Þ (cf. Fig. 2) has two Linear fuzzy number (LFN): A LFN B parameters b1 ,b2 , where b1 ob2 and is characterized by the membership function me, given by b 8 0 for x rb1 > > > < xb 1 ð7Þ meB ðxÞ ¼ b b for b1 r x rb2 > 2 1 > > :1 for x Zb2 According to above definitions following lemmas can easily be derived. Lemma 1. If e f ¼ ðf 1 ,f 2 ,f 3 Þ be a TFN with 0 o f 1 and b is a crisp number then nesðe f 4bÞ Z a iff ðbf 1 Þ=ðf 2 f 1 Þ r1a. Lemma 2. If e f ¼ ðf 1 ,f 2 ,f 3 Þ be a TFN and g be a crisp number then 8 gf 1 > > if f 2 Z g Z f 1 > > > < f 2 f 1 Ng ðe f Þ ¼ Pg ðe f Þ ¼ f 3 g > if f 2 r g r f 3 > > f f > > : 3 2 0 otherwise Lemma 3. If e f ¼ ðf 1 ,f 2 Þ be a LFN and c be a crisp number then 8 0 if f 1 Z c > > > < cf 1 if f 2 Z c Z f 1 f Þ ¼ Pc ðe fÞ¼ Nc ðe > f f > > 2 1 : 1 otherwise e a and is a-cut set: a-cut of a fuzzy number Ae in R is denoted by A½ defined as e a ¼ fx A R=m ðxÞ Z ag A½ e A

Let F(X) be the space of all compact and convex fuzzy sets on X. If

2.1. Comparison of intervals 2.1.1. Fuzzy preference ordering (FPO) of intervals There are several approaches in comparison of interval numbers. A detailed discussion about the merits and demerits of these methods is made by Sengupta and Pal (2009) in ‘Fuzzy Preference Ordering of Interval Numbers in Decision Problems’. According to Sengupta and Pal, fuzzy preference ordering scheme gives a complete interval ranking method and defines different sets of ‘pairs of intervals’ for which there exist strict and fuzzy preference relation and indifference between the interval-attributes. They used following assumptions for the maximization problem: 1. More profit is better than less profit. 2. More certainty is better than less certainty. 3. If more profit is associated with more uncertainty, a decision maker (DM) undergoes a trade-off between the two. 4. To a pessimistic DM, Assumption 2 is somewhat more important than Assumption 1 (obviously to an optimistic DM, Assumption 1 is somewhat more important than Assumption 2). According to this set of assumptions they ordered any pair of intervals A and B, as (A,B) if mðAÞ rmðBÞ and classified into two sets S1 and S2 as follows: 1. ðA,BÞ A S1 if wðAÞ Z wðBÞ. 2. ðA,BÞ A S2 if wðAÞ o wðBÞ. Then for a maximization problem:

 For ðA,BÞ A S1 unless A and B are identical, B is always the best choice.

 For ðA,BÞ A S2 fuzzy preference between A and B may be constructed. In order to develop fuzzy preference between the pair (A,B) in S2, a fuzzy set B0 as rejection of B in S2 is defined as B0 ¼ fðX,BÞ A S2 =X ¼ ½X L ,X R  ¼ /mðXÞ,wðXÞS, mðXÞ rmðBÞ, wðXÞ o wðBÞg with membership function mB0 ðX,BÞ (mB0 given by 8 1 > >   > < mðXÞðBL þwðXÞÞ mB0 ðX,BÞ ¼ max 0, mðBÞðBL þwðXÞÞ > > > : 0

being a function S2 -½0,1), if mðXÞ ¼ mðBÞ if mðBÞ Z mðXÞ Z ðBL þwðXÞÞ otherwise

According to above definition following conclusions are obvious (Fig. 3):

 If mB ðX,BÞ ¼ 1, then B is definitely rejected compared to X.  If mB ðX,BÞ ¼ 0, then B is definitely accepted compared to X.  If mB ðX,BÞ A ½0,1, then B is accepted/rejected according to DM’s 0 0

0

e ¼ ðb1 ,b2 Þ. Fig. 2. Linear fuzzy number B

preference.

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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2.1.2. Order relations of intervals (ORI) for maximization problems The definitions of order relations of intervals for maximization problems was developed by Mahato and Bhunia (2006) in the context of optimistic and pessimistic decision makers point of view. These definitions are as follows: Optimistic decision-making: For maximization problems, they defined the order relation Z omax between the intervals A ¼ ½AL ,AR  and B ¼ ½BL ,BR  as

 A Z omax B iff AR ZBR .  A 4 omax B iff AR ZBR and A aB. This implies that, A is superior to B and optimistic decision maker accepts the profit interval A. Pessimistic decision-making: For maximization problems, they defined the order relation Z pmax between the intervals A ¼ ½AL ,AR   /mðAÞ,wðAÞS and B ¼ ½BL ,BR   ½mðBÞ,wðBÞ as:

 If none of A and B is contained in other A Z pmax B if mðAÞ ZmðBÞ.  If one of A and B is contained in other then A 4 pmax B iff mðAÞ Z mðbÞ and wðAÞ o wðBÞ. In this case if mðAÞ Z mðBÞ and wðAÞ 4 wðBÞ then pessimistic decision cannot be taken. Here, the optimistic decision may be considered. Gupta et al. (2009) used this order relation of intervals from pessimistic decision maker’s point of view for solving their inventory model. Due to this reason, ORI is also used in this research paper according to pessimistic decision maker’s point of view.

Depending upon the limits of interval two types of fuzzy Riemann integral have been defined by Wu (2000). Fuzzy Riemann integral of type-I (Wu, 2000): Let e f ðxÞ be a closed and bounded fuzzy valued function on ½a,b and ½f L ða,xÞ,f R ða,xÞ be a-cut of ef ðxÞ 8x A ½a,b. If f L ða,xÞ and f R ða,xÞ are Riemann integrable Rb f ðxÞ dx is a closed fuzzy on ½a,b 8a, the fuzzy Riemann integral a e number and its a-cut set is given by ! "Z # Z b Z b b e f ðxÞ dx ½a ¼ f ða,xÞ dx, f ða,xÞ dx a

L

L

if bL ðaÞ 4 aU ðaÞ if bL ðaÞ r aU ðaÞ

Case2: If e f ðxÞ is non-positive and f L ða,xÞ and f R ða,xÞ are Riemann integrable on ½aR ðaÞ,bL ðaÞ and ½aL ðaÞ,bU ðaÞ respectively Re be f ðe x Þ de x is a closed fuzzy number 8a, the fuzzy Riemann integral e a and its a-cut set is given by i 8 hR R b ðaÞ 0 1 b ðaÞ > Z e < aLUðaÞ f L ða,xÞ dx, aUL ðaÞ f R ða,xÞ dx b e @ i f ðe x Þ de x A½a ¼ hR b ðaÞ > e a : a UðaÞ f L ða,xÞ dx,0 L

if bL ðaÞ 4 aU ðaÞ if bL ðaÞ r aU ðaÞ

2.3. Fuzzy differential equation (Buckley and Feuring, 2000) Considered the first-order ordinary differential equation dY ¼ f ðt,Y,KÞ, dt

