An EPQ Model with Stock Dependent Demand under Imprecise and Inflationary Environment using Genetic Algorithm

An EPQ Model with Stock Dependent Demand under Imprecise and Inflationary Environment using Genetic Algorithm

Chaman Singp

1Assistant Professor, Dept. of Mathematics,

A.N.D. College (University of Delhi), Delhi-110019

S.R. Singp

2Reader, Dept. of Mathematics, D.N.(P.G.) College, Meerut, U.P. – 250002

Abstract

In this paper a production inventory

number of mathematical models have been reported in the existing literature. Among them, to get the idea of the trends of recent research, one may refer to the

model for the newly launched product is developed incorporating the effect of inflation and time value of money under imprecise environment. It is common phenomenon in the supermarket that the stock level has a motivational effect on the customers; i.e. the demand rate may go up or down if the on-hand inventory level increases or decreases. To deal with such kind of situations, demand is considered as stock dependent. Items are produced at the rate, which is constant proportional to demand rate. Production is stopped when the stock-level reached to level Q and Q0 is the fixed stock-level below which inventory has no effect on demand. Model is formulated to maximize the total profit. It is assumed that all the cost parameters involved in the model are imprecise in nature and represented by triangular fuzzy numbers. Centroid Method is being used to defuzzified total fuzzy profit. A genetic algorithm with varying population size is used to solve the model. In this GA a subset of better children is included with the parent population for next generation and size of this subset is a percentage of the size of its parent set. Numerical example along with sensitivity analysis is presented to illustrate the model. Mathematica7.0 is being used to reach the optimal policies.

Keywords: Inflation, Stock-dependent demand

1. Introduction

   It is well known that the stock level has a motivational effect on the customers in a supermarket; i.e. the demand rate may go up or down if the on-hand inventory level increases or decreases. In corporate world such a situation is known as the stock-dependent demand. It generally arises for a consumer-goods type inventory. In this area a large

   works of Teng and Chang (1995), Sarkar et. al. (1997), Datta et al.(1998), Balki and Benkherouf (2004), Teng and Chang (2005), Wu et. al. (2006), Singh et al. (2007) and Singh et al. (2010).

   In classical inventory, models are developed on the basis of that all the parameters in total cost function are fixed and have the certain values. But in day-by- day changing market scenario there may be increase or decrease within a range of the values of the parameters. To deal with such type of irregularities, these parameters are considered as fuzzy in nature. When some parameters are fuzzy in nature, the resultant objective function also becomes fuzzy. To get an idea or trends in recent research one may refer to the work of Yao and Lee (1999), Yao et al. (2000), Chang et al. (2004), Maiti and Maiti (2007) and Singh and Singh (2010).

   Most of the inventory models unrealistically ignore the influence of inflation. This was due to the belief that inflation would not influence the inventory policy to any significant degree. This belief is unrealistic since the resource of an enterprise is highly correlated to the return on investment. The concept of the inflation should be considered especially for long-term investment and forecasting. To get an idea trends in recent research one may refer to the work of Buzacott (1975), Misra (1979), Chandra and Bahner (1985), Lieo et al. (2000), Mehta and Shah (2003) and Singh and Singh (2010).

   Use of Genetic Algorithm in complex decision making problem is well established by Michalewicz (1992). Extensive research work has been made to improve the performance of GA for single/multi-objective continuous/discrete optimization problems during last two decades.

   Michalewich (1992) proposed a GA named contractive mapping genetic algorithm (CMGA) where movement from old population to new population takes place only when average fitness of new population is better than the old one and proved the asymptotic convergence of the algorithm by Banach fixed point theorem. Bessaou and Siarry (2001) proposed a GA where initially more than one population of solutions are generated. Genetic operations are done on every population a finite number of times to find a promising zone of optimum solution. last and Eyal (2005) developed a GA with varying population size, where chromosomes are classified into young, middle-age and old according to their age and lifetime. Pezzellaa. Morgantia, and Ciaschettib (2008) developed a GA for the flexible Job-shop scheduling problem, which integrates different strategies for generating the initial population, selecting the individuals for reproduction and reproducing new individuals.

