A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

Abstract

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume (𝑠,𝑆) ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is above 𝑠, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool is obtained in the steady-state case. Various system performance measures in the steady state are derived, and the long-run total expected cost rate is calculated.

1. Introduction

In most of the inventory models considered in the literature, the demanded items are directly delivered from the stock (if available). The demands occurring during the stock-out period are either lost (lost sales case) or satisfied only after the arrival of ordered items (backlogging). In the latter case, it is assumed that either all (full backlogging) or any prefixed number of demands (partial backlogging) that occurred during the stock-out period are satisfied. The often quoted review articles Nahmias [1] and Raafat [2] and the recent review articles N. H. Shah and Y. K. Shah [3] and Goyal and Giri [4] provide excellent summaries of many of these modelling efforts. For some recent references, see Chakravarthy and Daniel [5], Yadavalli et al. [6], and Kalpakam and Shanthi [7, 8].

In the case of continuous review perishable inventory models with random life times for the items, most of the models assume instantaneous supply of order (Lian and Liu [9], Liu and Shi [10], and Kalpakam and Arivarignan [11], and Gürler and Özkaya [12]). When a server is assumed to process the demands in continuous review inventory systems, the concept of server vacation was also taken up in some works. Daniel and Ramanarayanan [13] introduced the concept of server vacation in an inventory model with two servers. In [14], they studied an (𝑠,𝑆) inventory system in which the server takes a rest when the level of the inventory is zero. The demands that occurred during stock-out period were lost. The interoccurrence times between two successive demands, the lead times, and the rest times are assumed to follow general distributions, which are mutually independent. Using renewal and convolution techniques, they obtained the steady-state transition probabilities. However, server vacations in Queueing systems have been widely studied in different contexts in the literature. We refer the readers to Doshi [15, 16], Takagi [17, 18], and Tian and Zhang [19] for details on various vacations systems and for additional references.

In the context of inventory system, any arriving demands during the server vacation either enter a pool or leave the system. These pooled customers are selected at a later time, preferably after the stock is replenished. This type of inventory problems is called inventory with postponed demands.

The concept of postponed demands in the inventory model has been introduced by Berman et al. [20]. They assumed that both demand and service rates are deterministic. Krishnamoorthy and Islam [21] considered an inventory system with Poisson demand, exponential lead time and the pooled customers are selected according to an exponentially distributed time lag. Sivakumar and Arivarignan [22] considered an inventory model in which the demand occurs according to a Markovian arrival process, lead time has a phase type distribution, life time for the items in the stock has exponential distribution, and the pooled customers are selected exponentially. Manuel et al. [23] dealt with an inventory system in which the positive and negative demands occur according to an independent Markovian arrival processes, the lead time of the reorder, the life time of the items, the interselection time of customers from the pool and the reneging time points of the customers in the pool are independent, and each has the exponential distribution. In all the above models, the authors assumed that the pool size is finite. Sivakumar and Arivarignan [24] considered an inventory system with independent Markovian arrival processes for positive and negative demands, exponential lead time for the reorders, exponential life times for each item in the stock, and the pool size is infinite.

In earlier works listed above, the server is assumed to be available even during the stock-out periods. In this work, we have introduced the concept of server vacation. It may be noted that this model also includes situations in which the server may not be availing vacation, but the server may attend some other work during the stock-out period and may return at random times to the inventory system. If the stock is still not replenished, the server goes for another vacation again or to attend the other work. Employees who are assigned extra work on additional remuneration basis can also be thought of such servers.

In this paper, we consider a continuous review perishable inventory system. The unit demands occur according to a Poisson process. The (𝑠,𝑆) ordering policy is implemented to replenish stock. When the inventory is completely depleted, the server is assumed to go on a vacation of random duration. At the end of vacation, if the stock is not replenished, the server again starts his/her vacation. The demands that occurred during stock-out period are offered postponement of their demand after the stock is replenished. Those demands that accept the postponement joins a pool of finite capacity. We assume that fraction 𝑝 of demands join the pool and the rest leave the system. The pooled demands are selected once the stock is replenished, and the server resumes service. But the selection from the pool will be stopped if the inventory falls below the reorder level, 𝑠.