Yð0Þ ¼ C,

ð9Þ

where K ¼ ðK 1 ,K 2 , . . . K n Þ is a vector of constants, and t is in some interval (closed and bounded) I which contains zero. Let (9) has a unique solution Y ¼ gðt,K,CÞ

for t A I, K A K  Rn , c  R

ð10Þ

e be another TFN, e 2, . . . K e n Þ is a vector of TFNs and C e ¼ ðK e 1,K When K then (9) reduces to fuzzy differential equation e dY e ,K e Þ, ¼ f ðt, Y dt

e e ð0Þ ¼ C Y

ð11Þ

assuming that derivative (Buckley and Feuring, 2000) of the e ðtÞ exists. Then according to Buckley unknown fuzzy function Y and Feuring (2000)

2.2. Fuzzy Riemann integral

a

and its a-cut set is given by i 8 hR R b ðaÞ 0 1 b ðaÞ > Z e < aUL ðaÞ f L ða,xÞ dx, aLUðaÞ f R ða,xÞ dx b e @ i f ðe x Þ de x A½a ¼ h R b ðaÞ > e a : 0, a UðaÞ f R ða,xÞ dx

a

R

eÞ be a Fuzzy Riemann Integral of type-II (Wu, 2000): Let e f ðx bounded and closed fuzzy valued function defined on the closed e and e e, b fuzzy interval ½a f ðxÞ be induced by e f ðe x Þ. ½f L ða,xÞ,f R ða,xÞ be a-cut of ef ðxÞ and ef ðxÞ is either nonnegative or non-positive. Case1: If e f ðxÞ is nonnegative and f ða,xÞ and f ða,xÞ are L

eÞ e ðtÞ ¼ gðt, K e ,C Y

ð12Þ

e ðtÞ½a ¼ ½Y L ðt, aÞ,Y R ðt, aÞ satisfies the is solution of (11) if its a-cut Y following conditions: 8 dY L ðt, aÞ dY R ðt, aÞ > > and are continuous on I  ½0,1 > > > dt dt > > > > dY L ðt, aÞ > > is an increasing function of a for each t A I < dt ð13Þ dY R ðt, aÞ > > > is a decreasing function of a for each t A I > > dt > > > > dY L ðt,1Þ dY R ðt,1Þ > > r , 8t A I : dt dt e ðtÞ is obtained using fuzzy where membership function of Y extension principle (5).

R

Riemann integrable on ½aR ðaÞ,bL ðaÞ and ½aL ðaÞ,bU ðaÞ respectively Re be f ðe x Þ de x is a closed fuzzy number 8a, the fuzzy Riemann integral e a

3. Notations and assumptions for the model 3.1. Notations To develop the proposed model, the notations adopted in this paper are as follows: ch cp cpRM c3 Te i N t1i

Fig. 3. Membership function of rejection of B for the pair of intervals (X,B) in S2 .

holding cost per unit per unit time. production cost per unit. remanufacturing cost per unit. setup cost in each cycle. duration of ith cycle having a-cut Te i ðaÞ ¼ ½T iL ðaÞ,T iR ðaÞ. number of cycles during the planning horizon. duration of production in ith cycle. It increases with i (as demand increases in each cycle) and is of the form t 1i ¼ t 1 þ ði1Þl, where t1 (duration of production in first cycle) and l (any parameter) are decision variables.

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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t0 e t 2i e Q 1i e Q 2i e Q 3i ei ðtÞ q sp

e Z

price discount period in each cycle with t 0 ot 1 . remanufacturing time period in ith cycle with a-cut et 2i ½a ¼ ½t2iL ðaÞ,t 2iR ðaÞ. e ½a ¼ inventory level in ith cycle at t0 with a-cut Q 1i ½Q 1iL ðaÞ,Q 1iR ðaÞ. e ½a ¼ inventory level in ith cycle at t1i with a-cut Q 2i ½Q 2iL ðaÞ,Q 2iR ðaÞ. inventory level in ith cycle at t 1i þ e t 2i with a-cut Qe 3i ½a ¼ ½Q 3iL ðaÞ,Q 3iR ðaÞ. inventory level at any time t in ith cycle with a-cut qei ðtÞ½a ¼ ½qiL ðaÞ,qiR ðaÞ. selling price in each cycle which is a mark-up of production cost cp. m1, m2 are mark-ups during price discount period and normal period respectively, i.e. ( m1 cp for 0 o t rt 0 , m1 4 1 sp ¼ m2 cp for t 0 o t r Te i , m2 4 m1 total profit during the planning horizon with a-cut ½Z L ðaÞ,Z R ðaÞ.

3.2. Assumptions In addition, the following assumptions are imposed:

5

the item is met from produced fresh units. After the interval of time t1i, production stops and remanufacturing of produced defective units starts. It takes a time interval e t 2i to remanufacture that units. As production rate is fuzzy in nature total produced defective units are obviously fuzzy quantity so, remanufacturing time is also fuzzy in nature. Remanufactured units are sold as fresh units. After the time duration Tei , production for next cycle e , e 11 =ðm1 cp Þg ¼ d starts. According to the assumptions and letting D 1 g g g e e e e e f A 11 =ðm1 cp Þ ¼ a 1 , D 11 =ðm2 cp Þ ¼ d 2 , A 11 =ðm2 cp Þ ¼ a2 , change of inventory level can be represented by the following differential equation: 8 bmði1Þt 0 bmt f a e ½d > rK e  for 0 o t rt 0 1 f 1e > > > > bmit f < e 0 f r K ½d2 a  for t 0 o t r t 1i ei ðtÞ dq 2e ð14Þ ¼ bmit 0 f a > K 11 ½d dt f e  for t 1i ot r t 1i þ e t 2i > 2 2 > > > : ½d f a febmit0  for t þ e t o t r Te 2

2

1i

2i

i

e , q e ei ð0Þ ¼ 0, q e i ðt 0 Þ ¼ Q with boundary conditions q 1i e i ðt 1i Þ ¼ Q 2i and e ,d e ,a e . Here a-cuts of d e i ðt 1i þ e e2 are respectively q t 2i Þ ¼ Q 1 2 e 1 and a 3i e ½a ¼ ½d ðaÞ,d ðaÞ, d e ½a ¼ ½d ðaÞ,d ðaÞ, a e1 ½a ¼ ½a1L ðaÞ,a1R ðaÞ d 1 1L 1R 2 2L 2R e2 ½a ¼ ½a2L ðaÞ,a2R ðaÞ. and a Solving above differential equation (according to Buckley and ei ðtÞ is obtained and given by Feuring, 2000), a-cut of the solution q e i ðtÞ½a ¼ ½qiL ða,tÞ,qiR ða,tÞ, q

ð15Þ

where e is the production rate which is assumed as a triangular fuzzy 1. K e ¼ ðK 1 ,K 2 ,K 3 Þ) with a-cut K e ½a ¼ ½K L ðaÞ,K R ðaÞ. number (K 2. r is the reliability of production process. 3. K11 is remanufacturing rate. e is the planning horizon of the model which is also assumed as 4. H e ¼ ðH1 ,H2 ,H3 Þ) with a-cutH½ e a ¼ a triangular fuzzy number (H ½HL ðaÞ,HR ðaÞ. e i is the demand of the item in ith cycle which increases with 5. D time during price discount period and decreases with selling price. But after discount period, it depends only on selling price and is of the form: 8 e 11 ebmði1Þt0 ebmt e 11 A > D > > for 0 ot r t 0 > < ðm1 cp Þg e i ðtÞ ¼ D e 11 ebmit0 e 11 A > >D > > for t 0 ot r Te i , : ðm2 cp Þg e 11 are estimated from expert’s opinion. Here e 11 and A where D e 11 ð ¼ ðD111 ,D112 ,D113 ÞÞ m ¼ m2 m1 . It is also assumed that D e e 11 ½a ¼ and A 11 ð ¼ ðA111 ,A112 ,A113 ÞÞ are TFNs with a-cut D e 11 ½a ¼ ½A11L ðaÞ,A11R ðaÞ respectively. b, g ½D11L ðaÞ,D11R ðaÞ and A are the parameters so chosen to best fit the demand function and g is called price elasticity of the demand function. 6. b1 and b2 are the confidence level for fuzzy constraint on the whole planning horizon. Symbols is used on the top of the above notations to represent fuzzy parameters. Here t0, t1 l and m1 are decision variables.