   In this paper an EPQ model of an item is developed considering the power form stock- dependent demand under imprecise and inflationary environment. It is assumed that the production rate is demand dependent. Two situations were discussed in this paper (I) Q Q0 and (II) Q > Q0, where Q is the stock-level at the time production is stopped and Q0 is the fixed stock-level. Model is formulated to maximize the total profit. A genetic algorithm with varying population size is used to solve the model.

2. Genetic Algorithm

   Genetic Algorithm is exhaustive search algorithm based on the mechanics of natural selection and genesis (crossover, mutation, etc.). It was developed by Holland, his colleagues and students at the University of Michigan. Because of its generality and other advantages over conventional optimization methods, it has been successfully allied to different decision making problems.

   In natural genesis, we know that chromosomes are the main carriers of hereditary factors. At the time of reproduction, crossover and mutation take place among the chromosomes of parents. In this way, hereditary factors of parents are mixed-up and carried over to their offspring. Again, Darwinian principle states that only the fittest animals can survive in nature. So, a pair of parents normally reproduces a better offspring.

   The above-mentioned phenomenon is followed to create a genetic algorithm for an optimization problem. Here, potential solutions of the

   problem are analogous with the chromosomes, and the chromosome of better offspring with the better solution of the problem. Crossover and mutation among a set of potential solutions to get a new set of solutions are made, and it continues until terminating conditions are encountered. Michalewich proposed a genetic algorithm named Contractive Mapping Genetic Algorithm (CMGA) and proved the asymptotic convergence of the algorithm by Banach fixed point theorem. In CMGA, a movement from the old population to a new one takes place only if an average fitness of the new population is better than the fitness of the old one. In the algorithm, pc, pm are probability of crossover and probability of mutation respectively, T is the generation counter and P(T) is the population of potential solutions for the generation T . M is an iteration counter in each generation to improve P(T) and M0 is the upper limit of M . Initialize (P(1)) function generate the initial population P(1) (initial guess of solution set) at the time of initialization. Objective function value due to each solution is taken as fitness of the solution. Evaluate (P(T)) function evaluates fitness of each member of P(T ) .

   1. GA Algorithm:

1. Set generation counter T = 0 and maximum generation M = 0

2. Initialize probability of crossover pc, probability of mutation pm , upper limit of iteration counter M0, population size N .

3. Initialize (P(T )) .

4. Evaluate (P(T)) .

5. While (M < M0 ) .

6. Select N solutions from P(T) for mating pool using Roulette-Wheel process.

7. Select solutions from P(T) , for crossover depending on pc.

8. Make crossover on selected solutions.

9. Select solutions from P(T ) , for mutation depending on pm.

10. Make mutation on selected solutions for mutation to get population P1 (T) .

11. Evaluate (P1 (T ))

12. Set M = M +1

13. If average fitness of P1 (T) > average fitness of P(T) then

14. Set P(T +1) = P1(T )

1. Set T = T +1

2. Set M = 0

3. End if

4. End While

5. Output: Best solution of P(T)

6. End algorithm.

1. 1. Triangular fuzzy number:

   Let A

   (k1 , k2 , k3 ) is a triangular fuzzy number,

   M k1 k2 k3

   k 1 ( )

   where k k , k k and k k . Such

   k 3 3 2 1

   1 1 2 3 2

   that 0

   1 k, 0

   2 and

   1, 2 are determined

   1. ASSUMPTIONS AND NOTATIONS

      by the decision maker based on the uncertainty of the problem. And then A can be represented as

      A AL ( ), AU ( ) AL ( ), AU ( ) subje

      The mathematical model in this paper is developed on the basis of the following assumptions

      and notation.

      l l r r

      ct to the constraint 0 k1 k2

      k3 . And the

      1. Notations: Following notations have been

   membership function of the triangular fuzzy numbers is defined as follows.