The rest of the paper is organized as follows. In Section 2, we describe the mathematical model of the problem under study. 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, and, in the final section, we perform sensitivity analyses on the total expected cost rate in terms of numerical illustrations.

2. Problem Formulation

We consider an inventory system with a maximum stock of 𝑆 unit items. The items are assumed to have identical and independent exponential life-time distributions with parameter 𝛾(>0). When an item perishes, it is removed from the stock. Another process which removes items from the stock is the demand process. The demand is for single item, and the demand occurrences form a Poisson process with rate 𝜆(>0). A server processes the unit demands. The stock is replenished by the server by implementing (𝑠,𝑆) ordering policy, where the lead time is assumed to have independent exponential distribution with parameter 𝜇(>0). If the stock is depleted before the supply of reorder, the server goes on a vacation duration which has exponential distribution with parameter 𝛽(>0). At the end of a vacation period, if the stock is still empty, the server takes another vacation. The demands that occur during stock-out period are offered postponement at a later time. It is assumed that the demands accept this postponement or leave the system according to a Bernoulli distribution with probability 𝑝. The demand which accepts an offer of postponement may join a pool of finite capacity fixed as 𝑁.

The demands from the pool are selected if the server is on service and the inventory level is above 𝑠 such that the interval time between successive selections is exponentially distributed with parameter 𝛼_𝑛(>0) where 𝑛 is the number of customers in the pool.

Notation.𝐞: a column vector of appropriate dimension containing all ones.𝟎: a zero matrix of appropriate dimension.𝟎_(𝑖×𝑗): a zero matrix with 𝑖 rows and 𝑗 columns.

3. Analysis

Let 𝑋(𝑡)and𝐿(𝑡), respectively, denote the number of demands in the pool and the on-hand inventory level at time 𝑡. We define a binary variable 𝑌(𝑡) (which takes 0and1) to denote the status of the server, respectively, “on vacation’’ and “on service.’’ Similarly, the binary variable 𝑍(𝑡) will indicate the status of the reorder to replenish the stock: 𝑍(𝑡)=0or1, respectively, if   “no order is pending’’ or “an order is pending.’’

The values taken by these random variables are listed in the following.(1)

We consider the stochastic process {𝑋(𝑡),𝑌(𝑡),𝑍(𝑡),𝐿(𝑡),𝑡≥0}. Since the components carry sufficient information on the sample path of the process and from the assumptions made on the input and output processes, this process can easily be seen to be a Markov process.

The state space of the stochastic process {𝑋(𝑡),𝑌(𝑡),𝑍(𝑡),𝐿(𝑡),𝑡≥0} is the collection of all quadruples ⟨𝑖₁⟩=(𝑖₁,𝑖₂,𝑖₃,𝑖₄) where each entry is selected from each column as we move from left to right; we may cross vertical lines but not horizontal ones. These quadruples can be ordered in the lexicographic order in each box separated by the horizontal lines.

We shall write the transition rate matrix in a block-partitioned form where 𝑖₁th row (block) corresponds to the collection of quadruples ⟨𝑖₁⟩ (those with leftmost entry 𝑖₁), 𝑖₁=0,1,2,…,𝑁. It may be recalled that the pool size (𝑖₁) may increase or decrease by one unit (except the end states) when a demand joins or leaves the pool. Hence, the transition rate matrix is a tridiagonal one.