4. Mathematical formulation of the model Here it is assumed that the life time of the product, i.e., e is imprecise in nature. N full planning horizon of the model, H, e and length of each cycle is denoted cycles are completed during H by Te (ith cycle). As such, only constraint of the problem is PN i e e i ¼ 1 T i r H. At the beginning of ith cycle production starts and continues for an interval of time t1i. During that time demand of

8 a1L bmði1Þt0 > e ð1ebmt Þ ðrK L d1R Þt þ > > bm > > < bmit 0 ðtt 0 Þ qiL ða,tÞ ¼ Q 1iL þ ðrK L d2R Þðtt 0 Þ þa2L e > bmit0 > ðtt 1i Þ > Q 2iL þ ðK 11 d2R Þðtt 1i Þ þ a2L e > > : Q 3iL d2R ðtt 1i t 2iR Þ þ a2L ebmit0 ðtt 1i t 2iR Þ

for 0 o t r t 0 for t 0 o t r t 1i for t 1i o t r t 1i þ e t 2i for t 1i þ e t 2i o t r Te i

ð16Þ 8 a1R bmði1Þt0 > ð1ebmt Þ > > ðrK R d1L Þt þ bme > > < bmit0 ðtt 0 Þ qiR ða,tÞ ¼ Q 1iR þ ðrK R d2L Þðtt 0 Þ þ a2R e > bmit0 > Q þ ðK d Þðtt Þ þ a e ðtt 1i Þ 11 2L 2R > 1i 2iR > > : Q 3iR d2L ðtt 1i t 2iL Þ þ a2R ebmit0 ðtt 1i t 2iL Þ

for 0 o t r t 0 for t 0 o t r t 1i for t 1i o t r t 1i þ e t 2i for t 1i þ e t 2i o t r Te i

ð17Þ 9 a1L bmði1Þt0 e ð1ebmt0 Þ > = bm a1R bmði1Þt0 ; e ð1ebmt0 Þ > Q 1iR ¼ ðrK R d1L Þt 0 þ bm Q 1iL ¼ ðrK L d1R Þt 0 þ

Q 2iL ¼ Q 1iL þ ðrK L d2R Þðt 1i t 0 Þ þ a2L ebmit0 ðt 1i t 0 Þ Q 2iR ¼ Q 1iR þðrK R d2L Þðt 1i t 0 Þ þ a2R ebmit0 ðt 1i t 0 Þ Q 3iL ¼ Q 2iL þ ðK 11 d2R Þt 2iL þ a2L ebmit0 t 2iL

e Q 3i

) ð19Þ

)

Q 3iR ¼ Q 2iR þðK 11 d2L Þt 2iR þa2R ebmit0 t 2iR t 2i þ Te i ¼ t 1i þ e

ð18Þ

9 > > > > > > > > =

f a bmit 0 d 2 f 2e Q 3iL T iL ¼ t 1i þt 2iL þ d2R a2L ebmit0 > > > > > > Q 3iR > > T iR ¼ t 1i þ t 2iR þ ; bmit 0 d2L a2R e

ð20Þ

ð21Þ

ei ðtÞ½a ¼ ½qiL ða,tÞ,qiR ða,tÞ satisfies all the four conditions of Here q ei ðtÞ½a is a valid solution of (14). (13) in the interval ½0, Te i , so q Ti g i ¼ c Re e i ðtÞdt ¼ Holding cost during ith cycle is ch HOC h 0 q e e e e ch ½I 1 þ I 2 þ I 3 þ I 4  (say), where eI 1 ¼

Z

t0 0

  e1 bmði1Þt e Þt þ a e d 0 e ðr K ð1ebmt Þ dt 1 bm

ð22Þ

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P. Guchhait et al. / Engineering Applications of Artificial Intelligence ] (]]]]) ]]]–]]]

eI 2 ¼

Z

t1i t0

eI 3 ¼

Z

e Þðtt Þ þ a e þðr K e d e2 ebmit0 ðtt 0 Þ dt ½Q 2 0 1i

t1i þe t 2i t1i

eI 4 ¼

Z e T t 2i t1i þe

ð23Þ

f ½a ¼ ½cp SP ðaÞ,cp SP ðaÞ, where cp SP i iL iR ) SP iL ðaÞ ¼ m1 SR1L ðaÞ þ m2 SR2L ðaÞ

ð34Þ

SP iR ðaÞ ¼ m1 SR1R ðaÞ þm2 SR2R ðaÞ e Þðtt Þ þ a e þ ðK 11 d e2 ebmit0 ðtt 1i Þ dt ½Q 2 1i 2i

ð24Þ

bmit0 e ðtt e e d ½Q ðtt 1i e t 2i Þ dt 2 1i t 2i Þ þ a2L e 3i

ð25Þ

Let a-cut set of the above four integrals are eI 1 ½a ¼ ½I1L ðaÞ,I1R ðaÞ, eI 2 ½a ¼ ½I2L ðaÞ,I2R ðaÞ, eI 3 ½a ¼ ½I3L ðaÞ,I3R ðaÞ, and eI 4 ½a ¼ ½I4L ðaÞ,I4R ðaÞ respectively, then according to Wu (2000)  9 e1L bmði1Þt t 22 a ebmt0 1 > > 0 > e t0 þ I1L ðaÞ ¼ ðrK L d1R Þ þ > = bm 2 bm   ð26Þ 2 bmt 0 e1R bmði1Þt > t2 a e 1 > 0 > I1R ðaÞ ¼ ðrK R d1L Þ þ e t0 þ > ; bm 2 bm 9 2 > ðt t 0 Þ2 > 0Þ > I2L ðaÞ ¼ Q 1iL ðt 1i t 0 Þ þ ðrK L d2R Þ 1i þa2L ebmit0 ðt1i t = 2 2 2 2 > ðt t 0 Þ 0Þ > > þ a2R ebmit0 ðt1i t I2R ðaÞ ¼ Q 1iR ðt 1i t 0 Þ þ ðrK R d2L Þ 1i ; 2 2 9 > > > =

e , during the whole planning horizon is given by Then total profit, Z f i cp K g i C 3  e t 1i cpRM ch HOC e t 1i ð1rÞK e ¼ ½cp SP Z

Z L ðaÞ ¼ cp SP iL cp K R t 1i ð1rÞK R t 1i cpRM ch HOC iR c3 Z R ðaÞ ¼ cp SP iR cp K L t 1i ð1rÞK L t 1i cpRM ch HOC iL c3 So, the problem reduces to

