Design and Analysis of Inventory Model for Quadratic Trapezoidal Type Demand under Partial Backlogging

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Design and Analysis of Inventory Model for Quadratic Trapezoidal Type Demand under Partial Backlogging

Pravat Kumar Sukla

  1. S. College , Koksara, Kalahandi, odisha

    India

    Rajani Ballav Dasp

    Anjan Kumar Rana2

    Pragati College, Bhabanipatana, Kalahandi, Odisha,

    India

    Ex Reader in mathematics Revenshaw University, at present Visiting Faculty Revenshaw

    University, Cuttack. India

    Abstract:- In this paper, we consider the inventory model for perishable items with quadratic trapezoidal type demand rate, that is, the demand rate is a piecewise quadratic function. The model consider allows for shortages and the demand is partially backlogged. The model is solved analytically by minimizing the total inventory cost. The result is illustrated with numerical example for the model.

    Keywords: Quadratic ttrapezoidal demand. Deterioration. Shortages. Partial backlogging

    1. INTRODUCTION

      Deteriorating items are very common thing in our daily life situation. In recent years, many researchers have studied inventory models for deteriorating items, however, academia has not reached a consensus on the definition of the deteriorating items. According to the study of Wee (1993), deteriorating items refers to the items that become decayed, damaged, evaporative, expired, invalid, devaluation and so on through time. According to the definition, deteriorating items can be classified in to two categories. The first category refers to the items that become decayed, damaged, evaporative, or expired through time, like meat, vegetables, fruit, medicine, flowers and so on; the other category refers to the items that lose part or total value through time because of new technology or the introduction of alternatives, like computer chips, mobile phones, fashion and seasonal goods and so on. The inventory problem of deteriorating items was first studied by Whitin (1957), he studied fashion items deteriorating at the end of the storage period. Then Ghare and Schrader (1963) concluded in their study that the consumption of the deteriorating items was closely relative to a negative exponential function of time. Various authors (Deng et al. (2007), Cheng and Wang (2009), Cheng et al. (2011), Hung (2011)) studied inventory models for deteriorating items in various aspects.

      In world business market, demand has been always one of the most key factors in the decisions relating to the inventory and production activities. There are mainly two

      categories demands in the present studies, one is deterministic demand and the other is stochastic demand. Various formations of consumption tendency have been studied, such as constant demand (Padmanabhan and Vrat (1990), Sukla (2012), , Sukla and sahu (2008) Chung and Lin (2001), Benkherouf et al. (2003), Chu et al (2004)), level-dependent demand (Giri and Choudhuri (1998), Chung et al. (2000), Bhattacharya (2005), Wu et al. (2006)), price dependent demand (Wee and Law (1999), Abad (1996, 2001)), time dependent demand (Resh et al. (1976), Henery (1979), Sachan (1984), Dave (1989), Teng

      (1996), Teng et al. (2002), Skouri and Papachristos (2002), Panda, Sahoo, and Sukla (2012) ), Panda, Sahoo, and Sukla (2013) , Sett et al. (2013), Shah, , Chaudhari and Jani (2015), Shah, , Chaudhari and Jani (2016), Mishra et al. (2013)) and time and price dependent demand (Wee (1995)). Among them, ramp type demand is a special type of time dependent demand. Hill (1995), one of the pioneers, developed an inventory model with ramp type demand that begins with a linear increasing demand until to

      the turning point, denoted as , proposed by previous

      researchers, then it becomes a constant demand. There has been a movement towards developing this type of inventory system for minimum cost and maximum profit problems. Several authors: Mandal and Pal (1998) focused on deteriorating items. Wu et al. (1999) were concerned with backlog rates relative to the waiting time. Wu and Ouyang (2000) tried to build an inventory system under two replenishment policies: starting with shortage or without shortage. Panda, Sahoo and sukla (2013), Wu (2001) considered the deteriorated items satisfying Weibull distribution. Giri et al (2003) dealt with more generalized three parameter Weibull deterioration distribution. Deng (2005) extended the inventory model of Wu et al. (1999) for the situation where the in-stock period is shorter