We also note that when the number of demands in the pool is 𝑖(𝑖=0,1,2,…,𝑁−1), a new demand joining the pool will shift the current set of quadruple ⟨𝑖⟩ to ⟨𝑖+1⟩ and the transition rate does not depend upon 𝑖. Therefore, all the entries in the upper diagonal of the rate matrix are equal to a constant block matrix, which we shall denote by 𝐵₀. The lower diagonal entries correspond to the transition from states of ⟨𝑖⟩,𝑖=1,2,…,𝑁, to states ⟨𝑖−1⟩. This is due to the selection of a demand from the pool, and the rate of selection of any one of the demands is 𝛼_𝑖. Hence, these transition rates depend on row indices. Therefore, the entries of the lower diagonal depend on the row index. We shall denote them by 𝐶_𝑖. In the case of diagonal entries, again, the transition rate may depend upon the pool size (row index), if it involves selection of a demand from the pool. Hence, the entries here also depend upon the respective row index. We shall denote them by 𝐴_𝑖.

Hence, the infinitesimal generator matrix can be written as𝑁𝑁𝑁⋮012−−21⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐴0₀123⋯𝑁−2𝑁−1𝑁𝐶𝐵₀00⋯⋯00₁0𝐴𝐶₁₂𝐵₀0⋯⋯00⋯⋮⋯⋮𝐴⋯⋮₂⋯𝐵⋮₀⋯⋯⋮𝐴⋯⋮_𝑁−2⋮0⋮0⋯⋯⋯⋯⋯𝐶_𝑁−1𝐵₀⋯𝐴_𝑁−1𝐵₀⋯⋯⋯⋯⋯0𝐶_𝑁𝐴_𝑁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2) We shall determine the submatrix 𝐵₀, the elements of which correspond to the transitions from ⟨𝑖⟩to⟨𝑖+1⟩,𝑖=0,1,2,…,𝑁−1. This corresponds to the case that one demand has joined the pool. In a Markov process, at most one event can occur in a small interval. Hence, for the transitions from (𝑖₁,𝑖₂,𝑖₃,𝑖₄)to(𝑖₁+1,𝑖^2,𝑖^3,𝑖^4), we have 0 when 𝑖^2≠𝑖₂,𝑖^3≠𝑖₃,𝑖^4≠𝑖₄ and 𝑝𝜆 when they are equal and 𝑖₂=0. The matrix 𝐵₀ is given by(3)

Next, we consider the submatrix 𝐶_𝑖, the elements of which give the transitions from ⟨𝑖⟩ to ⟨𝑖−1⟩. This corresponds to the event that a demand from the pool is selected, and this indicates that the server is “on service’’ and that “ no order is pending.’’ Therefore, possible value is 𝑖^2=𝑖₂=1. As a result of a selection from the pool, an item from the inventory is used to meet this demand. Hence, 𝐿(𝑡) moves from 𝑖₄ to 𝑖₄−1,if𝑖₄=𝑠+2,𝑠+3,…,𝑆 (with 𝑖^3=0) and from 𝑠+1 to 𝑠 (with 𝑖^3=1). Hence, the submatrix 𝐶_𝑖,𝑖=1,2,…,𝑁, is given by(4)

where 𝐶^0(𝑖)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝛼00⋯00^𝑖0𝛼0_𝑖⋯00⋮0⋮0⋯⋯⋮⋮𝛼0_𝑖⋮0𝐶_1(𝑖)0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝00⋯0𝛼^𝑖⋮000⋮000⋯⋯⋯⋮⋮000⋮000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(5)

Let us consider the submatrix 𝐴_𝑖,𝑖=1,2,…,𝑁. In this case, there is no addition to nor selection from the pool. Firstly, we note the transitions(i)from (𝑖₁,0,1,𝑖₄) to (𝑖₁,1,0,𝑖₄)or(𝑖₁,1,1,𝑖₄),(ii)from (𝑖₁,1,0,𝑖₄) to (𝑖₁,0,0,𝑖₄)or(𝑖₁,0,1,𝑖₄),(iii)from (𝑖₁,1,1,𝑖₄) to (𝑖₁,0,0,𝑖₄),

as these involve more than one occurrence.

Secondly, we observe that, only in the case of diagonal entries 𝑎((𝑖₁,1,0,𝑖₄),(𝑖₁,1,0,𝑖₄)) (i.e., the intensity of passage), the rate 𝛼_𝑖 (the rate of selection from the pool) will appear.