   A (x) : R [0,1]

   used in this paper

   c1 fuzzy holding cost of the inventory item, $ / per unit / per unit time

   c2 fuzzy deterioration cost, $ / per unit

   0 , x k1, x k3

   c3 fuzzy ordering cost, $ / per order

   A (x)

   L(x)

   x k1

   k2 k1

   , k1 x k2

   c1r fuzzy holding cost for raw material

   R(x)

   k3 x

   , k x k

   inventory, $ / per unit / per unit

   k3 k2

   ~ (x)

   A

   c 2r fuzzy deterioration cost for raw material inventory, $ / per unit

   2 3

   1

   c3r fuzzy ordering cost for raw material

   0 k1 k2

   k3 x

   inventory, $ / per order

   c r fuzzy purchasing cost for raw material inventory, $ / per unit

   Figure 1: Membership Function

   3.2. Centroid Method:

   This procedure (also called center of area, center of gravity) is the most prevalent and physically appealing of all the defuzzification methods it is given by the algebraic expression as follows:

   (x).xdx

   p fuzzy production cost, $ / per unit

   s fuzzy selling price, $ / per unit deterioration rate

   i inflation rate

   d discount rate

   M A

   A

   A (x)dx

   r = d – i

   Q maximum inventory level of the

   Where denotes an algebraic integration. Hence the centroid for A is given as

   production cycle

   Q0 constant inventory level

   T length of the ordering cycle

   With the boundary conditions

   I1 0 0and I2 T

   0 . Solution of equation

   I(t) inventory level at time t [0, T]

   3.3.2. Assumptions: Model is developed under the

   1. and (2) are as follows

      following assumptions

      1. The inventory system involves only one item and the planning horizon is infinite.

         I1 t

         (a 1)D 1 e t

         D

         , 0 t T1

         (3)

      2. Production rate is demand dependent i.e. P(t) = a D(t)

         I2 t

         e(T t )

         1 , T1 t T

         (4)

      3. The demand rate D(t) is deterministic and its functional form is given by

         D t I t , Q Q0, D , 0 Q Q0

         From equation (3) and (4), using the condition

         I1(T1) I2 (T1) , we have

         Inventory Level

         Where > 0, 0 < < 1, D Q0 , and

         both and are known as scale and shape Q0

         parameters, respectively.

      4. Shortages are not allowed. Q

2. Model Formulation

   Let the production is stopped when the 0

   stock-level reached to level Q, Q0 is the Constant

   T1 T

   Time

   stock-level. Then there may arise the following two cases

   Figure 2: Case I (Q Q0)

   1. Q Q0 (II) Q > Q0

      T 1 ln a eT 1 1

      1. 1

         1. Case I: Q Q0

            This is the classical EPQ model for deteriorating items with constant demand rate. Production is started at the time t = 0, with zero inventory level and

            Maximum inventory level is

            (a 1)D T

            stopped at the time t = T1

            when the inventory level

            Q I1 T1

            1 e 1

      2. reaches to the level Q. After that inventory level decreases due to the combined effect of demand and deterioration up to the time T, at which inventory level reaches to the zero level. Inventory level at any

         time can be described by the following differential

         Present worth of fuzzy holding cost of the inventory held is

         equation and graphically represented in the figure-1.

         H C

         c1D (a 1)

         1 e rT1

         (a 1)

         e ( r)T1 1

         I' t

         I t

         (a 1)D,

         0 t T

         (1)

         r ( r)

         1 1 1

         eT

         e ( r)T1

         e ( r)T 1 e rT

         e rT1

      3. I' t

         I t D,

         T t T

         (2)

         ( r) r

         2 2 1

         Present worth of fuzzy production cost is

         P C

         p aD r

         1 e rT1

      O Cr

      c3r

      (15)

      Present worth of fuzzy deterioration cost is

      Present worth of fuzzy holding cost for the raw

      material is

      D C

      c D

      (a 1) 1 e rT1

      (a 1)

      e ( r)T1 1

      2 r ( r)

      Inventory Level

      eT ( r)T ( r)T 1 rT rT

      e 1 e e e 1 (9)

      Q

      ( r) r

      r

      Present worth of fuzzy sales revenue is

      S R

      sD r

      1 e rT

      (10)

      0 T1 Time

      Present worth of fuzzy ordering cost is

      Figure 3: Raw material inventory

      O C

      c3

      (11)

      c1raQ

      eT1

      ( r)T 1 rT

      HC r

      0 1 e

      1 1 e 1

      (16)

      4.1.1. Raw Material Inventory Model

      Initially, vendor purchases the raw material in lots and produces the finished goods. Vendor starts production at time (t = 0), and the raw material

      ( r) r

      Present worth of fuzzy deterioration cost for the raw material is

      reaches the zero level at the time (t = T1) due to the combined effect of production and deterioration. The

      eT1

      DCr c raQ0

      1 e ( r)T1 1 1 e rT1

      (17)

      vendors raw materials inventory system at any time t can be represented by the following differential equation and has shown in figure 1.