      than . Manna and Chaudhuri (2006) set up a model

      where the deterioration is dependent on time. Panda et al. (2007) constructed an inventory model with a comprehensive ramp type demand. Deng et al. (2007)

      contributed to the revision of Mandal and Pal (1998), and Wu and Ouyang (2000). Panda et al. (2008) examined the cyclic deterioration items. Wu et al. (2008) studied the maximum profit problem with the stock-dependent selling rate. They developed two inventory models all related to the conversion of the ramp type demand, and then examined the optimal solution for each case. However, in a realistic product life cycle, demand is increasing with time during the growth phase. Then, after reaching its peak, the demand becomes stable for a finite time period called the maturity phase. Thereafter, the demand starts decreasing with time and eventually reaching zero or constant.

      In this work, we extend Hills ramp type demand rate to quadratic trapezoidal type demand rate. Such type of demand pattern is generally seen in the case of any fad or seasonal goods coming to market. The demand rate for such items increases quadratic-ally with the time up to certain time and then ultimately stabilizes and becomes constant, and finally the demand rate approximately decreases to a constant, and then begins the next replenishment cycle. We think that such type of demand rate is quite natural and useful in real world market situation. One can think that our work may provide a solid foundation for the future study of this kind of important inventory models with quadratic trapezoidal type demand rate and preservation technology

    2. ASSUMPTION AND NOTATIONS

      • k2 is the inventory holding cost per unit per unit of time.

      • k3 is the shortage cost per unit per unit of time.

      • S is the maximum inventory level for the ordering cycle, such that S=I(0).

      • Q is the ordering quantity per cycle.

      • A1(t1) is the average total cost per unit time under

        the condition t1 1 .

      • A2(t1) is the average total cost per unit time, for

        1 t1 2 .

      • A3(t1) is the average total cost per unit time, for

        2 t1 T

    3. MATHEMATICAL AND THEORETICAL RESULTS

      Here, we consider the deteriorating inventory model with demand rate is trapezoidal type quadratic function. Replenishment occurs at time t =0 when the inventory level attains its maximum. For t [0, t1 ] , the inventory level reduces due to both demand and deterioration. At time t1, the inventory level reaches zero, then shortage is allowed to occur during the interval (t1, T), and all of the demand during the shortage period (t1, T) is completely backlogged. The total amount of backlogged items is replaced by the next replenishment. The rate of change of the inventory during the stock period [0, t1] and shortage period (t1, T) is governed by the following differential equations:

      The fundamental assumption and notations used in this paper are given as follows: The demand rate, R(t), which is positive and consecutive, is assumed to be a quadratic

      dI (t) I (t) R(t) 0 , 0 t t

      dt 1

      dI (t)

      , (2)

      trapeoidal type function of time, that is

      R(t) 0 , t1 t T , (3)

      b t c t 2 , t , dt

      1 1 1

      with boundary condition I(0)=S and I(t1)=0. One can think

      R(t) R0 ,

      1 t 2 ,

      about t1, t1 may occur within

      [0, ] or [ , ]

      b t c t 2 , t T

      1 1 2

      2 2 2

      (1)

      or[ 2 , T ] . Hence in this paper we are going to discuss all three possible cases.

      Chose b , c1, b and c2 such a way that b t c t 2 should

      Case 1:

      0 t

      1 2 2 2 1 1

      not be negative for 2 t T . Where 1 is the time point changing from the increasing quadratic demand to constant demand, and 2 is the time point changing from

      The quadratic trapezoidal type market demand and constant rate of deterioration, the inventory level gradually diminishes during the period [0, t1] and ultimately reaches to zero at time t=t1. Then, from equations (2) and (3), we

      the constant demand to the decreasing demand.

      • Replenishment rate is infinite, thus replenishment

        have

        dI (t) I (t) b t c t 2 0 , 0 t t

        (4)

        is instantaneous. 1 1 1

      • I(t) is the inventory level at any time t, dt

        0 t T .

        dI (t) b t c t 2 0 , t t

        (5)

      • T is the fixed length of each ordering cycle.

        dt 1 1 1 1

      • is the constant rate of deterioration,

        dI (t) R

        0 ,

        t

        (6)

        0 1.

      • t1 is the time when the inventory level reaches

        zero.