Hence, the block-partitioned submatrix 𝐴_𝑖 will have the following structure:(6)

The following cases are pertaining to the event that the server is on vacation before and after the transition.(i)In the case of column vector 𝑎₀₁, the reorder status changes from “no order is pending’’ to “an order is pending.’’ Thus, 𝑖₄=0 and, after transition, 𝑖^4=0. No transition is possible from other states 𝑖₄=1,2,…,𝑄. Thus, we obtain 𝑎₀₁=(𝛽,0,…,0)^.(7)(ii)The row vector 𝑎₁₀ considers the change of reorder status (from “order is pending’’ to “no order is pending’’). Thus, a supply of 𝑄 items is received. This changes 𝑖₂=0,𝑖₃=0,𝑖₄=0 to 𝑖^2=0,𝑖^3=1,𝑖^4=𝑄. Hence, 𝑎₁₀=(0,0,…,0,𝜇).(8)(iii)The constant 𝑎₁₁ is a diagonal entry, and, considering all transitions from (𝑖₁,0,1,0) to other states, we get 𝑎₁₁=−(𝑝𝜆+𝜇)if𝑖¹−𝜇if𝑖₁=1,2,…,𝑁−1,=𝑁,(9) or simply 𝑎₁₁=−(𝑝𝜆𝛿_𝑖1𝑁+𝜇), where 𝛿_𝑖1𝑁=1−𝛿_𝑖1𝑁.(iv)The nondiagonal entry of 𝐴₀₀, namely,𝑎((𝑖₁,0,0,𝑖₄),(𝑖₁,0,0,𝑖^4)) is equal to 𝑖₄𝛾 for 𝑖^4=𝑖₄−1 and zero for other nondiagonal elements. For the diagonal entry (considering the transitions to other states), we get 𝐴₀₀_𝑖4,𝑖4=−𝑝𝜆+𝛽+𝑖⁴−𝛽+𝑖₄𝛾if𝑖₄=1,2,…,𝑄,𝑖₁≠𝑁,𝛾if𝑖₄=1,2,…,𝑄,𝑖₁=𝑁,(10) or, simply, −(𝑝𝜆𝛿_𝑖1𝑁+𝛽+𝑖₄𝛾)if𝑖₄=1,2,…,𝑄, for all𝑖₁.

Using arguments similar to the above, we can express the matrix 𝐴_𝑖,𝑖=1,2,…,𝑁 as follows:(11)

where(i)𝑢^𝑘=(0,0,…,1,…,0,0) a vector having 1 at 𝑘th place and 0 at other places,(ii)𝐷^(𝑖)00=𝑄⋮𝑄012−1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−(𝑝0𝜆⋮𝛾000+𝛽)−(𝑝1𝜆2+⋮000𝛾𝛾+⋯𝛽)⋱⋯⋯⋯⋯⋯−𝑄(𝑝−𝜆+1(𝑄𝑄⋮000−𝛾1)𝛾+𝛽𝑄)−(𝑝𝜆+⋮0000𝑄𝛾+𝛽)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(12)(iii)𝐹₀₀=𝑠𝑠⋮⋮𝑄++0112⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑠⋮⋮𝛽0000+1⋮⋮𝑠𝛽0000+2⋱⋱⋯⋯⋯⋯⋯⋯⋮⋮𝑄𝛽0000𝑄⋮⋮00000+1⋱⋱⋯⋯⋯⋯⋯⋯⋮⋮00000𝑆⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(13)(iv)𝐹₀₁=𝑠⋮⋮𝑄−01𝑠1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1⋮⋮𝛽0000⋮⋮200000⋯⋱⋯⋯⋯⋯⋯⋮𝑠−⋮⋮𝛽00001𝑠⋮⋮𝛽0000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(14)(v)𝐻^(𝑖)00=𝑠𝑠⋮⋮++𝑆12⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝑠𝜆++1(𝑠+1)𝛾+𝛼^𝑖𝑠+2⋯𝑆−1𝑆𝜆+(𝑠+2)𝛾−𝜆+(𝑠+02)𝛾+𝛼_𝑖⋯00⋮⋮0⋮⋮0⋯⋯⋮⋮𝜆+⋮⋮0𝑆𝛾−𝜆+⋮⋮𝑆0𝛾+𝛼_𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(15)(vi)𝐻₀₁=𝑠𝑠⋮++𝑆12⎛⎜⎜⎜⎜⎜⎜⎝⋮10002⋮000⋯⋯⋯⋯⋯𝑠−⋮0001𝜆+𝑠(𝑠⋮00+1)𝛾⎞⎟⎟⎟⎟⎟⎟⎠,(16)(vii)𝐻₁₀=⋮1𝑠⎛⎜⎜⎜⎜⎝𝑠⋮00+1⋱⋯⋯⋯⋯00𝑄⋮𝑄𝜇0+1⋱⋯⋯⋯⋮𝜇0𝑆⎞⎟⎟⎟⎟⎠,(17)(viii)𝐻₁₁=⋮12𝑠⎛⎜⎜⎜⎜⎜⎜⎝−(𝜆1𝜆++⋮0𝛾2𝛾+𝜇)−2(𝜆+⋮002𝛾⋯+𝜇)𝑠⋯⋯⋯−⋮1𝜆+⋮00𝑠𝑠𝛾−(𝜆+⋮00𝑠𝛾+𝜇)⎞⎟⎟⎟⎟⎟⎟⎠.(18)

