International Journal of Mathematics and Statistics Invention (IJMSI)
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI......

International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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  • 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 1 Issue 2ǁ December. 2013ǁ PP 01-15 Perishable Inventory System with a Finite Population and Repeated Attempts K. Jeganathan And N. Anbazhagan Department of Mathematics, Alagappa University, Karaikudi-630 004, Tamil Nadu, India. ABSTRACT : In this article, we consider a two commodity continuous review perishable inventory system with a finite number of homogeneous sources of demands. The maximum storage capacity of S i units for the i th commodity (i = 1,2) . The life time of items of each commodity is assumed to be exponentially distributed with parameter  i (i = 1,2) . The time points of primary demand occurrences form independent quasi random distributions each with parameter  i ( i = 1,2). A joint reordering policy is adopted with a random lead time for orders with exponential distribution. When the inventory position of both commodities are zero, any arriving primary demand enters into an orbit. The demands in the orbit send out signal to compete for their demand which is distributed as exponential. We assume that the two commodities are both way substitutable. The joint probability distribution for both commodities and number of demands in the orbit is obtained for the steady state case. Various system performance measures are derived and the results are illustrated with numerical examples. KEYWORDS: Retrial Demand, Positive Lead-Time, Finite Population, Perishable Inventory, Substitutable, Markov Process, Continuous Review. I. INTRODUCTION The analysis of perishable inventory systems has been the theme of many articles due to its potential applications in sectors like food industries, drug industries, chemical industries, photographic materials, pharmaceuticals, blood bank management and even electronic items such as memory chips. The often quoted review articles ([21], [23]) and the recent review articles ([24], [16]) provide excellent summaries of many of these modelling efforts. Most of these models deal with either the periodic review systems with fixed life times or continuous review systems with instantaneous supply of reorders. One of the factors that contribute the complexity of the present day inventory system is the multitude of items stocked and this necessitated the multicommodity systems. In dealing with such systems, in the earlier days models were proposed with independently established reorder points. But in situations were several product compete for limited storage space or share the same transport facility or are produced on (procured from) the same equipment (supplier) the above strategy overlooks the potential savings associated with joint ordering and, hence, will not be optimal. Thus, the coordinated, or what is known as joint replenishment, reduces the ordering and setup costs and allows the user to take advantage of quantity discounts [17]. Inventory system with multiple items have been subject matter for many investigators in the past. Such studies vary from simple extensions of EOQ analysis to sophisticated stochastic models. References may be found in ([7], [17], [20], [22], [25], [28]) and the references therein. Multi commodity inventory system has received more attention on the researchers on the last five decades. In continuous review inventory systems, Ballintfy [11] and Silver [25] have considered a coordinated reordering policy which is represented by the triplet ( S , c , s ) , where the three parameters S i , c i and s i are specified for each item i with s i  c i  S i , under the unit sized Poisson demand and constant lead time. In this policy, if the level of i th commodity at any time is below s i , an order is placed for S i  s i items and at the same time, any other item j (  i ) with available inventory at or below its can-order level c j , an order is placed so as to bring its level back to its maximum capacity S j . Subsequently many articles have appeared with models involving the above policy and another article of interest is due to Federgruen et al. [14], which deals with the general case of compound Poisson demands and non-zero lead times. A review of inventory models under joint replenishment is provided by Goyal and Satir [15]. www.ijmsi.org 1|Pa ge