        1

        1

      • t * is an optimal point.

      • k0 is the fixed ordering cost per order.

      • k1 is the cost of each deteriorated item.

      dt 0 1 2

      dI (t) b t c t 2 0 , t T (7)

      dt 2 2 2

      Now solving the differential equations (4) (7) with the condition I(t1)=0 and continuous property of I(t), we get

      b t c t 2 b 2c t 2c

      c t 2 b 2c t 2c

      I (t) 1 1 1 1 1 1 1 1 e

      (t1 t )

      1 1 1 1 1 1 e (t1 t )

      2

      3

      t1

      2 3

      b t c t 2

      b 2c t 2c

      0

      b t c t 2

      b 2c t 2c

      dt

      1 1 1 1 1

      1 1 1 1 1

      , 0 t t1

      2 3

      (8)

      b t

      • c t 2

        3

        3

        b 2c t

        2 3

        4

        4

        2c

        I (t) (t 2 t 2 ) b1 (t 3 t 3 ) c1 , t

        t

        1 1 1 1

        2

        1 1 1

        1 (e t1 1)

        1 2 1 3 1

        b

        1 (9)

        c

        b1

        2c1

        c1

        b1 2 c1 t 3

        I (t) R t (t 2 2 ) 1 (t 3 2 3 ) 1 ,

        2

        3 t1

        2

        2 t1

        3 1

        (14)

        0 1 1 2 1 1 3

        1 t 2

        I (t) (t 2 2 ) b1 (t 3 2 3 ) c1

        (10)

        The total shortage quantity during the interval [t1, T], say BT, is

        1 1 2 1 1 3

        ,

        b c

        T

        BT I (t)dt

        (t 2 2 ) 2 (t 3 2 3 ) 2 t1

        2 2 2 3

        1 2 T

        2 t T

        (11)

        I (t)dt I (t)dt I (t)dt

        The beginning inventory level can be computed as

        t1 1 2

        b 2c

        1 b c

        2

        3

        1

        2 1 3

        S I (0) 1 1 (et1 1)

        (12)

        (t 2 t 2 )

        t1

        1 (t 3 t 3 ) 1 dt

        b t c t 2 2c t

        t

        1 1 1 1 1 1 e 1 2

        2

        R t (t 2 2 ) b1 (t 3 2 3 ) c1

        0

        1 1 2 1

        1 3 dt

        The total number of items which is perish in the interval [0, t1], say DT, is

        1

        2 2 b1 3

        3 c1

        t1 t1

        2

        T (t1 1 ) 2 (t1 21 ) 3

        DT S R(t)dt S (b1t c1t

        )dt

        b

        dt

        c

        0 0 2 (t 2 2 ) 2 (t 3 2 3 ) 2

        b 2c t

        2 2

        2 3

        1 1 (e 1 1)

        2 3

        b1 t 2 ( t ) b1 ( 3 t 3 )

        b t c t 2 2c t

        b t 2

        (13)

        c t 3

        2 1 1 1 6 1 1

        1 1 1 1 1 1 et1 1 1 1 1 c c

        2 2 3

        1 t 3 (

        t ) 1 ( 4 t 4 )

        3 1 1 1 12 1 1

        The total amounts of inventory carried during the R b

        interval [0, t1], say CT, is

        0 ( 2 2 ) 1 t 2 (

        )

        t1 2 1 2 2 1 2 1

        CT I (t)dt

        b1 2 (

        ) c1 t 3 (

        )

        0 2 1 2

        1 3 1 2 1

        2c1 3 (

        ) b1 t 2 (T

        ) b1 2 (T )

        3 1 2

        1 2 1

        2 2 1 2

        c1 (t 3 2 3 )(T

        ) b2 (T 3 3 ) b2 2 (T

        ) c2 (T 4 4 ) 2c2 3 (T )

        (15)

        3 1 1 2 6

        2 2 2

        2 12

        2 3 2 2

        The average total cost per unit time for 0 t1 1 is given by

        The total back order amount at the end of the cycle is

        1 b1 (t*2 2 ) c1 (t*3 2 3 )

        A1 (t1 ) T [k0 k1 DT k2CT k3 BT ] (16)

        1 2 1 1 3 1

        1

        (21)

        The first order derivative of A1 (t1 ) with respect

        b2 (T 2 2 ) c2 (T 3 2 3 )

        to t is as follows:

        2 2 3 2

        1 Therefore, the optimal order quantity, denoted by

        1

        1

        Q* , is Q* S * , where

        S * denote the

        T

        T

        dA1 (t1 ) 1 k

        k2 (et1 1) k (t

        T )

        optimal value of S.