4. Steady-State Analysis

It can be seen from the structure of the infinitesimal generator 𝐴 that the time-homogeneous Markov process {(𝑋(𝑡),𝑌(𝑡),𝑍(𝑡),𝐿(𝑡));𝑡≥0} on the finite state space 𝐸 is irreducible. Hence, the limiting distribution𝜙^{(𝑗1,𝑗2,𝑗3,𝑗4)}=lim_𝑡→∞P𝑍r(𝑋𝑡)(=𝑡)𝑗=₃𝑗₁,𝑌(𝑡)=𝑗₂,𝐿(0)],𝐿(𝑡)=𝑗₄∣𝑋(0),𝑌(0),𝑍(0),(19) exists. Let𝜙^(𝑖1)=𝜙𝜙^{(𝑖1,0,0,0)}_{(𝑖1,0,1,0)},𝜙^{(𝑖1,0,0,1)},…,𝜙^{(𝑖1,0,0,𝑄)},𝜙,𝜙^{(𝑖1,1,0,𝑠+1)}_{(𝑖1,1,0,𝑠+1)},𝜙^{(𝑖1,1,0,𝑠+2)},…,𝜙^{(𝑖1,1,0,𝑆)},𝜙_{(𝑖1,1,1,1)},𝜙^{(𝑖1,1,0,𝑠+2)},…,𝜙^{(𝑖1,1,0,𝑆)},Φ=𝜙₍₀₎,𝜙^{(𝑖1,1,1,2)},…,𝜙^{(𝑖1,1,1,𝑠)},,𝜙⁽¹⁾,…,𝜙^(𝑁).(20) Then, the vector of limiting probabilities Φ satisfiesΦ𝐴=𝟎,Φ𝐞=1.(21)

From the structure of 𝐴, it is seen that the Markov process under study falls into the class of birth and death process in a Markovian environment as discussed by Gaver et al. [25]. Hence, using the same arguments, we can calculate the limiting probability vectors. For the sake of completeness, we provide the algorithm here.

Algorithm. (1) Determine recursively the matrices 𝐹𝐹_0𝑖1=𝐴₀,=𝐴_𝑖1+𝐶_𝑖1−𝐹^𝑖−1₁⁻¹𝐵⁰,𝑖₁=1,2,…,𝑁.(22)

(2) Compute recursively the vectors 𝜙_(𝑖1) using 𝜙^(𝑖1)=𝜙^(𝑖1+1)𝐶_𝑖1+1−𝐹^𝑖−1₁,𝑖¹=𝑁−1,𝑁−2,…,0.(23)
(3) Solve the system of equations 𝜙^(𝑁)𝐹_𝑁=𝟎,(24)_𝑁𝑖₁₌₀𝜙^(𝑖1)𝐞=1.(25) From the system of (24), vector 𝜙^(𝑁) could be determined uniquely, up to a multiplicative constant. This constant is obtained by using (23) and (25).