  • 2. Perishable Inventory System With A Finite.. Kalpakam and Arivarignan [17] have introduced ( s , S ) policy with a single reorder level s defined in terms of the total number of items in the stock. This policy avoids separate ordering for each commodity and hence a single processing of orders for both commodities has some advantages in situation where in procurement is made from the same supplies, items are produced on the same machine, or items have to be supplied by the same transport facility. Krishnamoorthy and Varghese [18] have considered a two commodity inventory problem without lead time and with Markov shift in demand for the type of commodity namely ’‘commodity-1’’, ‘‘commodity-2’’ or ‘‘both commodity’’, using the direct Markov renewal theoretical results. Anbazhagan and Arivarignan ([3], [4], [5], [6]) have analyzed two commodity inventory system under various ordering policies. Yadavalli et al. [29] have analyzed a model with joint ordering policy and varying order quantities. Yadavalli et al. [30] have considered a two-commodity substitutable inventory system with Poisson demands and arbitrarily distributed lead time. All the models considered in the two-commodity inventory system assumed lost sales of demands during stock out periods. Traditionally the inventory models incorporate the stream of customers (either at fixed time intervals or random intervals of time) whose demands are satisfied by the items from the stock and those demands which cannot be satisfied are either backlogged or lost. But in recent times due to the changes in business environments in terms of technology such as Internet, the customer may retry for his requirements at random time points. The concept of retrial demands in inventory was introduced in [9] and only few papers ([2], [26], [27], [31] ) have appeared in this area. Moreover product such as bath soaps, body spray, etc., may have different flavours and the customer would be willing to settle for one only when the other is not available. These aspects provided the motivation for writing this paper. We will focus on the case in which the population under study is finite so each individual customer generates his own flow of primary demands. For the analysis of finite source retrial queue in continuous time, the interested reader is referred to Falin and Templeton [12], Artalejo and Lopez-Herrero [10], Falin and Artalejo [13], Almasi et al. [1] and Artalejo [8] and references therein. The rest of the paper is organized as follows. In the next section, we describe the mathematical model. The steady state analysis of the model is presented in section 3 and some key system performance measures are derived in section 4. In section 5, we calculate the total expected cost rate in the steady state. Several numerical results that illustrate the influence of the system parameters on the system performance are discussed in section 6. The last section is meant for conclusion. II. MATHEMATICAL MODEL We consider a continuous review perishable inventory system with a maximum stock of S i units for the i th commodity (i = 1,2) and the demands originated from a finite population of sources N . Each source th is either free or in the orbit at any time. The primary demand for i commodity is of unit size and the time points of primary demand occurrences form independent Quasi-random distributions each with parameter  i (i = 1,2) . The items are perishable in nature and the life time of items of each commodity is assumed to be exponentially distributed