        1

        1

        dt1

        3 1

        Case-II, 1 t1 2

        1 1

        1 1

        2

        2

        (b1t1 c t )

        The necessary condition for

        (17)

        A1 (t1 ) to be

        For the time period t1 [1 , 2 ] , then, the differential equations governing the inventory model can be expressed as follows:

        minimized, is

        dA1 (t1 ) 0 , that is

        dI (t) I (t) b t c t 2 0 , 0 t

        dt1

        1 k

        dt 1 1

        1

        (22)

        k 2 (e t1 1) k (t

        T )

        dI (t)

        T 1

        3 1

        (18)

        I (t) R0 0 , 1 t t1

        dt

        (23)

        (b t

        c t 2 ) 0

        dI (t)

        1 1 1 1 R0 0 , t1 t 2

        (24)

        This implies that dt

        1

        1

        k2 t

        dI (t) 2

        k

        (e

        1 1) k3 (t1 T ) 0

        (19)

        dt b2t c2t

        0 , 2 t T

        (25)

        Let p(t ) k k2 (et1 1) k (t

        T ) (20)

        Solving differential equations (22) to (25), using

        1 1 3 1

        I(t1)=0, we get

        R

        b b t c t 2

        b 2c t 2c

        Since

        I (t) 0 et1 1 e1 et 1 1 1 1 1

        k

        2

        2 3

        p(0) k T 0, p(T ) k 2 (eT 1) 0

        and

        3 1

        2c1 1 e ( 1 t ) 2c1 e ( 1 t ) , 0 t

        k

        2 3 1

        p(t ) k 1 et1 k

        0 , it implies

        (26)

        1 1 3

        that p(t1) is a strictly monotonically increasing function and equation (19) has a unique solution at

        t * , for t* (0, T ) . Therefore, we have

        I (t) R0 (e (t1 t ) 1) ,

        1

        t t1

        (27)

        1 1 I (t) R (t

        t), t1 t 2

        (28)

        Property-1

        The constant deteriorating rate of an inventory model with quadratic trapezoidal type demand rate under the time interval 0 t1 1 ,

        A (t ) attains its minimum at t t* , where

        0 1

        I (t) R t b2 (t 2 2 ) c2 (t 3 2 3 )

        0 1 2 2 3 2

        , 2 t T

        1 1 1 1

        (29)

        p(t* ) 0 if t*

        . On the other hand,

        The beginning inventory can be computed as

        1 1 1

        A (t ) attains is minimum at t* if

        1 1 1 1

        t* .

        1 1

        S I (0)

        R0 et1

        • b1

        2

        e1

      • b1

        2

        2c1

        3

        2c1 1 e1

        2

        2c1 e1

        3

        (30)

        The total amount of items which is perish within the time interval [0, t1] is

        t

        t

        R t b2 (t 2 2 )

        2 T 0 1 2

        2 T 0 1 2

        2

        1 R0 (t1 t)dt

        dt

        DT S R(t)dt

        t1 2 c2 (t 3 2 3 )

        0

        S (b1t c1t )dt R dt

        S (b1t c1t )dt R dt

        1 t1

        3

        R

        2

        0

        0

        2 R t (

        t ) 0 ( 2 t 2 )

        0 1

        0 1 2 1

        2 2 1

        R0 et1 b1

        2c11 2c1 e1 R (t

        )

        • R t (T

        ) b2 (T 3 3 )

        2

        2 3

        0 1 1

        0 1 2 6 2

        1 2 2 3

        (31)

        b2 2 (T ) c2 (T 4 4 )

        b1 c

        2 1

        2 1

        3

        3

        1 1

        2 2 2 12 2

        2 3

        (33)