ð27Þ

ð28Þ

Determine t 0 ,t 1 , l and m1 to 9 maximize ½Z L ðaÞ,Z R ðaÞ > > = N X e > Te i r H subject to > ;

ð37Þ

As constraints of the problem is fuzzy in nature, using fuzzy chance constraint (Maiti, 2011), the above problem can be written respectively in optimistic and pessimistic senses as follows:

9 > > > > > > > > > > > > > > > > > > > > > =

Determine t 0 ,t 1 , l and m1 to 9 maximize ½Z L ðaÞ,Z R ðaÞ > > ! = N X e Zb > subject to pos Te i r H 1> ; i¼1

I4R ðaÞ ¼ Q 3iR ðT iR t1i t2iR Þ > > " # > > > > T 2iR ðt1i þ t2iR Þ2 > > d2L ðt 1i þ t 2iL ÞT iR  þ ðt1i þ t2iL Þðt 1i þ t 2iR Þ > > > 2 2 > > " #> > > 2 2 > > T ðt þ t Þ 1i 2iR iR bmit 0 > ðt1i þ t 2iL ÞT iR  þ ðt 1i þ t2iL Þðt1i þt 2iR Þ > þa2R e > ; 2 2

Determine t 0 ,t 1 , l and m1 to 9 maximize ½Z L ðaÞ,Z R ðaÞ > > ! = N X e e T i r H Z b2 > subject to nes > ; i¼1

I4L ðaÞ ¼ Q 3iL ðT iL t1i t 2iL Þ " # T2 ðt þ t Þ2 d2R iL ðt 1i þ t 2iR ÞT iL  1i 2iL þðt1i þt 2iR Þðt 1i þ t 2iL Þ 2 2 " # 2 T ðt þ t Þ2 þa2L ebmit0 iL ðt1i þ t2iR ÞT iL  1i 2iL þ ðt 1i þ t2iR Þðt1i þ t2iL Þ 2 2

ð29Þ g i ½a ¼ ½c HOC iL ðaÞ,c HOC iR ðaÞ, Let a-cut of holding cost is ch HOC h h then clearly ) HOC iL ðaÞ ¼ I1L ðaÞ þ I2L ðaÞ þI3L ðaÞ þ I4L ðaÞ ð30Þ HOC iR ðaÞ ¼ I1R ðaÞ þI2R ðaÞ þI3R ðaÞ þ I4R ðaÞ f i , where Sell revenue in ith cycle is cp SP 2

f i ¼ cp 4m1 cp SP

Z

t0 0

bmði1Þt0 bmt e a fd e g dt þ m2 1 e1 e

f 1 þ m2 SR f2 ¼ cp ½m1 SR

Z e Ti

ðsayÞ

t0

3 bmit 0 e a fd g dt 5 2 e2 e

ð31Þ

f 1 and SR f 2 are SR f 1 ½a ¼ ½SR1L ðaÞ,SR1R ðaÞ and SR f 2 ½a ¼ Let a-cut of SR ½SR2L ðaÞ,SR2R ðaÞ respectively, then according to Wu (2000) 9 ð1ebmt0 Þ > > > SR1L ðaÞ ¼ d1L t 0 a1R ebmði1Þt0 = bm ð32Þ bmt 0 > ð1e Þ> > SR1R ðaÞ ¼ d1R t 0 a1L ebmði1Þt0 ; bm and SR2L ðaÞ ¼ d2L ðT iL t 0 Þa2R ebmit0 ðT iL t 0 Þ SR2R ðaÞ ¼ d2R ðT iR t 0 Þa2L ebmit0 ðT iR t 0 Þ

ð36Þ

e ½a ¼ ½Z L ðaÞ,Z R ðaÞ, where Let a-cut set of Z

i¼1

t 22iL 2 2 > bmit 0 t 2iR > > Þ I3R ðaÞ ¼ Q 2iR t 2iR ðK 11 d2L þ a2R e ; 2

I3L ðaÞ ¼ Q 2iL t 2iL ðK 11 d2R þa2L ebmit0 Þ

f ¼ cp K f Þ for ith cycle is given by PC e t 1i . Let the Production cost ðPC i i f a-cut set of production cost is PC i ½a ¼ ½PC iL ðaÞ,PC iR ðaÞ, so ) PC iL ðaÞ ¼ cp K L ðaÞt 1i ð35Þ PC iR ðaÞ ¼ cp K R ðaÞt 1i

) ð33Þ

f , i.e., SP f ½a ¼ ½SP ðaÞ,SP ðaÞ, then from Eq. (31) Let a-cut of SP i i iL iR with the help of Eqs. (32) and (33), a-cut of sell revenue, i.e.,

ð38Þ

and

ð39Þ

where b1 and b2 are predefined possibility and necessity levels respectively, which are entirely determined by the DM, their significance is discussed later. e 11 , H, e e, D e 11 , A Lemma 4. For any types of fuzzy numbers K P PN e e posð N i ¼ 1 T i r HÞ Z b1 , iff i ¼ 1 T iL ðb1 Þ rH R ðb1 Þ. e 11 , H e are fuzzy numbers, Te i are also valid fuzzy e, D e 11 , A Proof. As K PN e numbers, which implies i ¼ 1 T i is a valid fuzzy number. So meT ðxÞ r b1 for x rT iL ðb1 Þ and mHe ðxÞ r b1 for x Z HR ðb1 Þ. So i PN i ¼ 1 T iL ðb1 Þ is least value of real number for which membership P value of N Te i is b and HR ðb Þ is greatest value of real number i¼1

1

1

e is b . Hence the result holds for which membership value of H 1 from definition (1). & e 11 , H, e e, D e 11 , A Lemma 5. For any types of fuzzy numbers K P PN e e nesð N i ¼ 1 T i r HÞ Z b2 , iff i ¼ 1 T iR ð1b2 Þ rH L ð1b2 Þ. PN e Proof. It is discussed in Lemma 4 that i ¼ 1 T i is a valid fuzzy number. Again from definition of necessity (2), we have ! ! N N X X e Z b ) 1pos e Zb Te i r H Te i 4 H nes 2

2

i¼1

i¼1

) pos

N X

e Te i 4 H

! r1b2

i¼1

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

P. Guchhait et al. / Engineering Applications of Artificial Intelligence ] (]]]]) ]]]–]]]

P For any real number x 4 N i ¼ 1 T iR ð1b2 Þ, membership value of PN e i ¼ 1 T i , is less than 1b2 and for any real x o H L ð1b2 Þ, membere is less than 1b . Hence result holds from ship value of H 2

definition (1).

&

4.1. Optimistic model (OM) Using Lemma 4, the problem (38) reduces to Determine t 0 ,t 1 , l and m1 to 9 maximize ½Z L ðaÞ,Z R ðaÞ > > = N X T iL ðb1 Þ rHR ðb1 Þ > subject to > ; i¼1

ð40Þ

4.2. Pessimistic model (PM) Using Lemma 5, the problem (39) reduces to Determine t 0 ,t 1 , l and m1 to 9 maximize ½Z L ðaÞ,Z R ðaÞ > > = N X T iR ð1b2 Þ rHL ð1b2 Þ > subject to > ; i¼1

ð41Þ

From (40) and (41) it is clear that increase of b1 and b2 imposes more restriction on constraints of (40) and (41) respectively. If the DM is most optimistic, he/she will go for a value of b1 near to 0 for maximum possible profit (least feasible) from the planning horizon and invites more uncertainty in his/her decision. But DM will go for a value of b1 near to 1 for most feasible profit from the planning horizon. And if the DM is much pessimistic he/she will go for a value of b2 near to 1 to secure the worst possible profit from the planning horizon, but he/she will go for a value of b2 near to 0 for most feasible profit from the planning horizon. The models are solved using ICGA which is discussed in the next section.