      ( r) r

      0

      The item cost includes the loss due to deterioration as well as the cost of the item sold. Because the order is

      Ir ' t

      Ir t

      aQ ,

      0 t T1

      (12)

      done at t = 0. The present worth of fuzzy item cost is

      On solving the above equation using the boundary condition I T1 0 , we have

      IC r

      crQr

      craQ T

      0 e 1 1

      (18)

      Ir t

      aQ (t T )

      0 e 1

      1 , 0 t T1 (13)

      The present worth of total cost during the cycle is the sum of fuzzy ordering cost O Cr , fuzzy holding cost

      H C r , fuzzy deterioration cost D C r and fuzzy item

      The maximum inventory level of the raw material,

      i.e. the order quantity per order from outside

      cost IC r . Thus for the raw materials, the present

      supplier worth of total fuzzy cost is

      aQ T

      T C 1r

      O C r

      H C r

      D C r

      IC r

      (19)

      Qr Ir 0

      0 e 1 1

      (14

      Present worth of total fuzzy profit is

      Since the order is done at the time t = 0, thus the

      T P

      S R

      P C

      H C

      D C

      O C

      T C

      (20)

      present worth of fuzzy ordering cost is

      1 1r

      Remark (1):

      T P1 is the function of T1 only. Now

      e(1 )(T t)

      1

      (1 ) , T t T

      our problem is to find the optimal value of

      T1 in

      1 1 2 (27)

      order to maximize the total profit T P1(T1) subject

      1

      (1 )

      to the inequality constraint Q Q0. Mathematically we have

      I t Q(1 ) e(1 )(T3 t)

      3 0

      (28)

      Maximizing

      T P1(T1)

      , T2

      t T3

      Subject to Q0

      Q 0 , (21)

      I4 t

      D e(T t )

      1 , T3 t T

      (29)

      1. Case II: Q Q0

      From equation (18), using the condition

      In this case production is started at time t = 0, with zero inventory level and stopped at time t = T2 when the inventory level reached the level Q, where

      I1 T1 Q0 , we have

      1 (a 1)D

      Q Q0 . Initially the demand and the production rate are constant up to the time t = T1 at which inventory level reaches to the level Qo after that demand becomes stock dependent so as the production rate up to the time t = T2. after that inventory level decreases due to the combined effect of the demand and the deterioration and reaches the zero level at the time t =

      T. Inventory level at any time can be described by the following differential equation and graphically represented in the figure-2.

      T1 ln

      Q

      (30)

      (a 1)D Q0

      I' t I t (a 1)D, 0 t T

      Q0

      (22)

      1 1 1

      I' t I t (a 1) I t ,T t T (23)

      Time

      2 2 2 1 2

      '

      0 T1

      T2 T3 T

      I3 t I3 t I3 t , T2 t T3 (24)

      Figure 4: Case II (Q > Q0)

      I' t I t D, T t T

      (25)

      4 4 3

      From equation (19) and (20), using the condition

      With the boundary conditions

      I2 T2

      I3 T2

      , we have

      I1 0 0, I2 T1 Q0 , I3 T3 Q0 and

      I4 T 0 . Solution of equations (22), (23), (24)

      T 1 ln

      1 ae(1 )T2

      0

      and (25) are as follows

      3 (1 ) ( Q(1 ) )

      I1 t

      (a 1)D 1 e t

      , 0 t T1

      (26)

      Q(1 )

      (a 1) e

      0

      (1 )T1

      (31)

      I t (a 1)

      Q(1 )

      (a 1)

      From equation (21) using the condition

      2 0

      I4 T3

      Q0 , we have

      4

      T 1 ln 1

      (1 )

      ae

      (1 )T2

      e ( r)T1

      1 c

      (a 1)

      1

      (1 )

      (1 ) ( Q0 ) 2

      (1 )