        2c

        The total amount of inventory carried during the time interval [0, t1] is

        2 3 (T ) 3 2 2

        t1

        CT I (t)dt

        0

        Now, the average total cost per unit time under the

        condition 1 t1 2 , can be obtained as

        1

        1 t1

        A2 (t1 ) T [k0 k1 DT k2CT k3 BT ] (34)

        I (t)dt I (t)dt

        0 1

        The first order derivative of A2 (t1 ) with respect to

        t1 isgiv

        R t b

        t

        b t c t 2

        0 e

        1 1 e

        1 e

        1 1

        dA2 (t1 ) R0 k

        k2 (et1 1) k (t

        T )

        2

        dt T 1

        3 1

        b

        b

        1

        1

      • 2c1t

        2

        2c1

        3

        dt

        1

        (35)

        0

        The required necessary condition for

        A2 (t1 ) to be

        2c11 e ( 1 t ) 2c1 e ( 1 t )

        dA (t )

        2 3

        minimized is 2 1 0 , that is

        dt

        R (t t )

        R (t t )

        t

        1

        k2 t

        1

        0 (e 1 1) dt

        k1 (e 1 1) k3 (t1 T ) 0

        (36)

        1

        k

        b R t b

        Let p(t ) k 2 (et1 1) k (t

        T ) ,

        3

        3

        2

        2

        3

        3

        1 0 e 1 1 e 1

        1 1 3 1

        b 2c 2c 2c

        (37)

        1 1 1 1 e 1 1 1 e 1

        k

        2 3

        4 4

        since p(t ) k 1 et1 k

        0 , which

        R0 R0 (t )

        (32)

        1 1 3

        2 1 1

        implies that p(t1 ) is strictly monotonically

        The total amount of shortage during the interval [t1, T]

        T

        BT I (t)dt

        t1

        2 T

        I (t)dt I (t)dt

        increasing function during the interval

        1 t1 2 .

        Property-2

        The constant deteriorating rate of an inventory model with quadratic trapezoidal type demand function during the time interval 1 t1 2 ,

        t1 2

        A (t ) attains its minimum at t* if

        2 1 1 1

        t* and A (t )

        attains its minimum

        1 1 2 1

        1

        1

        2

        2

        at t*

        if 2

        t* .

        1

        1

        Now, we can calculate the total amount of back- order quantity at the end of the cycle is

        R t * b2 (T 2 2

        • c2

          t1

          3 3

          2 0 1 2

          ) (T

          2 3

          2 2 )

          (38)

          DT S R(t)dt

          Therefore, the optimal order quantity denoted by 0

          Q* is Q* S* , where

          2

          S * denotes the

          1

          S (b t c t 2 )dt

          2 t1

          R dt b t c t 2 )dt

          optimal vale of S.

          Case-III 2 t1 T

          1 1

          0

          b 2c t 2c

          b t c t2

          0 2 2

          1 2

          For the time interval t

          [

          ,T ) , then, the

          I (t) 1 1 1 1 1

          1 2 2 3

          3

          3

          differential equations governing the inventory model can be expressed as follows:

          2

          b t c t

          b t c t

          b

          b

          2 1 2 1 2

          2

          2

          2c2t1

          2

          2c2 e (t1 t )

          dI (t) I (t) b t c t 2 0 , 0 t

          (39)

          dt 1 1 1

          b 2c b t 2

          dI (t)

          1 1 2 1

          c2t

        • b2

          2c2t1 2c2 et1

          I (t) R0 0 , 1 t 2

          dt

          (40)

          2 3 2 3

          dI (t) I (t) b t c t 2 0 , t t

          (41)

          b1 2c11 2c1 1

          dt 2 2 2 1

          2 3

          e

          e

          dI (t) 2

          b2 2c2 2

          2c2

          b1 2

          c1 3

          dt b2t c2t

          0 , t1 t T

          (42)

          2

          3 e

          2

          2

          1 3 1

          Solving the differential equations (39)- (42) with I(t1)=0, b c

          we can get

          R ( ) 2 (t 2 2 ) 2 (t 3 3 )

          b 2c

          2c

          2c

          b 2c

          0 2 1 2 1

          2 3 1 2

          2 2 2 2 e ( 2 t ) 1 1 1 1 e ( 1 t ) ,

          (48)