5. Interval compared Genetic Algorithm (ICGA) GAs are exhaustive search algorithms based on natural selection and genesis (Holland, 1975; Michalewicz, 1992) and have been extensively used/ modified to solve complex decision making problems in different fields of science and technology. A GA normally starts with a set of potential solutions (called initial population) and individual solutions are called chromosomes. Crossover and mutation operations are used among the potential solutions to get a new set of solutions and it continues until terminating conditions are encountered. Behavior and performance of a GA are directly affected by the interaction between the parameters, i.e., diversity of initial population, selection process of chromosomes for mating pool, probability of crossover (pc), probability of mutation (pm), etc. Here, a new type GA, developed by Guchhait et al. (2010), with varying population size, where diversity of the chromosomes in the initial population is maintained using entropy originating from information theory and chromosomes are classified into young, middle age and old (in fuzzy sense) according to their age and lifetime. Following comparison of fuzzy numbers, using possibility theory (Dubois and Prade, 1980; Liu and Iwamura, 1998), crossover probability is measured as a function of parent’s age interval (a fuzzy rule based on parents age limit is also used for this purpose). In this GA a subset of better children is included with the parent population for next generation and maximum size of this subset is a percentage of the size of its parent set. To control memory overflow at the runtime of the GA, an upper limit of population

7

size is imposed (Maxsize). Chromosomes with age exceeding lifetime are discarded from the population at the beginning of every iteration. Present ICGA terminates when difference between maximum fitness (Maxfit) of chromosome and average fitness (Avgfit) of the population becomes negligible (less than a small value E), i.e., when fitness of all chromosomes in P(t) are almost equal. If the algorithm does not terminates (converge) under above condition then to exit from infinite loop the algorithm will terminate after Maxgen iterations. In the algorithm pm is considered as a function of iteration counter t and initial value of pm, pm ð0Þ is considered as very high and it gradually decreases with the increase of iteration. Actually mutation operator changes the value of a gene xi, what brings the diversity among the population. Similarly to the temperature of simulated annealing, the mutation allows the exploration of the search space so that the algorithm ‘‘visits’’ several optima of the function. This exploration depends on two parameters, the mutation mode of the genes and the probability pm of applying this operator. The theoretical convergence towards the global optimum of a GA, operating with a constant probability of crossover (pc), is ensured, if the probability of mutation pm(t) follows a given decreasing law, in function of the generation number t (Bessaou and Siarry, 2001). From a practical viewpoint, like in the case of simulated annealing, here a fast decreasing rate (otherwise, a prohibitive number of generations would be necessary to ensure convergence of the GA towards the global optimum) is used and its functional form is presented in the next section (Section 5.1(h)). The outline of the proposed algorithm is presented below. In the algorithm Check_constraint (Xi) check whether solution Xi satisfies the constraints of the problem or not. It returns 1 if constraints are satisfied by Xi otherwise it returns 0. If a solution does not satisfy the problem constraint then it is discarded. ZðX i ðtÞÞ represents the fitness of solution Xi. Fitness of a solution is obtained using FPO or ORI on intervals. The ratio of the number of solutions dominated by a solution Xi and popsize is taken as fitness of the solution Xi. Proposed ICGA: 1. Initialize popsize, iteration counter t ¼0, Maxsize, Maxgen, E and pm ð0Þ. 2. Randomly generate initial population P(t), where diversity in the population is aintained using entropy originating from information theory. 3. Find fitness of each solution of P(t). 4. Set Maxfit¼Maximum fitness in P(t) and Avgfit¼Average fitness of P(t). 5. While (MaxfitAvgfit Z E and t o Maxgen) do 6. Increase age of each chromosome. 7. For each pair of parents do e c for the selected pair of 8. Determine probability of crossover p parents using fuzzy rule base and possibility theory. ec . 9. Perform crossover with probability p 10. For each offspring perform mutation with probability pm(t). 11. Store offsprings into offspring set PC(t). 12. End do 13. Select from P(t) all individuals with age not grater than their lifetime and insert into Pðt þ 1Þ 14. Select a percent of better offsprings from the offspring set PC(t) and insert into Pðt þ 1Þ, such that maximum size of it does not exceed Maxsize. 15. Calculate value of pm ðt þ 1Þ. 16. t ¼ t þ 1. 17. Find fitness of each new solution Xi(t) of P(t). 18. Set Maxfit¼Maximum fitness in P(t) and Avgfit¼Average fitness of P(t). 19. End While

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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20. Output: Best chromosome of P(t). 21. End Algorithm

6. Different procedures of the ICGA (a) Representation of solutions: A ‘n dimensional real vector’, X i ¼ ðxi1 ,xi2 , . . . ,xin Þ, is used to represent ith solution, where xi1, xi2,y, xin represent n decision variables of the decision making problem under consideration. (b) Initialization: popsize number of such solutions X i ¼ ðxi1 , xi2 , . . . ,xin Þ, where i ¼ 1,2, . . . ,popsize are randomly generated by random number generator within the boundaries of each variable such that each Xi satisfies the constraints of the problem. A separate subfunction check_constraint(Xi) is developed to check feasibility of a solution Xi under constraints of the problem, which returns 1 if Xi satisfies the constraints otherwise returns 0. This solution set is taken as initial population Pð0Þ. Initialize(Pð0Þ) subfunction is used for this purpose. (c) Diversity preservation: At the time of generation of Pð0Þ diversity is maintained using entropy originating from information theory (Bessaou and Siarry, 2001). Following steps are used. (i) Probability, prjk, that the value of the ith variable of the jth solution is different from the ith variable of the kth solution is calculated using pr jk ¼ 1

9xji xki 9 Bui Bli

where ½Bui ,Bli  is the variation domain of the ith variable. (ii) Entropy of the ith variable, Ei ðM P Þ, i ¼ 1,2, . . . ,n, is calculated using the formula Ei ðM P Þ ¼