with parameter  i (i = 1,2) . The reorder level for the i th commodity is fixed as s i (1  s i  S i ) and an order is placed for both commodities when both the inventory levels are less than or equal to their respective th reorder levels. The ordering quantity for the i commodity is Q i (= S i  s i > s i  1, i = 1,2) items. The requirement S i  s i > s i  1 , ensures that after a replenishment the inventory level will always be above the respective reorder levels; otherwise it may not be possible to place reorder (according to this policy) which leads to perpetual shortage. The lead time is assumed to be exponentially distributed with parameter  > 0 . Both the commodities are assumed to be both way substitutable in the sense that at the time of zero stock of any one commodity, the other one is used to meet the demand. If the inventory position of both the commodities are zero thereafter any arriving primary demand enters into the orbit. These orbiting demands send out signal to compete for their demand which is distributed as exponential with parameter  (> 0) . In this article, the classical retrial policy is followed, that is, the probability of a repeated attempt is depend on the number of demands in the orbit. The retrial demand may accept an item of commodity- i with probability p i (i = 1,2) , where p 1  p 2 = 1 . We also assume that the inter demand times between the primary demands, lead times, life time of each items and retrial demand times are mutually independent random variables. www.ijmsi.org 2|Pa ge
  • 3. Perishable Inventory System With A Finite.. 2.1 Notations: e : a column vec tor of appropriat e dimension containing all ones 0 : Zero matrix  A ij : entry at ( i , j ) th position of a matrix A k  V i : k = i , i  1,  j j  ij : 1   ij 1 if j = i 0 otherwise  ij :  1, H (x) :   0, E 1 : {0,1,2, if x  0, otherwise .  , S 1} E 2 : {0,1,2,  , S 2 } E 3 : {0,1,2,  , N } E : E1  E 2  E 3 III. Analysis Let L 1 ( t ) , L 2 ( t ) and X ( t ) denote the inventory position of commodity-I, the inventory position of commodity-II and the number of demands in the orbit at time t , respectively. From the assumptions made on the input and output processes it can be shown that the triplet {( L 1 ( t ), L 2 ( t ), X ( t )), t  0} is a continuous time Markov chain with state space given by E. To determine the infinitesimal generator  = a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )) , ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )  E , of this process. Theorem 1: The infinitesimal generator of this Markov process is given by, a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 , ) ) = www.ijmsi.org 3|Pa ge
  • 4. Perishable Inventory System With A Finite..  ( N  i 3 )(  1   2 ),      i 3 ,          i 3 p 1 ,      i 3 p 2 ,           ( N  i 3 )(  1   2 )  i 1  1 ,     ( N  i )(    )  i  , 3 1 2 2 2       (( N  i 3 )(  1   2 )  i 2     i 0 i 3   H ( s 2  i 2 )  ),   2 j1 = 0 , j 3 = i 3  1, j2 = 0, i 1 = 0, i3  V 0 i 2 = 0, j 2 = i 2  1, j 1 = i1 , i2  V1 i 1 = 0, S 2 N 1 , j 3 = i 3  1, i3  V 1 , N , or j 1 = i 1  1, i1  V 1 S 1 S 1 i2  V1 S 1 , i2  V1 s 1 S 1 i1  V 0 2  1 2 s 2 S 2 i2  V1 , S 2 2 , i2  V 0 i 1 = 0, j 1 = i1 , i1  V 1 S 1 2 i3  V 0 , i3  V 0 , N , , N , j3 = i3 , i3  V 0 , N , j3 = i3 , N , j3 = i3 , i3  V 0 , j2 = i2 , , N j3 = i3 , , j2 = i2 , S , j 3 = i 3  1, i3  V 1 , j 2 = i 2  1, j 1 = i1 ,   (( N  i 3 )(  1   2 )  i1  1  i 2  2   i 3    H ( s 1  i1 ) H ( s 2  i 2 )),     0    S j2 = i2 , i2  V 0 , j 1 = i1 , S i3  V 1 , j2 = i2  Q i2  V 0 , j 1 = i 1  1, i1  V 1 2 j 2 = i 2  1, j 1 = i1  Q 1 , i1  V 0 S N j 3 = i 3  1, j2 = i2 , , j 1 = i1 , i1  V 1 i3  V 1 i 2 = 0, j 1 = i 1  1, i1  V 1 j 3 = i 3  1, j2 = i2 , , i2  V 0 S 2 N , j3 = i3 , , i3  V 0 , N otherwise Proof: The infinitesimal generator a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )) of this process can be obtained using the following arguments: 1: Let i1 > 0, i 2 > 0, i 3  0 . A primary demand from any one of the ( N  i 3 ) sources or due to perishability takes the inventory level ( i1 , i 2 , i 3 ) to ( i1  1, i 2 , i 3 ) with intensity ( N  i 3 )  1  i1 1 for I-commodity or ( i1 , i 2 , i 3 ) to ( i1 , i 2  1, i 3 ) with intensity ( N  i 3 )  2  i 2  2 for II-commodity. The level (0, i 2 , i 3 ) , and ( i1 ,0, i 3 ) , respectively, is taken to (0, i 2  1, i 3 ) , with intensity ( N  i 3 )(  1   2 )  i 2  2 , and ( i1  1,0, i 3 ) with intensity ( N  i 3 )(  1   2 )  i1 1 . 