          2

          3

          3

          2

          The total amount of inventory carried during the

          0 t 1

          (43)

          time interval [0, t1] is

          t1

          b t

        • c t 2 b

          C I (t)dt

          2 1 2 1 2 T

          2

          2

          I (t) R0

          e (t1 t ) 0

          2c t 2c

          1 2 t1

          2 1 2

          2 3

          I (t)dt

          0

          I (t)dt

          1

          I (t)dt

          2

          b2 2c2 2 2c2 e ( 2 t ) ,

          t

          (44)

          2 3 1 2

          b 2c t 2c b t c t 2

          b t c t 2 b 2c t

          2c

          b 2c t

          2c b t c t 2 ,

          1 1 1 1

          3

          3

          e

          e

          1

          1

          2

          2

          2

          I (t) 2 1

          2 1 2

          2 1

          2 e (t1 t ) 2

          2 2 2 2

          b t

        • c t

          b 2c t

          2c

          2 2

          3

          2 3

          2 1 2 1 2

          2

          2 1 2

          2 3

          (t1 t )

          2 t t1

          b c

          (45)

          1

          b 2c

          2c

          dt

          I (t)

          2 (t 2 t 2 ) 2 (t 3 t 3 ) , t

          t T

          0 2

          2 2

          2 e ( 2 t )

          2 1 3 1 1

          2

          3

          (46)The total amount of inventory level at the beginning

          2c

          b 2c

          can be computed as

          1 1 1 1 e ( 1 t )

          b 2c

          3 2

          S I (0) 1 1

          2 3

          R b t

        • c t 2 b

      2c t

      2c

      2 0 2 1 2 1 2 2 1 2 e (t1 t )

      b2t1 c2t1 b2

      2c2t1

      2c2

      t1

      2

      2 2

      3

      2 2

      3

      e

      dt

      1 b2 2c2 2 2c2 e ( 2 t )

      b2 2c2 2

      2c

      2

      2

      e

      2c

      2

      b1 2c11 1

      1

      1

      /p>

      2

      e

      e

      3

      2

      3

      3

      2

      t1 2 2

      2

      2

      (47)

      b2t1 c2t1 b2 2c2t1 2c2 e (t1 t ) b2 2c2t 2c2 b2t c2t dt

      The total amount of items which is perish within the time interval [0, t1] is

      2

      2 3

      2 3

      b t c t2

      2c t b

      2c et1 1

      k1

      t

      2 1 2 1 2 1 2 2

      since p (t1 )

      k1 e 1

      k3 0 , which

      3

      3

      2

      b 2c 2c e2 1

      implies that p(t1 ) is strictly monotonically

      2 2 2 2

      increasing function within the interval

      2 3

      t [ , T ] .

      2c b 2c e1 1 b 2 c 3 b

      2c

      1 2

      c 2

      1 1 1 1 1 1 1 1 1 1 1 1 1

      3

      2 2

      3 2

      3 2

      Property-3

      In this case, the inventory model under the

      R0 ( ) b2 (t 2 2 ) c2 (t 3 3 )

      condition 2 t1 T , A3 (t1 ) attains its

      2 1

      2 1

      2 3 1 2

      minimum at t t* , where

      1 1

      1 1

      b2

      2c2 (t

      ) c2 (t 2 2 )

      (49)

      p(t* ) 0 if t* . On the other hand,

      2

      3 1

      2 2 1 2

      1 2 1

      A (t )

      attains its minimum

      Total quantity of shortage during the time interval [t1, T] is 3 1

      1

      1

      1

      1

      2

      2

      2

      2

      T at t*

      if t*

      .Now, we can calculate the

      BT I (t)dt

      total back-order quantity at the end of the cycle is

      1

      1

      2 (t t )

      2 (t t )

      t b2 (T 2 t * 2 ) c2 (t *3 T 3 ) .