M P 1 X

MP X

pr jk log pr jk

j ¼ 1 k ¼ jþ1

where MP is the size of the current population size. (iii) Average entropy of the current population is calculated by the formula EðM P Þ ¼

n 1 X  E ðM P Þ n i¼1 i

If Avgfit oZðX i Þ,lifetimeðX i Þ ¼

Minlt þMaxlt KðZðX i ÞAvgfitÞ þ , 2 MaxfitAvgfit

where Maxlt and Minlt are maximum and minimum allowed lifetime of a solution, K ¼ ðMaxltMinltÞ=2. Maxfit, Avgfit, Minfit represent the best, average and worst fitness of the current population. According to the age, a solution can belongs to any one of age intervals—young, middle-aged or old, whose membership functions are presented in Fig. 4. Membership functions of age intervals are defined by mimicking the natural growth process of an animal species. For any species duration of young, middle aged and old stage are imprecise in nature and there is an overlapping between these intervals. Here for a species it is assumed that young, middle age and old life intervals are [0, 40], [25, 85] and [75, 100] respectively and similarly membership functions are defined. (f) Crossover: e c Þ: Probability of I. Determination of probability of crossover (p e c , for a pair of parents ðX i ,X j Þ is determined as crossover p below: (i) At first age intervals (young, middle-aged, old) of Xi and Xj are determined by making possibility measure of fuzzy numbers- young, middle-aged, old with respect to their age using Lemmas 2 and 3 (Section 2). (ii) After determination of age intervals of the parents, their e c ) is determined as a linguistic crossover probability (p variable (low, medium or high) using a fuzzy rule base as presented in Table 1. Membership function of these linguistic variables are presented in Fig. 5. Membership functions of age intervals and crossover probabilities are defined by mimicking the natural cross over process of an animal species. Following natural phenomenon it is assumed that probability of crossover is higher between two middle aged animals, i.e., between middle aged chromosomes. Also it is natural that crossover frequency between two young or between two old or between one old and one young chromosome are very low. But a medium crossover frequency is observed between a middle aged and a old or between a middle aged and an young chromosome. Following this phenomenon fuzzy

(iv) Incorporating the above three steps, a separate sub-function check_diversity (Xi) is developed. Every time a new solution Xi is generated, the entropy between this one and previously generated individuals is calculated. If this information quantity is higher than a threshold, ET, fixed at the beginning, Xi is included in the population otherwise Xi is again generated until diversity exceeds the threshold, ET. This method induces a good distribution of initial population. (d) Evaluation: As goal of the paper is to optimize interval valued objective function under constraint, the ratio of the number of solutions dominated by a solution Xi and popsize of current population is taken as fitness of the solution Xi. Let it be ZðX i Þ. A solution Xi dominates a solution Xj if interval objective function (a-cut of profit) due to solution Xi is better than that due to Xj. This comparison is made using FPO on intervals or ORI. (e) Calculation of lifetime: At the time of birth, age of each solution is set to zero. Following Michalewicz (1992), at the time of birth lifetime of Xi is computed using the following formula: KðZðX i ÞMinfitÞ If Avgfit Z ZðX i Þ,lifetimeðX i Þ ¼ Minlt þ AvgfitMinfit

Fig. 4. Membership functio of age intervals.

Table 1 Fuzzy rule base for crossover probability. Parent 2

Young Middle-aged Old

Parent 1 Young

Middle-aged

Old

Low Medium Low

Medium High Medium

Low Medium Low

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

P. Guchhait et al. / Engineering Applications of Artificial Intelligence ] (]]]]) ]]]–]]]

9

7. Numerical illustration

Fig. 5. Membership functions of crossover probabilities.

rule base of crossover probability (Table 1) and membership function of crossover probability (Fig. 5) is prepared. e c ), a ran(iii) After determination of crossover probability (p dom number c is generated in the range [0, 1] and if e c Þ 4 b (cf. Lemma 1(Section 2)) then crossover nesðc o p operation is made on Xi, Xj, where bð0 o b o 1Þ is a predefined necessity level. For the proposed model it is assumed that b ¼ 0:5. II. Crossover process: For each pair of coupled solutions Xi, Xj, a random number cn is generated from the range [0,1]. Then two children C1 and C2 are obtained by the formula: C 1 ¼ cn X i þð1cn ÞX j

and

C 2 ¼ cn X j þ ð1cn ÞX i

If C1 and C2 satisfies the constraints of the problem, they are included in PC(t). (g) Mutation: (i) Selection for mutation: For each solution Xi of PC(t), generate a random number r from the range [0,1]. If r o pm ðtÞ, the solution is taken for mutation, where pm(t) is the probability of mutation. (ii) Mutation process: To mutate a solution X ¼ ðx1 ,x2 , . . . ,xn Þ, select a random integer r in the range [1,n]. Then replace xir by randomly generated value within the boundary ½Blr ,Bur  of rth component of Xi. (h) Reduction process of pm: Let pm ð0Þ is the initial value of pm. Then probability of mutation in tth generation pm(t) is calculated by the formula pm ðtÞ ¼ pm ð0Þexpðt=a1 Þ, where a1 is calculated so that the final value of pm is small enough (10 2 in our case). So a1 ¼Maxgen/log[pm ð0Þ=102 ], where Maxgen is the expected number of generations that the GA can run for convergence. (i) Selection of offsprings: Maximum population growth in a generation is assumed as forty percent. So not all offsprings are taken into the parent set for next generation. At first offspring set is arranged in descending order in fitness. Then better solutions are selected and entered into parent set such that population size does not exceeds Maxsize. (j) Termination condition: Algorithm terminates when difference between maximum fitness (Maxfit) of chromosome, i.e., fitness of the best solution of the population and average fitness (Avgfit) of the population becomes negligible ( o E). In other words when fitness of all chromosomes in P(t) are almost equal. If the algorithm does not terminates (converge) under above condition then to exit from infinite loop the algorithm will terminate after Maxgen iterations. (k) Implementation: With the above function and values the algorithm is implemented using C-programming language.

Here different models are solved using RCGA (Gupta et al., 2009), MOGA (Bera et al. 2012) and the proposed ICGA. Parametric values of RCGA are same as taken by Gupta et al. (2009) and that of MOGA are same as taken by Bera et al. (2012). Different parametric values of the ICGA used for solving the models are given below: Initial population size popsize ¼30. Maximum population size Maxsize¼150. Initial value of probability of mutation pm ð0Þ ¼ 0:9. Expected value of maximum number of generation to converge the ICGA, Maxgen ¼100 and E ¼ 0:0001. Number of variables in a solution, n ¼4. Two examples are used to illustrate the model. Parametric values for two examples are presented below: Example 1. cp ¼1.1, C pRM ¼ 0:4, r ¼0.95, b1 ¼ 0:9, b2 ¼ 0:1, e ¼ ðK 1 ,K 2 ,K 3 Þ ¼ ð92,93,94Þ, ch ¼0.1, m2 ¼ 2.2, g ¼ 1:5, K 11 ¼ 100, K e 11 ¼ ðA111 ,A112 ,A113 Þ ¼ ð20, e 11 ¼ ðD111 ,D112 ,D113 Þ ¼ ð90,95,100Þ, A D e 20:5,22Þ, H ¼ ðH1 ,H2 ,H3 Þ ¼ ð20,20:5,21:2Þ, C 3 ¼ 25, b ¼0.3. Example 2. cp ¼1.1, C pRM ¼ 0:4, r ¼0.90, b1 ¼ 0:9, b2 ¼ 0:1, ch ¼0.1, e ¼ ðK 1 ,K 2 ,K 3 Þ ¼ ð70,71,71:5Þ, D e 11 ¼ m2 ¼2.4, g ¼ 1:5, K 11 ¼ 80, K e 11 ¼ ðA111 ,A112 ,A113 Þ ¼ ð30,30:5, ðD111 ,D112 ,D113 Þ ¼ ð100,105,109Þ, A e ¼ ðH1 ,H2 ,H3 Þ ¼ ð30,30:5,31:2Þ, C 3 ¼ 30, b¼0.2. 32Þ, H For the above examples results are obtained for both optimistic and pessimistic models using ICGA with FPO and presented in Table 2. For both the models and both the examples, it is observed that initially profit increases with N, reaches a maximum limit and then decreases as N increases. When N, i.e., number of cycle is small, set-up cost is small and holding cost is very high. But when N increases, total setup cost increases and holding cost decreases. Initial gain due to decrease of holding cost dominates loss due to increase of setup cost and as a result, profit increases with N. But after a certain limit of N, loss due to increase of setup cost dominates gain due to decrease of holding cost and so after that limit of N, profit gradually decreases as N increases. For OM and PM models, maximum profit are obtained for N ¼4 and N ¼3 respectively in case of Example 1. On the other hand, for Example 2, maximum profits are obtained for N ¼5 and N¼ 4 for OM and PM models respectively. In case of OM, as possibility constraint is imposed on planning horizon, effective planning horizons for both examples are larger than the corresponding values of PM. As a result, optimum number of cycles (N) in OM is larger than that of PM. Again, as total profit is optimized, profit in OM is more than that of PM. But there exists much risk in OM than that of PM. If the DM is more pessimistic he/she will adopt for PM. On the other hand, an optimistic DM will adopt for OM. Taking N¼4 for OM, and N¼3 for PM, Example 1 is solved using ICGA with FPO of intervals for a ¼ 0:5 due to different values of popsize and Maxsize and presented in Tables 3 and 4 respectively. It is observed that for solving the models performance of ICGA with FPO is best with popsize¼30 and Maxsize¼150. However, these parametric values of popsize and Maxsize are taken for ICGA to solve the proposed models. It is also observed that ICGA takes less iterations to converge with popsize¼40 or 50 and Maxsize¼200 than popsize¼ 30 and Maxsize¼150. But with popsize¼30 and Maxsize¼ 150, ICGA reflects better results. As our models are profit maximization models, popsize¼30 and Maxsize¼150 are taken for ICGA for further investigations. It should be noted that increase of popsize and Maxsize increases the total number of objective function evolution. As a result, though for popsize¼30 and Maxsize¼150 takes some more iterations to converge it takes less number of objective function evolution compared to popsize¼40 or 50 and Maxsize¼ 200.