2: If the inventory position of both the commodities are zero then any arriving primary demand enters into the orbit. Hence a transition takes place from (0,0, i 3 ) to (0,0, i 3  1) with intensity ( N  i 3 )(  1   2 ) , www.ijmsi.org 4|Pa ge
  • 5. Perishable Inventory System With A Finite.. 0  i3  N  1 . 3: Let i1 > 0, i 2 > 0, i 3  1 . A demand from orbit takes the inventory level ( i1 , i 2 , i 3 ) to ( i1  1, i 2 , i 3  1) with intensity i 3 p 1 for Icommodity or ( i1 , i 2 , i 3 ) to ( i1 , i 2  1, i 3  1) with intensity i 3 p 2 for II-commodity. The level (0, i 2 , i 3 ) , and ( i1 ,0, i 3 ) , respectively, is taken to (0, i 2  1, i 3  1) and ( i1  1,0, i 3  1) with intensity i 3 . 4: From a state ( i1 , i 2 , i 3 ) with ( i1 , i 2 )  ( s 1 , s 2 ) , i 3  0 a replenishment by the delivery of orders for both commodities takes the inventory level to ( i1  Q 1 , i 2  Q 2 , i 3 ) , Q 1 = S 1  s 1 , Q 2 = S 2  s 2 , with intensity of this transition  . We observe that no transition other than the above is possible. Finally the value of a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 ) ) is obtained by a (( i1 , i 2 , i 3 ), ( i1 , i 2 , i 3 )) =   j j j 1 2 3 ( i ,i ,i )  ( j , j , j ) 1 2 3 1 2 3 a (( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 )) □ Hence we get the infinitesimal generator a ( ( i1 , i 2 , i 3 ), ( j1 , j 2 , j 3 ) ). In order to write down the infinitesimal generator  in a matrix form, we arrange the states in lexicographic order and group ( S 2  1)( N  1) states as: < q >= (( q ,0,0), ( q ,0,1),  , ( q ,0, N ), ( q ,1,0), ( q ,1,1),  , ( q ,1, N ),  , ( q , S 2 ,0), ( q , S 2 ,1),  , ( q , S 2 , N )) for q = 0,1,  , S 1 . Then the rate matrix  has the block partitioned form with the following sub matrix [  ] i j 1 1 at the i1 - the row and j1 -th column position. [  ]i j 1 1 A , i  1 B , =  i1 C ,  0, j 1 = i1 , i1  V 0 S j1 = i1  1, i1  V 1 S j 1 = i1  Q 1 , i1  V 0 1 otherwise 1 1 s . where [C ]i j 2 2 [ H ]i j 3 3  W =   0,  j2 = i2  Q 2 , otherwise i2  V 0 s 2 .  ( N  i 3 )(  1   2 ),  =   (( N  i 3 )(  1   2 )   ),  0,  www.ijmsi.org N 1 j 3 = i 3  1, i3  V 0 j 3 = i3 , i3  V 0 , otherwise , N . 5|Pa ge
  • 6. Perishable Inventory System With A Finite.. [ A0 ]i H   Fi =  2  H i2  0,  j 2 2 For [ B i ]i i 2 = 0, j 2 = i 2  1, i2  V1 S i2  V1 S j2 = i2 , otherwise 2 , 2 , . i1 = 1,  , S 1 V i 1  = M i 1  0,  j 2 = 0, 1 j 2 2 [ Ai ]i 1 G i  2 Ji =  1  Li i 1 2  0,  j 2 2 j2 = i2 , i2  V1 S j 2 = i 2  1, i2  V1 S j2 = i2 , i 2 = 0, j2 = i2 , i2  V1 otherwise 2 , 2 , 2 , . otherwise S . i1 = 1,2,  , S 1 For [V i ] i 1 j 3 3  i 3 ,  =  ( N  i 3 )(  1   2 )  i1  1 ,  0,  j 3 = i 3  1, i3  V 1 , j 3 = i3 , i3  V 0 , N N otherwise . i1 = 1,2,  , S 1 For i 1 ]i [ Fi ]i 2 For  0,  j 3 = i 3  1, i3  V 1 , j3 = i3 , i3  V 0 , otherwise N N . j 3 3  i 3 ,  =  ( N  i 3 )(  1   2 )  i 2  2 ,  0,  j 3 = i 3  1, i3  V 1 , j 3 = i3 , i3  V 0 , otherwise N N . i 2 = 1,2,  , S 2 For i j 3 3  p 1 ,  =  ( N  i 3 )  1  i1  1 , i 2 = 1,2,  , S 2 For [H i 2 = 0, i1 = 1,2,  , S 1 For [M j2 = i2 , 2 ]i j 3 3   (( N  i 3 )(  1   2 )  i 2   =  i 3  H ( s 2  i 2 )  ), 2  0,   i3  V 0 , N j 3 = i3 , otherwise . i 2 = 1,2,  , S 2 www.ijmsi.org 6|Pa ge