      (1 )T 1 (1 )

      Q0

      (a 1) e 1 ln 1 Q0

      (32)

      1 rT rT 1

      e 1 e 2

      Maximum inventory levels is

      r (1 )( r )

      Q(1 ) (1 )T ( r )T

      S (a 1)

      e(1 )(T T )

      (1 )

      Q

      0

      1

      (1 )

      (a 1)

      0 1 e (a 1)

      1 e 1

      1

      (1 )

      1 2 (33)

      Present worth of fuzzy holding cost is

      e ( r )T2

      eT3

      c2 Q(1 )

      0

      1

      e ( r)T2 e ( r)T3

      H C

      c1(a 1)D 1 1 e rT1

      1 e ( r)T1 1

      ( r) ( r)

      r ( r)

      1 eT3

      ( r)T ( r)T

      (a 1)

      1

      (1 ) 1

      rT rT 1

      e 3 e 2

      (1 ) Q(1 )

      c1

      e 1 e 2 0

      r (1 )( r )

      Q(1 )

      eT

      c2D e

      ( r)T3

      e ( r)T

      0

      (a 1)

      1 e(1 )T1

      e ( r )T1

      e ( r )T2

      ( r)

      1

      (1 )

      eT3

      1 e rT

      e rT3

      (35)

      c1

      Q(1 ) e ( r)T2

      e ( r)T3 r

      0 ( r)

      1 eT3

      Present worth of fuzzy production cost is

      paD (a 1)

      (1 )

      (1 )( r)

      (1 )

      P C 1 e

      rT1

      pa

      Q0 r

      c D

      1 rT rT 1

      Q(1 )

      e ( r)T3

      e ( r)T2

      1 e 1 e 2 0 1

      r (1 ) ( r ) (a 1)

      eT ( r)T

      ( r)T

      1 rT rT

      (1 )T1

      ( r )T1

      ( r )T2

      e 3 e e e 3

      ( r) r

      Present worth of fuzzy deterioration cost is

      (34)

      e e e (36)

      Present worth of fuzzy sales revenue is

      D C c

      (a 1)D 1 1 e

      rT1 1

      S R

      sD

      1 e rT1

      s

      (a 1)

      (1 )

      2 r ( r) r

      1

      1 e rT

      e rT 1

      2

      r (1 ) ( r )

      Q

      (1 )

      0

      1 e(1 )T1 e ( r )T1

      e ( r )T2

      (T

      t)

      a (a 1) (1 )

      I t e 2 1

      (a 1)

      T

      r2

      0

      0

      s Q(1 )

      (1 ) e 3

      e ( r)T2

      Q(1 )

      (a 1) e(1 )T1

      eT2 e t

      ( r)T

      ( r)

      1 e (1 2)T3

      (1 )(a 1)

      (1 )t

      e 3

      0

      (1 ) ( 2 r) Q(1 )

      e , T1

      t T2

      (42)

      e( 2 r)T3

      e( 2 r)T2

      sD

      e rT3

      e rT

      Using the condition

      Ir1 T1

      Ir 2

      T1 , maximum

      r

      (37)

      inventory level of the raw material, i.e. the order quantity per order from outside supplier

      Present worth of fuzzy ordering cost is

      aQ

      0

      Qr

      eT1 1

      a (a 1)

      (1 )

      O C

      c3

      (38)

      4.2.1. Raw Material Inventory Model

      T T

      Q(1 )

      (a 1)

      (1 )T

      Initially, vendor purchases the raw material in lots and produces the finished goods. Vendor starts production at time (t = 0), and the raw material reaches the zero level at the time (t = T2) due to the combined effect of production and deterioration. The vendors raw materials inventory system at any time t

      e 2 e 1

      eT2 eT1

      0 e 1

      (1 )(a 1)

      (43)

      can be represented by the following differential equation and has shown in figure 1.