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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Table 2 m1

TL

TR

ZL

ZR

1.5361 1.5701 1.5831 1.5298 1.5823

20.5695 20.5693 20.5641 20.5692 20.5697

20.9366 20.9383 20.9354 20.9425 20.9418

247.1154 277.1775 281.7979 271.0132 254.5267

455.2501 473.8390 475.2709 461.6284 441.6097

0.3365 0.1448 0.1215 0.1375

1.6046 1.5248 1.5408 1.5837

17.4206 17.3646 17.3661 17.3759

20.4487 20.4472 20.4491 20.4487

232.1632 252.9945 247.6491 234.4835

417.3066 429.9964 421.2614 404.6314

(b) Results of OM and PM (for example-2) for different N ða ¼ 0:5Þ OM 3 2.7819 3.3130 0.4219 4 1.9890 2.3560 0.1967 5 1.7138 1.8470 0.2088 6 1.1901 1.4040 0.1555 7 0.9152 1.0234 0.1378

1.6015 1.6761 1.5832 1.5821 1.6777

30.5693 30.5686 30.5699 30.5691 30.5666

31.0601 31.0584 31.0713 31.0681 31.0580

495.0992 509.8253 511.7275 496.0853 476.2899

775.6859 777.3683 782.0996 758.6992 730.2903

1.6491 1.5839 1.6610 1.6320 1.6125

26.4212 26.3237 26.2988 26.2332 26.2277

30.4498 30.4480 30.4474 30.4500 30.4496

379.5875 447.6065 462.4118 454.8984 441.0920

634.0948 698.5248 708.3880 700.4273 683.2287

Model

N

t0

t1

l

(a) Results of OM and PM (for example-1) for different N ða ¼ 0:5Þ OM 2 2.0997 2.6047 0.4257 3 1.0377 1.4861 0.2685 4 0.8682 1.1369 0.1267 5 0.5422 0.8286 0.1054 6 0.3553 0.5666 0.1053 PM

PM

2 3 4 5

1.9618 1.0837 0.6408 0.4260

2 3 4 5 6

2.2748 2.1007 1.9896 1.7082 1.3321

2.3096 1.4884 0.9891 0.6305

3.9145 2.8270 2.1940 1.8365 1.3834

1.0740 0.4127 0.1858 0.1004 0.1204

Table 3 Results of OM (for example-1) for different popsize ða ¼ 0:5Þ. Maxsize

100 125 150 175 200

Popsize 10

20

30

40

50

[281.0522,475.2022] [280.7881,475.1140] [280.9455,475.1953] [280.8271,475.0667] [280.5845,475.1357]

[281.4602,474.8924] [281.2979,474.9410] [281.4673,474.1752] [281.0905,474.2645] [282.1308,474.9908]

[281.9526,473.9015] [281.2908,475.0056] [281.7979,475.2709] [281.2780,474.8954] [281.9453,474.5864]

[280.4928,474.5765] [281.0445,474.7211] [281.4955,474.7933] [281.4543,475.1269] [280.9930,474.9493]

[281.3990,474.4037] [281.9135,474.2552] [280.8986,474.0516] [281.6877,475.1596] [281.7263,475.1334]

Table 4 Results of PM (for example-1) for different popsize ða ¼ 0:5Þ. Maxsize

100 125 150 175 200

Popsize 10

20

30

40

50

[251.3640,428.4877] [251.6497,428.7024] [251.5120,428.1588] [251.4329,428.7953] [251.3213,428.8296]

[249.3821,428.8558] [251.7678,430.9630] [252.1101,429.3313] [251.6501,428.4162] [251.8793,428.3167]

[252.8261,430.3489] [252.7230,429.7885] [252.8401,430.5376] [252.7159,429.6383] [252.3711,430.2070]

[250.6759,428.6179] [252.1631,429.3747] [252.5582,429.7815] [251.1426,429.9731] [252.4748,430.1778]

[252.0455,429.5662] [252.2176,429.6028] [251.6760,429.4774] [252.3694,429.4464] [251.6797,429.3304]

Taking N ¼4 and N ¼ 3 for OM and PM respectively, Example 1 is solved using RCGA, MOGA and ICGA using ORI and FPO of intervals due to different values of a and presented in Table 5. It is observed that for all the cases performance of ICGA with FPO is best for solving the models. Due to this reason all the models for different cases are solved using this approach. a-cuts ½Z L ðaÞ,Z R ðaÞ of observed profit (Ze ) for both the models due to Example 1 are plotted in Figs. 6 and 7. It is interesting to note that both the figures represent almost triangular fuzzy numbers for fuzzy profit. Assuming these fuzzy quantities to be perfect triangular numbers, the corresponding membership functions me ðxÞ and me ðxÞ for OM and PM respectively are Z OM Z PM formulated as 8 x193:82 > > < 181:64 meZ ðxÞ ¼ 582:69x > OM > : 207:50

for 193:82 rx r 375:46 for 375:46 rx r 582:96

and 8 x171:72 > > < 166:70 meZ ðxÞ ¼ 527:80x > PM > : 189:38

for 171:72 r x r 338:42 for 338:42 r x r 527:80

For the assumed parametric values maximum possible profit from OM of Example 1 is a TFN (193.82, 375.46, 582.96) due to b1 ¼ 0:9, provided optimum values of decision variables are incorporated by the DM for their company. This implies that for the given parametric values of Example 1, profit will never exceed 582.96 but may fall below 193.82 for the possibility label 0.9. It happens because optimistic DM assumes that demand of the item will last up to HR ðb1 Þ, which is maximum possible planning horizon due to possibility label 0.9. According to this assumption maximum possible profit is 582.96. If the demand of the item does not last up to HR ðb1 Þ then total fuzzy profit will decrease and so profit may fall below 193.82. As optimistic DMs take risks, so their organizations may have