  • 7. Perishable Inventory System With A Finite.. [G i ]i 2 [ J i ]i For [ Li i 1 2 j 3 3 j 3 = i 3  1, i3  V 1 , j 3 = i3 , i3  V 0 , N N otherwise . i1 = 1,2,  , S 1 For 1  i 3 p 2 ,  =  ( N  i3 )  2  i 2  2 ,  0,    (( N  i 3 )(  1   2 )  i1  1  i 3   =   H ( s 1  i1 ))  0,  j 3 3 i3  V 0 , N j 3 = i3 , otherwise . i1 = 1,2,  , S 1 ; i 2 = 1,2,  , S 2 , ]i   (( N  i 3 )(  1   2 )  i1  1  i 2   =  i 3   H ( s 1  i1 ) H ( s 2  i 2 )), j 3 3 2   0,  i3  V 0 , N j 3 = i3 , otherwise . W =  I N 1 It may be noted that the matrices A i , B i , i1 = 1,2,  , S 1 , A 0 and C are square matrices of order 1 ( S 2  1)( N  1) . The sub matrices V i , M 1 H i 2 1 i 1 , J i , L i i , i1 = 1,2,  , S 1 , i 2 = 1,2,  , S 2 , W , H , F i , 1 1 2 2 , G i , i 2 = 1,2,  , S 2 , are square matrices of order ( N  1) . 2 It can be seen from the structure of  that the homogeneous Markov process {( L 1 ( t ), L 2 ( t ), X ( t )) : t  0} on the finite space E is irreducible, aperiodic and persistent non-null. Hence the limiting distribution  ( i ,i ,i ) 1 2 3 = lim Pr [ L 1 ( t ) = i1 , L 2 ( t ) = i 2 , X ( t ) = i 3 | L 1 (0), L 2 (0), X (0)], exists. t  Let Π = (  (0) , (1) , ,  partitioning the vector,   (i ) 1 = ( ( i ,0) 1 (S ) 1 (i ) 1 , ), into as follows: ( i ,1) 1 , ( i ,2) 1 , ,  (i ,S ) 1 2 ), i1 = 0,1,2,  , S 1 which is partitioned as follows:  ( i ,i ) 1 2 = ( ( i , i ,0) 1 2 , ,  ( i ,i , N ) 1 2 ), i1 = 0,1,2,  , S 1 ; i 2 = 0,1,2,  , S 2 . Then the vector of limiting probabilities Π satisfies Π  = 0 and Π e = 1. (1) www.ijmsi.org 7|Pa ge
  • 8. Perishable Inventory System With A Finite.. Theorem 2: The limiting distribution Π is given by,  (i ) 1 (Q ) 1 =  i , i1 = 0,1,  , S 1 , 1 where Q i  i 1 1  (  1) 1 1 B A  1 B  B i  1 A i , i1 = 0,1,  , Q 1  1, Q Q 1 Q 1 1 1 1 1 1   I, i1 = Q 1 ,   =  S i 1 1 2 Q  i 1 1 1 1  (  1) 1 1  B Q A Q  1 B Q  1  B s  1  j A s  j CA S  j 1 1 1 1 1 1 1 1 1  j =0 1  1 1  B S  j A S  j 1 B S  j 1  B i  1 A i ,  1 1 1 1 1 1 1 1  i1 = Q 1  1,  , S 1 ,    (2)   (Q ) 1 The value of   (Q ) 1 can be obtained from the relation   (  1)   Q s 1 1 1  B 1 Q 1 AQ 1 1 BQ 1 1  Bs 1 1 1 j 1 As 1  j 1 CA j =0 1   BS 1  j 1 1 AS 1  j 1 1 BS 1  j 1 1  BQ 1 1 AQ 2 1 1 B (3)  (  1) Q 1 1 B Q AQ 1 1 1 BQ 1 1 1 1 S  j 1 1 Q 1 1  AQ 1   B 1 A 0 C = 0, and  (Q ) 1     (  1)  i = Q 1 1 1  S 1  Q1  1   (  1)  i1 = 0   2 Q  i 1 1 1 S i 1 1  Q i 1 1 1 B Q AQ 1 B j =0 1 1 Q 1 AQ   BS 1 1 1 1 BQ 1 BQ 1 1  j 1 AS 1  Bi  j 1 1 BS 1  I 1 1  Bs 1 1 1 1 Ai 1 1 1 1 j 1  j 1 1 As 1  j 1  Bi 1 CA 1 1 Ai 1 1 (4) S  j 1 1 e = 1. Proof: The first equation of (1) yields the following set of equations :  ( i  1) 1 Bi 1  Bi 1  Bi 1  1  ( i  1) 1 1  ( i  1) 1 1  (i ) 1 A i = 0, i1 = 0,1,  , Q 1  1, (5) 1 (i ) 1 Ai   (i Q ) 1 1 1 Ai   1 (i ) 1 (i ) 1 Ai   1 (i Q ) 1 1 (i Q ) 1 1 C = 0, i1 = Q 1 , (6) C = 0, i1 = Q 1  1,  , S 1  1, (7) (8) C = 0, i1 = S 1 . Solving the above set of equations we get the required solution. www.ijmsi.org □ 8|Pa ge
  • 9. Perishable Inventory System With A Finite.. IV. SYSTEM PERFORMANCE MEASURES In this section some performance measures of the system under consideration in the steady state are derived. 4.1 Expected inventory level Let  i and  i denote the average inventory level for the first commodity and the second commodity 1 2 respectively in the steady state. Then S S 1 N 2   i i = ( i ,i ,i ) 1 2 3 (9) 1 1 i =1 i = 0 i = 0 1 2 3 and S S 1 N 2   i  i = ( i ,i ,i ) 1 2 3 (10) 2 2 i = 0 i =1i = 0 1 2 3 4.2 Expected reorder rate Let  r denote the mean reorder rate in the steady state. Then s 2  (N r = 1  i 2 0 N  2  ( s 1  1)  1 )  ( s  1, i ,0) 1 2 i =0 2 s  1  (N   i 0 N  1  ( s 2  1)  2 )  2 ( i , s  1,0) 1 2 1 i =0 1 s  N 2   (( N  i 3 )  1  ( s 1  1)  1   i 2 0 (( N  i 3 )  2  i 3 p 2 )  i 3 p 1 )  ( s  1, i , i ) 1 2 3 i = 0 i =1 2 3 (11) s  N 1   (( N  i 3 )  2  ( s 2  1)  2   i 0 (( N  i 3 )  1  i 3 p 1 )  i 3 p 2 )  ( i , s  1, i ) 1 2 3 1 i = 0 i =1 1 3 4.3 Expected perishable rate Let  p and  p denote the expected perishable rates for the first commodity and the second 1 2 commodity respectively in the steady state. Then S  p S 1 N 2   i = 1 1 1  ( i ,i ,i ) 1 2 3 (12) i =1 i = 0 i = 0 1 2 3 and S  p = 2 S 1 N 2   i 2  2 ( i ,i ,i ) 1 2 3 (13) i = 0 i =1i = 0 1 2 3 4.4 Expected number of demands in the orbit Let  o denote the expected number of demands in the orbit. Then S o = 1 S 2 N   i  ( i ,i ,i ) 1 2 3 3 (14) i = 0 i = 0 i =1 1 2 3 www.ijmsi.org 9|Pa ge