      Since the order is done at the time t = 0, thus the

      present worth of fuzzy ordering cost is

      I ' t I t aQ , 0 t T

      (39)

      O C r

      c3r

      (44)

      r1 r1 0 1

      Ir 2 ' t

      Ir 2 t

      a I (t) , T

      t T2

      Inventory Level

      2 1

      I ' t

      I t

      a (a 1)

      Q(1 )

      r 2 r 2 0

      Qr

      (a 1) e(1 )(T t)

      (1 )

      , T t T

      1

      1 2 (40)

      On solving the above equation using the boundary

      0 T1 T2 Time

      condition Ir1

      0 Qr and Ir2 T2

      0 , we have

      Figure 5: Raw material inventory

      0

      Ir1

      t Qr

      aQ

      e t

      aQ

      0 , 0 t T1 (41)

      Present worth of fuzzy holding cost for the raw material is

      aQ 1

      ( r)T

      e ( r )T1

      e ( r )T2

      (46)

      H C r

      c1r

      Qr 0 1 e 1

      ( r)

      aQ

      rT a (a 1)

      (1 )

      eT2

      The item cost includes the loss due to deterioration as well as the cost of the item sold. Because the order is

      0 1 e 1

      done at t = 0. The present worth of fuzzy item cost is

      r ( r)

      ( r)T ( r)T 1 rT rT

      IC r

      c rQr

      (47)

      e 1 e

      2 e 1 e 2

      r

      The present worth of total cost during the cycle is the

      Q(1 )

      (a 1)

      (1 )T

      eT2

      ( r)T ( r)T

      sum of fuzzy ordering cost O C , fuzzy holding cost

      0 e 1 e 1 e 2 r

      (1 )(a 1) ( r)

      H C r , fuzzy deterioration cost D C r

      and fuzzy item

      1 e

      ( r )T

      e ( r )T

      cost

      IC r . Thus for the raw materials, the present

      1 2

      ( r )

      worth of total fuzzy cost is

      (45)

      T C 2r

      O C r

      H C r

      D C r

      IC r

      (48)

      Present worth of fuzzy deterioration cost for the raw material is

      Present worth of total fuzzy profit is

      aQ

      p>T P2

      S R

      P C

      H C

      D C

      O C

      T C 2r

      (49)

      D C r

      c2r

      Qr 0 1 e ( r)

      ( r)T1

      Remark (2):

      T P2 is the function of T2 only. Now

      0

      aQ

      1 e rT

      a (a 1)

      1

      (1 )

      eT2

      our problem is to find the optimal value of T2 in order

      to maximize the total profit

      TP2 (T2 ) subject to the

      r ( r)

      inequality constraint

      Q Q0 . Mathematically we

      e ( r)T1 e ( r)T2 1 e rT1 e rT2

      r

      have

      Maximizing

      T P2 ( T2 )

      Q(1 )

      (a 1)

      (1 )T

      eT2

      0 e 1

      Subject to Q Q0 0 , (50)

      (1 )(a 1) ( r)

      e ( r)T e ( r)T 1

      1 2

      ( r )

3. Numerical Illustrations

5.1. Case I: Q0 Q :

The following numerical data are used to illustrate the model.

a 1.2, 25, 0.25, Q0

150, c1

0.45, 0.50, 0.60 , c2

0.35, 0.40, 0.50 , c3

130, 200,300 ,

c1r

0.45, 0.50, 0.60 , c2r

0.35, 0.40, 0.50 , p

4,5, 7 , c3r

80,100,125 ,

cr

0.8,1.0,1.2 ,s

10,12,15 , d 0.1,i 0.05, r 0.05, 0.01.

For the above assumed parametric values the results are obtained using GA and results are presented in Table 1.

Table 1: Results for above assumed numeric data

It is found that optimal cycle length T = 4.95,

maximum inventory level Q = 64.15 units, production time T1 = 4.14 and the present value of the optimal profit is 823.67. At the beginning of each cycle the manufacturer initially purchases 401.20 units of the raw material and produce the finished goods items at a rate P for the period T1 = 4.14. During this period inventory of raw material

decreases and reaches to the zero level at time t =

4.14, on the other hand inventory of finished goods items build up to the time t = 4.14 and inventory level reaches 64.15 units. At this time the manufacturer stops production and inventory depleted due to the combined effect of demand and deterioration. Inventory level reaches zero level at time t = 4.95, then production for next cycle starts.