      T b c

      3 2 1 3 1

      2 3 3 dt

      2 3 3 dt

      2 2 2

      1

      t1

      3 (t t1 )

      Therefore, the optimal order quantity, denoted by Q* , is Q* S * , where S * denotes the

      b2 t 2 (T t ) b2 (T 3 t

      2 1 1 6

      c t3

      2 1 (T t1)

      3

      3 ) c2 (T 4 t 4 )

      1 12 1

      (50)

      3

      optimal value of S.From the above three cases, we can derive the following results

      Result-1

      An inventory model having constant deteriorating

      Then, the total average cost per unit time under the time interval 2 t1 T , can be written as

      rate with quadratic trapezoidal type demand, the optimal replenishment time is * and

      A (t ) 1 [k k D k C k B ] (51)

      t

      t

      1

      1

      A (t ) attains its minimum at t t* if and only

      3 1 T 0 1 T 2 T 3 T

      1 1 1 1

      if t* . On the other hand, A (t ) attains its

      The first order derivative of

      A (t ) with respect to t is as 1 1 2 1

      3 1 1

      minimum at t t* if and only if t*

      follows:

      1 1 1 1 2

      and A (t ) attains its minimum at t t* if and

      dA3 (t1 ) 1 k k2 et1 1) k (t T )(b t c t 2 ) 3 1 1 1

      dt T 1

      (

      3 1 2 1 2 1

      only if

      t* , where t * is the unique solution of

      1

      2 1 1

      The required necessary condition for minimized is

      dA (t )

      (52)

      A3 (t1 ) to be

      equation p(t1 ) 0 .

      Example 1

      We can consider suitable values of the following

      3 1 0 , that is

      dt

      parameters as follows: T= 20 weeks, 1 = 6

      1 weeks,

      2 =15 weeks, b1=10 unit, c1= 5 unit,

      1 k

      k2 (et1 1) k (t

      T )

      b2=20 unit, c2= 2 unit, 0.1, k0=$220, k1= $3

      T

      T

      1

      1

      3 1

      (53)

      per unit, k2=$12 per unit, k3=$4 per unit. By Using MATHEMATICA 8.0 the above data, we can find

      1 1 1 1

      1 1 1 1

      (b t c t 2 ) 0

      p(1 ) =168.1206>0, the optimal replenishment

      time t * =3.41 weeks, the optimal order quantity

      This implies that 1

      t

      t

      k

      Q*, for each ordering cycle, is 3576.478 unit and

      k1 2 (e 1 1) k3 (t1 T ) 0

      (54)

      the minimum cost A (t * )=$4688.2

      1 1

      Let p(t ) k k2 (et1 1) k (t

      T ) ,(55)

      1 1

      1 1

      3 1

      parameter

      b1

      Table

      1 1

      1 1

      0

      0

      t1

      Q

      (

      +50

      3.53

      3674.30

      4894

      + 25

      3.50

      3664.02

      4883

      + 20

      3.45

      3584.112

      4730

      +10

      3.41

      3576.478

      4688

      -10

      3.30

      3576.478

      4626

      0

      0

      t1

      Q

      (

      +50

      3.53

      3674.30

      4894

      + 25

      3.50

      3664.02

      4883

      + 20

      3.45

      3584.112

      4730

      +10

      3.41

      3576.478

      4688

      -10

      3.30

      3576.478

      4626

      A t)

      .2

      .5

      .1

      .2

      .4

      + 50 3.86 3665.4 4867.3

      + 25 3.77 3584.7 4856.9

      + 20 3.66 3563.2 4852.2

      +10 3.512 3533.4 4847.3

      -10 3.487 3489.7 4842.2

      -20 3.415 3477.8 4834.6

      -25 3.330 3465,4 4830.7

      -50 3.311 3443.6 4822.4

      -20

      3.27

      3560.903

      4610.7

      -25

      3.20

      3284.369

      4577.6

      + 50

      4.265

      2824.5

      3346.8

      -50

      3.17

      3225.332

      4566.9

      + 25

      3.880

      2753.8

      3384.2

      .