5. System Performance Measures

In this section, we give some system performance measures, which are useful in the qualitative interpretation of the model under study.

5.1. Expected Inventory Level

Let 𝜂_𝐼 denote the mean inventory level in the steady-state. Since 𝜙^{(𝑖,𝑘,𝑝,𝑚)} denotes the steady state probability when the number of demands in the pool is 𝑖, the server status is 𝑘, the status of reorder is 𝑝, and the inventory level is 𝑚, the mean inventory level is given by𝜂_𝐼=^𝑁_𝑖=0^𝑄_𝑚=1𝑚𝜙^{(𝑖,0,0,𝑚)}+^𝑆_𝑚_=𝑠+1𝑚𝜙^{(𝑖,1,0,𝑚)}+^𝑠_𝑚₌₁𝑚𝜙^{(𝑖,1,1,𝑚)}.(26)

5.2. Expected Reorder Rate

Let 𝜂_𝑅 denote the expected reorder rate in the steady state. We note that a reorder is triggered(i)when the inventory level drops from 𝑠+1 to 𝑠 if the server is in service or(ii)when the server returns from the vacation, he/she finds the inventory level less than or equal to 𝑠 and there is no order pending.

This leads to𝜂_𝑅=^𝑁_𝑖=0+_𝑁𝜙^{(𝑖,1,0,𝑠+1)}𝜆+(𝑠+1)𝛾+𝛿_𝑖0𝛼_𝑖_𝑠𝑖=0_𝑚=0𝛽𝜙^{(𝑖,0,0,𝑚)},(27) where 𝛿_𝑖𝑗 is the Kronecker delta function.

5.3. Expected Perishable Rate

Let 𝜂_PR denote the expected reorder Perishable in the steady state. This is given by𝜂_PR=^𝑁_𝑖=0^𝑄_𝑚=1𝑚𝛾𝜙^{(𝑖,0,0,𝑚)}+^𝑆_𝑚_=𝑠+1𝑚𝛾𝜙^{(𝑖,1,0,𝑚)}+^𝑠_𝑚₌₁𝑚𝛾𝜙^{(𝑖,1,1,𝑚)}.(28)

5.4. Mean Shortage Rate

Let 𝜂_SR denote the mean shortage rate in the steady state. It may be noted that a shortage occurs only during the server vacation. The customers who arrive during server vacation prefer to leave the system either by their own choice or the pool has reached its maximum capacity:𝜂_SR=^𝑁−1_𝑖=0𝑞𝜆^𝑄_𝑚₌₀𝜙^{(𝑖,0,0,𝑚)}+𝜙^(𝑖,0,1,0)+𝜆_𝑄_𝑚₌₀𝜙^{(𝑁,0,0,𝑚)}+𝜙^(𝑁,0,1,0).(29)

5.5. Expected Number of Customers in the Pool

Let 𝜂_𝑃 denote the expected number of customers in the pool in the steady state. Since 𝜙^(𝑖) is the steady state probability vector for 𝑖 customers in the pool with each component specifying the probability for the particular combination of the status of the server, status of the reorder, and the inventory level, the expected number of customers in the pool is given by𝜂_𝑃=^𝑁_𝑖=1𝑖𝜙^(𝑖)𝐞.(30)

5.6. Fraction of Time When the Server is on Vacation

Let 𝜂_𝐹 denote the fraction of time when the server is on vacation. This is given by𝜂_𝐹=^𝑁_𝑄𝑖=0_𝑚=0𝜙^{(𝑖,0,0,𝑚)}+^𝑁_𝑖=0𝜙^(𝑖,0,1,0).(31)