5.1.1. Sensitivity analysis: variation of the total profit w.r.t. different parameters.

Table 2: Present value of total profits of the model due to different production rates

Table 3: Present value of total profits of the model due to different demand parameter

Table 4: Present value of total profits of the model due to different demand parameter

Table 5: Present value of total profits of the model due to different r

500

500

1000

2 4 6 8 10

increases stock level and as demand is stock dependent, and hence increases the demand of the items, which in turn increases the profit. But increase of stock level is also increases the holding cost and deterioration cost. Initially profit due to increased demand dominates the loss due to increases holding cost and deterioration cost. But after certain level of production rate if production rate, the increases holding cost and the deterioration cost dominate the profit due to increased demand and after that level of production rate if production rate increases then profit decreases.

Figure 6: Total profit with respect to T1

6.1. Observations:

For the above parametric values optimal profit due to different production rates are obtained and presented in Table 2. It is observed that optimal profits increases with production rate and attains a maximum limit and then decreases as production increases. It happens because increase of production rate initially

5.2. Case II: Q Q0

Results are obtained for the different values of demand parameters and presented in the table 3 and

4. It is observed that profit increases with increase in demand parameters, which agrees with reality.

Results are obtained for the above parametric values and different values of resultant effect of inflation and discount rate r, and presented in the table 5. It is observed that profit decreases with increase of r, which agrees with reality.

The following numerical data are used to illustrate the model.

a 1.2, 25, 0.25, Q0

150, c1

0.45, 0.50, 0.60 , c2

0.35, 0.40, 0.50 ,

c3

130, 200,300 , c1r

0.45, 0.50, 0.60 , c2r

0.35, 0.40, 0.50 , p

4,5, 7 ,

c3r

80,100,125 , cr

0.8,1.0,1.2 ,s

10,12,15 , d 0.1, i 0.05, r 0.05, 0.01.

For the above assumed parametric values the results are obtained using GA and results are presented in Table 6.

Table 6: Results for above assumed numeric data

It is found that optimal cycle length T = 59.82,

maximum inventory level Q = 913.08 units, production time T2 = 51.66 and the present value of the optimal profit is 42221.80. At the beginning of each cycle the manufacturer initially purchases 11345.10 units of the raw material and produce the finished goods items at a rate P for the period t =

51.66. During this period inventory of raw material decreases and reaches to the zero level at time t =

51.66, on the other hand inventory of finished goods

items build up to the time t = 51.66 and inventory level reaches 913.08 units. At this time production is stopped and then inventory depleted due to the combined effect of demand and deterioration. Inventory level of finished goods items reaches zero level at time t = 59.82, then production for next cycle starts.

5.2.1. Sensitivity analysis: variation of the total profit w.r.t. different parameters.

Table 7: Present value of total profits of the model due to different production rates

Table 8: Present value of total profits of the model due to differnt demand parameter

Table 9: Present value of total profits of the model due to different demand parameter

Table 10: Present value of total profits of the model due to different r

40 000

30 000

20 000

Results are obtained for the above parametric values and different values of resultant effect of inflation and discount rate r, and presented in the table 10. It is observed that profit decreases with increase of r, which agrees with reality.

10 000

10 000

20 40 60 80 100

7. Conclusions and Future Research

In this paper, an EPQ model has been considered under imprecise and inflationary environment and time discounting over an infinite

Figure 7: Total profit with respect to T2

1. Observations:

   For the above parametric values optimal profit due to different production rates are obtained and presented in Table 7. It is observed that optimal profits increase as production rate increases. It happens because increase in production rate increases stock level and as demand is stock dependent, increased stock level increases the demand of the item, which in turn increases the profit. But increase in stock level increases the holding cost. Profit due to increased demand dominates the loss due to increased holding cost. Thus increase in the production rate increases optimal profit.Results are obtained for the different values of demand parameters and presented in the table 8 and 9. It is observed that profit increases with increase in demand parameters, which agrees with reality.

   horizon. Some interesting observations are presented. Demand is taken as the power form stock dependent and the production is taken as the demand dependent. In this paper we discussed the following two cases (I) Q Q0 and (II) Q > Q0, where Q is the stock-level at the time production is stopped and Q0 is the fixed stock-level. Model is formulated to maximize the total profit. Also a genetic algorithm with varying population size is used to solve a production inventory model. It is found that this GA is efficient in solving the proposed inventory model. This GA can also be used to solve different decision making problems in different fields of science and technology. This inventory model can be extend incorporating price dependent demand, trade credit policy, two warehouse, etc.