  • 10. Perishable Inventory System With A Finite.. 4.5 Expected an arriving demand enters into the orbit The expected an arriving primary demand enters into the orbit is given by N 1 a =  (N  i 3 )(  1   2 )  (0,0, i ) 3 (15) i =0 3 4.6 The overall rate of retrials The overall rate of retrials for the orbit customers in the steady state. Then S  or = 1 S 2 N    i  ( i ,i ,i ) 1 2 3 (16) 3 i = 0 i = 0 i =1 1 2 3 4.7 The successful rate of retrials The successful rate of retrials for the orbit customers in the steady state. Then S  sr = 2  S N  i 3 (0, i , i ) 2 3  1  i =1i =1 2 3 S N  i 3 ( i ,0, i ) 1 3  i =1 i =1 1 3 S 1 N 2    i  ( i ,i ,i ) 1 2 3 (17) 3 i =1 i =1i =1 1 2 3 4.8 Fraction of successful rate of retrials Let  fr denote the fraction of successful rate of retrials is given by  fr =  sr (18)  or V. COST ANALYSIS To compute the total expected cost per unit time (total expected cost rate), the following costs, are considered. c h 1 : The inventory holding cost per unit item per unit time for I-commodity. c h 2 : The inventory holding cost per unit item per unit time for II-commodity. c s : The setup cost per order. c p 1 : Perishable cost of the I - commodity per unit item per unit time. c p 2 : Perishable cost of the II- commodity per unit item per unit time. c w : Waiting cost of an orbiting demand per unit time. The long run total expected cost rate is given by TC ( S 1 , S 2 , s 1 , s 2 , N ) = c h  i  c h  i  c s r  c p  1 1 2 2 1 p 1  cp  2 p 2  c w o . (19) Substituting the values of  ’s we get TC ( S 1 , S 2 , s 1 , s 2 , N )  S1 S 2 N = c h     i1 1  i1 = 1 i 2 = 0 i3 = 0   S1 S 2 N  c w     i3  i1 = 0 i 2 = 0 i3 = 1  ( i ,i ,i ) 1 2 3 ( i ,i ,i ) 1 2 3  S1 S 2 N  c p 2     i2 2   i1 = 0 i 2 = 1 i3 = 0    ch 2    S1 S 2 N     i2  i1 = 0 i 2 = 1 i3 = 0  ( i ,i ,i ) 1 2 3   S1 S 2 N   c p 1     i1  1   i1 = 1 i 2 = 0 i3 = 0   ( i ,i ,i ) 1 2 3         ( i ,i ,i ) 1 2 3      s2 c s    ( N  1   i 0 N  2  ( s 1  1)  1 )  2   i2 = 0  www.ijmsi.org ( s  1, i ,0) 1 2     10 | P a g e
  • 11. Perishable Inventory System With A Finite..  s1    ( N  2   i 0 N  1  ( s 2  1)  2  1  i =0 1 ( i , s  1,0) 1 2     (20)  s2 N     (( N  i 3 )  1  i 3 p 1  ( s 1  1)  1   i 0 (( N  i 3 )  2  i 3 p 2 )) 2  i = 0 i =1  2 3  s1 N     (( N  i 3 )  2  i 3 p 2  ( s 2  1)   i = 0 i =1 1 3 2       i 0 (( N  i 3 )  1  i 3 p 1 ))  ( i , s  1, i ) 1 2 3 1     Due to the complex form of the limiting distribution, it is difficult to discuss the properties of the cost function analytically. Hence, a detailed computational study of the cost function is carried out in the next section. VI. NUMERICAL ILLUSTRATIONS In this section we discuss some interesting numerical examples that qualitatively describe the performance of this inventory model under study. Our experience with considerable numerical examples indicates that the function TC ( S 1 , S 2 ), to be convex. Appropriate numerical search procedures are used to * * * obtain the optimal values of TC , S 1 and S 2 (say TC , S 1 and S 2 ). The effect of varying the system parameters and costs on the optimal values have been studied and the results agreed with what one would expect. A typical three dimensional plot of the total expected cost function is given in Figure 1 .In Table 1 gives * * the total expected cost rate as a function of S 1 and S 2 by fixing the parameters and the cost values: s 1 = 2, s 2 = 3, N = 10,  1 = 0.01,  2 = 0.02,  = 0.01, c p 1 = 0.4, c p 2 = 0.5, c w = 6, c h 1 = 0.01, c h 2 = 0.04,  1 = 0.2,  2 = 0.1,  = 0.02, c s = 12, p 1 = 0.4 and p 2 = 0.6 . * * From the Table 1 the total expected cost rate is more sensitive to the changes in S 2 than that of in S 1 . Some of the results are presented in Tables 2 through 6 where the lower entry in each cell gives the total * * expected cost rate and the upper entries the corresponding S 1 and S 2 . www.ijmsi.org 11 | P a g e