      +50 3.07 3888.237 4331.4

      +25 3.31 3424.66 4874.7

      +20 3.35 3234.050 3814.8

      +10 3.44 3021.12 3 3665.3

      c1 -10 3.73 2734.464 3396.0

      + 20 3.825 2757.2 3391.1

      +10 3.654 2631.8 3513.6

      k1 -10 3.057 2566.3 3544.2

      -20 2.879 2108.1 3573.2

      -25 2.533 2015.5 3604.9

      -50 1.865 1994.2 3604.9

      -20

      3.82

      2494.192

      3123.4

      -25

      4.26

      2124.375

      3068.8

      + 50

      2.936

      2724.6

      3674.2

      -50

      4.41

      1912.005

      2894.4

      +25

      3.833

      2675.7

      3624.1

      + 50 2.99 2887.633 4843.2

      +10 3.63 3476.768 4655.6

      b2 -10 3.69 3534.739 4596.7

      +20 3.714 2634.3 3600.5

      + 10 3.685 2536.3 3557.4

      k2 -10 3.612 2222.1 3487.2

      – 20 3.467 2185.2 3426.2

      -25 3.345 2105.4 3385.1

      -50 3.292 1538.7 3114.8

      +25

      3.48

      3331.537

      4771.3

      +20

      3.55

      3379.185

      4733.8

      -20

      3.76

      3581.35

      4534.2

      .

      -25

      3.88

      3636.423

      4412.5

      +50

      4.265

      2824.5

      3346.8

      -50

      4.75

      3675.547

      4385.4

      + 25

      3.884

      2753

      3384.2

      50 3.645 3658.4 4855.0

      25 3.583 3497.2 4839.7

      20 3.572 3484.5 4834.6

      10 3.557 3448.1 4826.4

      c2 -10 3.534 3436.3 4823.1

      + 20 3.825 2657.2 3391.1

      + 10 3.654 2631.8 3513.6

      3

      -20

      2.879

      2108.1

      3573.2

      -25

      2.533

      2015.5

      3604.9

      -50

      1.865

      19994.2

      3688.3

      3

      -20

      2.879

      2108.1

      3573.2

      -25

      2.533

      2015.5

      3604.9

      -50

      1.865

      19994.2

      3688.3

      k -10 3.057 2566.3 3544.2

      performed by changing the parameter -50%, -25%, -20, –

      -20

      3.475

      3401.2

      4818.5

      -25

      3.461

      3382.3

      4806.9

      -50

      3.391

      3232.4

      4806.9

      In the above table some sensitivity analysis of the model is

      -20

      3.475

      3401.2

      4818.5

      -25

      3.461

      3382.3

      4806.9

      -50

      3.391

      3232.4

      4806.9

      In the above table some sensitivity analysis of the model is

      10, 10%,20%, 25%, and 50%, taking one at time and keeping the remaining parameters unaltered.

      CONCLUSION 4

      In a realistic product life cycle, demand is increasing with time during the growth phase. Then, after reaching its peak, the demand becomes stable for a finite time period called the maturity phase. Thereafter, the demand starts decreasing with time. Therefore, in this paper, we study the inventory model for constant deteriorating items with quadratic trapezoidal demand rate. We proposed an inventory replenishment policy for this type of inventory model. From the market information, we find that the quadratic trapezoidal type demand rate is more realistic than ramp type demand rate, constant demand rate and other time dependent demand rate Our paper provides an interesting topic for the future study of such kind of important inventory models, and at the same time, the following problems can be considered for future research work (1) How about the inventory model starting with shortages? (2) How about the inventory model with time dependent deteriorating rate instead of constant deteriorating rate?

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BIOGRAPHY

1. Pravat kumar sukla is working as a lecturer in mathematics Panchayat samiti college koksara, kalahandi, odisha, INDIA. He has obtained his M.sc and Ph.D in mathematics from Sambalpur University, odisha, India .he has published 11 research papers in different national and international journals. His research interests include inventory control.

3. Rajani Ballav Dash is Ex Reader in mathematics Revenshaw University, at present Visiting Faculty Revenshaw University, Cuttack. He has a retired principal,

S.C.S (Autonomous) college, Puri, odisha Ex- Visiting Faculty Institute of mathematics and Applications, bbsr

,odisha.Ten students have completed Ph.D under guidance. He has published 50 research papers different national and international journals and NO of text books -14. His research interests include inventory control and Numerical Analysics, Operations Research (Theorical and applied), Graph Theory and fracture mechanics.

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