   Reference

   1. Balkhi, T.Z., Benkherouf, L. (2004), On an inventory model for deteriorating items with stock dependent and time varying demand

      rates. Computers and Operations Research 31, 223-240.

   2. Bessaou, M., Siarry, P. (2001), A genetic algorithm with real-value coding to optimize multimodal continuous function, Structural and Multi-disciplinary Optimization 23, 63-74.

   3. Buzacott, J.A. (1975), Economic order quantities with inflation, Operation Research Quarterly, 26, 553-558.

   4. Chandra M.J., Bahner, M.J. (1985), The effects of inflation and time value of money on some inventory systems. International Journal of Production Research, 23, 723-729.

   5. Chang H.C., Yao J.S., Quyang L.Y., ( 2004), Fuzzy mixture inventory model with variable lead-time based on probabilistic fuzzy set and triangular fuzzy number, Computer and Mathematical Modeling 39, 287-304.

   6. Datta T.K., Paul K., Pal A.K. (1998). Demand promotion by up-gradation under stock- dependent demand situation a model. International Journal of Production Economics, 55, 3138.

   7. Last, M., Eyal, S. (2005), A fuzzy-based lifetime extension of genetic algorithms, Fuzzy sets and Systems 149, 131-147

   8. Liao H.C., Tsai C-H, Su C.T. (2000), An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63, 207-214

   9. Maity M.K., Maiti M., ( 2007), Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm, European Journal of Operational Research 179, 352-371.

   10. Mehta N.J. and Shah N.H. (2003), An inventory model for deteriorating items with exponentially increasing demand and shortages under inflation and time discounting, Investigacao Operacional, 23, 103-111.

   11. Michalewicz, Z. (1992), Genetic algorithms + data structures = evolution programs. Berlin: Springer.

   12. Pezzella, F., Morgantia, G., Ciaschettib, G. (2008), A genetic algorithm for the flexibile job-shop scheduling problem, Computers and Operations Research 35, 3202-3212.

   13. R.B. (1979), A note on optimal inventory management under inflation. Naval Research Logistics, 26, 161-165.

   14. Sarkar, B.R., Mukherjee, S., Balan, C.V., (1997), An order level lot-size inventory model with inventory-level independent demand and deterioration. International Journal of Production Economics 48, 227-236.

   15. Singh, S. R., Singh, C., Singh, T.J., 2007. Optimal policy for decaying items with stock- dependent demand under inflation in a supply chain, International Review of Pure and Applied Mathematics, 3(2), 189-197.

   16. Singh, S. R., Kumari, R., Kumar, N., 2010. Replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial back logging with two- storage facility under inflation. International Journal of Operations Research and Optimization, 1(1), 161-179.

   17. Singh, S. R., Singh, C., (2010), Supply Chain Model with Stochastic Lead Time under Imprecise Partially Backlogging and Fuzzy Ramp-Type Demand for Expiring Items, International Journal of Operational Research 8(4), 511-522.

   18. Singh, S. R., Singh, C. (2010), Two echelon supply chain model with imperfect production, for Weibull distribution deteriorating items under imprecise and inflationary environment. International Journal of Operations Research and Optimization, 1(1), 9-25.

   19. Teng, J.T., Chang, C.T., (1995), EPQ model for deteriorating items with price and stock- dependent demand. Computers and Operations Research 31, 297-308.

   20. Teng, J.T. and Chang, C.T. (2005) Economic production quantity models for deteriorating items with price and stock dependent demand. Computers and Operational Research 32, 297- 308.

   21. Wu, K.S., Ouyang, L.Y., and Yang, C. T., (2006), An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging, International Journal of Production Economics 101, 369-384.

   22. Yao J.S., Lee H.M., (1999), Fuzzy inventory with or without backorder for fuzzy order quantity with trapezoidal fuzzy number, Fuzzy Sets and Systems 105, 311-337.

   23. Yao J.S., Chang S.C., Su J.S.,( 2000), Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity, European Journal of Operational Research 148, 401-409.