  • 12. Perishable Inventory System With A Finite.. s 1 = 2, s 2 = 3, N = 10,  1 = 0.01,  2 = 0.02,  = 0.01,  1 = 0.2,  2 = 0.1,  = 0.02, c s = 12, c h 1 = 0.01, c h 2 = 0.04, c p 1 = 0.4, c p 2 = 0.5, c w = 6, p 1 = 0.4, p 2 = 0.6 . Figure 1: A three dimensional plot of the cost function TC ( S 1 , S 2 ) Table 1: Total expected cost rate as a function of S 1 and S 2 S2 29 30 31 32 33 S1 88 89 90 91 92 52.866093 52.866088 52.866167 52.866586 52.867052 52.863418 52.863289 52.863315 52.863494 52.8638820 52.862517 52.862247 52.862134 52.862227 52.862361 52.863258 52.862850 52.862599 52.862502 52.862555 52.865527 52.864984 52.864598 52.864368 52.864288 6.1 Example 1 In the first example, we look at the impact of  1 ,  2 ,  1 , and  * 2 * on the optimal values ( S 1 , S 2 ) * and the corresponding total expected cost rate TC . For this, first by fixing the parameters and cost values as s 1 = 2, s 2 = 3, N = 10,  = 0.02 ,  = 0.01 , p 1 = 0.4 , p 2 = 0.6 , c h 1 = 0.01 , c h 2 = 0.04 , c s = 12 , c w = 6 , c p 1 = 0.4 and c p 2 = 0.5 . Observe the following from Tables 2 and 3 : 1. From the Table 2 , it is observed that the TC , S 1 and S 2 increase when  1 and  2 increase. The result * * * is obvious as  1 and  2 increase it has impact on higher re-ordering and the cost of carrying to orbit customers. Hence arrival rates are vital to this system. Also the TC * is more sensitive to changes in  1 than that of in  2 . 2. From the Table 3 , it is observed that if  1 and  * 2 * increase then S 1 and S 2 decrease, and the TC increases, in a significant amount. This results is obvious as  1 and  2 * increase, more items will be perished that finally incurred a substantial amount of costs to the system. From the observation it seems that the TC very sensitive to changes in  2 than that of in  1 . * is 6.2 Example 2 In this example, we study the impact of c s , c h 1 , c h 2 , c p 1 , c p 2 and c w on the optimal values * * (S1 , S 2 ) * and the corresponding TC . Towards this end, first by fixing the parameter values as s 1 = 2, s 2 = 3, N = 10,  1 = 0.01 ,  2 = 0.02 ,  = 0.02 ,  = 0.01 ,  1 = 0.2 ,  2 = 0.1 , p 1 = 0.4 and p 2 = 0.6 . Observe the following from Tables 4  6 : 1. The total expected cost rate increases when c h 1 , c h 2 , c s , c w , c p 1 and c p 2 increase monotonically. * * 2. As c h 1 and c h 2 increase, the optimal values S 1 and S 2 decrease monotonically. This is to be expected since c h 1 and c h 2 increase, we resort to maintain low stock in the inventory. * * 3. Similarly, when c w increases, the values of S 1 and S 2 increase monotonically. This is because if c w increases then we have to maintain high inventory to reduce the number of waiting customers in the orbit. www.ijmsi.org 12 | P a g e
  • 13. Perishable Inventory System With A Finite.. * * 4. As S 1 and S 2 increase monotonically, c s increases. This is a common decision that we have to maintain more stock to avoid frequent ordering. * * 5. If c p 1 and c p 2 increase monotonically then S 1 and S 2 decrease and TC * increases. We also note that the total expected cost rate is more sensitive to changes in c p 1 than that of in c p 2 . Table 2: Sensitivity of  1 and  2 on the optimal values 2 0.010 0.015 0.020 0.025 104 30 52.404997 104 30 52.677218 105 30 52.916048 106 30 53.128612 107 30 53.319961 104 30 52.616266 104 31 52.862134 105 31 53.080348 106 31 53.276383 107 32 53.454181 105 52.805787 105 53.030387 105 53.231281 106 53.413216 107 53.579277 0.030 1 0.005 0.010 0.015 0.020 0.025 104 52.166935 104 52.471973 105 52.735666 106 52.967782 107 53.174930 28 28 28 28 28 33 33 33 34 34 Table 3: Variation in optimal values for different values of  1 and  0.08 0.10 105 31 50.734492 99 31 51.777332 67 27 52.403399 49 26 52.802492 37 25 105 31 51.983923 95 31 53.098322 65 27 53.768192 47 24 54.198838 35 22 104 52.862134 90 54.023159 62 54.715274 44 55.153269 32 53.067357  0.06 54.498210 55.446462 105 52.977922 105 53.184492 106 53.370828 106 53.540578 107 53.696383 34 34 34 34 35 2 0.12 0.14 102 30 53.512248 