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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11

Table 5 Results of OM (N ¼ 4) and PM (N ¼3) for different values of a for example-1. Model

a

Approach

ZL

ZR

Model

a

Approach

ZL

ZR

OM

0.0

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

193.1496 190.6497 192.5479 190.6479 191.3558 193.8170

576.8455 582.2866 582.5204 577.7443 580.8503 582.6863

PM

0.0

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

162.0950 168.4585 170.0152 162.8979 167.8735 171.7199

529.7368 527.9319 527.4727 529.8842 525.7267 527.7996

0.25

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

236.3777 234.5475 236.6713 233.9217 234.9212 236.8937

525.0304 527.3299 528.0744 526.0974 527.1640 527.4778

0.25

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

209.4604 208.6953 209.5992 209.7867 208.7834 211.5172

475.1792 475.0484 478.3319 476.1990 474.2544 478.2272

0.50

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

280.7974 279.9612 281.0020 279.1778 279.9438 281.7979

472.6840 474.3781 474.8472 474.1228 474.4073 475.2709

0.50

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

248.4973 249.1805 252.3280 251.3858 250.1852 252.9945

428.0685 427.1669 429.7244 429.7278 428.6347 429.9964

0.75

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

326.5789 326.2917 327.1183 325.6920 326.2899 327.5325

422.3197 423.2913 423.6630 422.9752 423.3172 423.6698

0.75

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

291.3962 291.8605 294.0887 291.9132 292.4025 294.5635

380.9985 381.2690 382.8055 380.8630 381.1883 383.1093

0.999

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

368.4244 373.8777 375.1476 370.5126 373.8838 375.2652

368.8188 374.2650 375.4340 370.8902 374.2711 375.6502

0.999

RCGA(ORI) MOGA(ORI) ICGA(ORI) RCGA(FPO) MOGA(FPO) ICGA(FPO)

335.3642 335.7597 337.4657 335.5677 335.7590 338.4174

335.7219 336.1157 337.8200 335.9228 336.1149 338.7702

Fig. 6. Membership of fuzzy profit (OM) (Example 1). Fig. 7. Membership function of fuzzy profit (PM) (Example 1).

to run in loss at some situations. HR ðb1 Þ increases due to the decrease of b1 . Again increase of HR ðb1 Þ increases the upper limit of fuzzy profit but decrease of b1 decreases the feasibility of lasting of demand up to HR ðb1 Þ, which in turn decreases the possibility of obtaining upper limit of fuzzy profit. For this reason, optimistic DM normally takes value of b1 near to 1. On the other hand for the same parametric values of Example 1, profit from PM is a TFN (171.72, 338.42, 527.80) for b2 ¼ 0:1, provided optimum values of decision variables are incorporated by the decision maker (DM) for their company. This implies that for the given parametric values of Example 1 profit will never fall below 171.72 for the necessity label 0.1. It happens because pessimistic DM assumes that demand of the item will last up to HL ð1b2 Þ, which is lower limit of planning horizon due to necessity label 0.1. According to this assumption minimum possible profit is 171.72. If the demand of the item lasts more than HL ð1b2 Þ then total fuzzy profit can be increased, and so profit may exceed 527.80. As pessimistic DMs take no risks, so their organizations may not get extra profit in these situations. HL ð1b2 Þ decreases

due to the increase of b2 . Again decrease of HL ð1b2 Þ decreases the lower limit of fuzzy profit but increase of b2 increases the feasibility of lasing of demand up to HL ð1b2 Þ, which in turn decreases the risk of the company but decreases minimum possible profit. For this reason, pessimistic DM normally takes value of b2 near to 0. Similarly for Example 2, results of OM and PM are obtained due to different values of a and presented in Table 6. a-cuts e ) from both the models of ½Z L ðaÞ,Z R ðaÞ of observed profit (Z Example 2 are plotted in Figs. 8 and 9. In this case, also both the figures represent almost triangular fuzzy numbers for fuzzy profit. Assuming these fuzzy quantities to be perfect triangular numbers, the corresponding membership functions me ðxÞ and Z OM me ðxÞ for OM and PM respectively are formulated as Z PM

8 x385:14 > > < 259:70 meZ ðxÞ ¼ 929:41x > OM > : 284:57

for 385:14 r x r 644:84 for 644:84 r x r 929:41

Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017

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Table 6 Results of OM (N ¼ 5) and PM (N ¼4) for different values of a for example-2. Model

a

ZL

ZR

Model

a

ZL

ZR

OM

0.00 0.25 0.50 0.75 0.999

385.1420 446.4390 511.7275 577.4587 644.5653

929.4064 854.7289 782.0996 712.7175 645.1051

PM

0.00 0.25 0.50 0.75 0.999

345.5428 403.3900 462.4118 521.5925 583.5522

843.1351 774.2452 708.3880 644.4914 584.0439

the methodology presented here implied that it is applicable for any type of membership functions of fuzzy parameters. Here the analysis method and the developed soft computing algorithm are quite general and can be used to solve different fuzzy problems in the areas of inventory, supply-chain, portfolio, etc.

Acknowledgment The authors are heartily thankful to the Honorable Reviewers for their valuable comments to improve the quality of the paper. Also, first author expresses his heartfelt gratitude to his wife and son for their encouragement and dedication related to this paper. This research work is supported by University Grants Commission of India with Grant no. PSW-089/11-12.

References

Fig. 8. Membership function of fuzzy profit (OM) (Example 2).

Fig. 9. Membership function of fuzzy profit (PM) (Example 2).

and 8 x345:54 > > < 238:25 meZ ðxÞ ¼ 843:14x > PM > : 259:35

for 345:54 rx r 583:79 for 583:79 rx r 843:14

8. Conclusion Here for the first time a production inventory model is developed, where production rate and demand are imprecise in nature. Using fuzzy differential equation and fuzzy Riemann integration, an approach is proposed, where a-cut of fuzzy profit is optimized to get optimal decision. Here, planning horizon is also assumed to be imprecise. Inventory policies are derived in both pessimistic and optimistic seances. Depending on the values of possibility/necessity levels, the approximate values of time horizon and inventory values are evaluated. Some interesting observations are derived and discussions are made. An algorithm ICGA is developed that can optimize interval objective function and is used to solve the model. Sengupta and Pal’s (2009) fuzzy preference ordering on intervals is used to compare the intervals in determining fitness of a solution. Profit functions have been graphically presented as TFNs. The inventory policies presented here may be used by the practitioners. For simplicity TFN type membership functions for the fuzzy parameters are used in the examples. But

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Please cite this article as: Guchhait, P., et al., A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm.... Eng. Appl. Artif. Intel. (2012), http://dx.doi.org/10.1016/j.engappai.2012.10.017