87 30 54.718720 60 26 55.438513 42 24 55.880546 31 21 100 26 53.989839 86 26 55.247406 60 23 56.014391 41 23 56.468607 29 20 56.169518 56.753754 2 1 0.20 0.25 0.30 0.35 0.40 31 31 27 24 21 Table 4: Effect of varying c h 1 and c h 2 on the optimal values c h1 0.005 0.010 0.015 0.020 93 36 52.655217 91 33 52.761980 90 31 52.862134 89 29 52.956645 89 28 53.045806 87 35 52.730014 86 33 52.836322 85 31 52.936195 84 29 53.030349 84 28 53.119568 83 35 52.800966 82 33 52.907085 81 31 53.006699 80 29 53.100529 79 27 53.189435 0.025 ch2 0.02 0.03 0.04 0.05 0.06 98 36 52.576308 98 34 52.683493 97 32 52.784094 95 30 52.878746 94 28 52.968191 www.ijmsi.org 79 35 52.868637 77 32 52.974514 76 30 53.073884 76 29 53.1673640 75 27 53.256085 13 | P a g e
  • 14. Perishable Inventory System With A Finite.. Table 5: Influence of c w and c s on the optimal values 2 4 6 23 13 18.666668 23 17 18.732696 22 19 18.792123 20 20 18.845248 19 21 18.892830 56 23 35.866303 56 24 35.926358 56 25 35.984528 55 26 36.040683 55 27 36.094711 91 30 52.753058 90 30 52.808218 90 31 52.862134 90 32 52.915075 90 33 52.966959 cw 8 10 cs 8 10 12 14 16 127 36 69.441440 127 36 69.493279 127 37 69.544235 127 38 69.594647 126 38 69.644419 164 41 85.985889 164 42 86.035232 164 42 86.084255 164 43 86.132520 163 44 86.180477 Table 6: Variation in optimal values for different values of c p 1 and c p 2 c p2 0.2 0.5 0.8 1.1 1.4 c p1 2 4 6 8 1.0 175 43 51.737733 94 39 52.545497 63 36 53.058156 47 35 53.431616 37 34 53.723445 167 35 52.071328 90 31 52.862134 61 29 53.368609 45 28 53.739166 35 27 54.029803 VII. 162 29 52.346578 87 26 53.126890 59 25 53.628758 45 27 53.829995 35 23 54.287583 159 24 52.577109 86 23 53.353545 58 22 53.853509 44 21 54.221504 34 20 54.511543 159 20 52.769276 85 20 53.549672 57 19 54.049595 43 18 54.418675 34 18 54.700904 CONCLUSIONS In this paper we consider a finite source two commodity perishable inventory system with substitutable and retrial demands. This model is most suitable to two different items which are substitutable. The joint probability distribution for both commodities and number of demands in the orbit is obtained in the steady state case. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics. ACKNOWLEDGMENT N. Anabzhagan’s research was supported by the National Board for Higher Mathematics (DAE), Government of India through research project 2/48(11)/2011/R&D II/1141. K. Jeganathan’s research was supported by University Grants Commission of India under Rajiv Gandhi National Fellowship F.16-1574/2010(SA-III). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] Almási, B., Roszik, J., Sztrik, J., (2005). Homogeneous finite source retrial queues with server subject to breakdowns and repairs. Mathematical and Computer Modelling, 42, 673 - 682. Anbazhagan, N., Jinting, W., Gomathi, D., (2013). Base stock policy with retrial demands. Applied Mathematical Modelling, 37, 4464 - 4473. Anbazhagan, N., Arivarignan, G., (2000). Two-commodity continuous review inventory system with coordinated reorder policy. International Journal of Information and Management Sciences, 11(3),19 –30. Anbazhagan, N., Arivarignan, G., (2001). Analysis of two-commodity Markovian inventory system with lead time. The Korean Journal of Computational and Applied Mathematics, 8(2), 427 - 438. Anbazhagan, N., Arivarignan, G. (2003). Two-commodity inventory system with individual and joint ordering policies. International Journal of Management and Systems, 19(2), 129 - 144. Anbazhagan, N., Arivarignan, G., Irle, A., (2012). A two-commodity continuous review inventory system with substitutable items. Stochastic Analysis and Applications, 30, 129 - 144. Agarwal, V., (1984). Coordinated order cycles under joint replenishment multi-item inventories. Naval Logistic Research Quarterly, 131 - 136. Artalejo, J. R. (1998). Retrial queues with a finite number of sources. Journal of the Korean Mathematical Society, 35, 503 - www.ijmsi.org 14 | P a g e
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