6. Cost Analysis

The expected total cost per unit time (expected total cost rate) in the steady state for this model is defined to beTC(𝑠,𝑆,𝑁)=𝑐_ℎ𝜂_𝐼+𝑐_𝑝𝜂_𝑃+𝑐_pr𝜂_PR+𝑐_𝑠𝜂_𝑅+𝑐_sh𝜂_SR,(32) where𝑐_𝑠: setup cost per order,𝑐_ℎ: the inventory carrying cost per unit item per unit time,𝑐_𝑝: cost of a waiting customer in the pool per unit time, 𝑐_pr: perishable cost per unit item per unit time,𝑐_sh: cost per customer lost per unit time.

Hence, we get TC(𝑠,𝑆,𝑁)=𝑐_ℎ^𝑁_𝑖=0^𝑄_𝑚=1𝑚𝜙^{(𝑖,0,0,𝑚)}+^𝑆_𝑚_=𝑠+1𝑚𝜙^{(𝑖,1,0,𝑚)}+^𝑠_𝑚₌₁𝑚𝜙^{(𝑖,1,1,𝑚)}+𝑐^𝑝^𝑁_𝑖=1+𝑐_𝑠𝑖𝜙^(𝑖)𝐞^𝑁_𝑖=0+_𝑁𝜙^{(𝑖,1,0,𝑠+1)}𝜆+(𝑠+1)𝛾+𝛿_𝑖0𝛼_𝑖_𝑠𝑖=0_𝑚=0𝛽𝜙^{(𝑖,0,0,𝑚)}+𝑐_pr^𝑁_𝑖=0^𝑄_𝑚=0𝑚𝛾𝜙^{(𝑖,0,0,𝑚)}+^𝑆_𝑚_=𝑠+1𝑚𝛾𝜙^{(𝑖,1,0,𝑚)}+^𝑠_𝑚₌₁𝑚𝛾𝜙^{(𝑖,1,1,𝑚)}+𝑐_sh^𝑁−1_𝑖=0𝑞𝜆^𝑄_𝑚₌₀𝜙^{(𝑖,0,0,𝑚)}+𝜙^(𝑖,0,1,0)+𝜆_𝑄_𝑚₌₀𝜙^{(𝑁,0,0,𝑚)}+𝜙^(𝑁,0,1,0).(33) Due to the complex form of the limiting distribution, it is difficult to discuss the properties of the cost function TC(𝑠,𝑆,𝑁) analytically. Hence, a detailed computational study of the expected cost function is carried out in the next section.

7. Numerical Illustrations

To study the behaviour of the model developed in this work, several examples were performed and a set of representative results is shown here. Although we have not shown the convexity of TC(𝑠,𝑆,𝑁) analytically, our experience with considerable numerical examples indicates the function TC(𝑠,𝑆,𝑁) to be convex. In some cases, it turned out to be an increasing function of 𝑠, and we use simple numerical search procedure to get the optimal values of TC,𝑠,𝑆 and 𝑁 (say TC^∗,𝑠^∗,𝑆^∗, and 𝑁^∗). A typical three-dimensional plot of the expected cost function is given in Figure 1. The optimal cost value TC^∗=37.352776 is obtained at (𝑠^∗,𝑆^∗,𝑁^∗)=(11,63,11).

Figure 1 

A three-dimensional plot of the cost function.𝜆=14,   𝜇=4,   𝛾=1.2,   𝛽=1.3,   𝛼𝑖=𝑖∗𝛼,   𝑖=1,2,…,𝑁,   𝛼=4,   𝑝=0.85,   𝑐ℎ=0.3,   𝑐𝑠=15,   𝑐pr=0.3,   𝑐sh=5,𝑐𝑝=3.

We have studied the effect of varying the system parameters and costs on the optimal values, and the results agreed with what one would expect. Some of our results are presented in Tables 1 through 20 where the upper entries in each cell give the 𝑠^∗,𝑆^∗, and 𝑁^∗, respectively, and the lower entry gives the corresponding TC^∗.

Table 1 

Optimal values corresponding to various demand rates and the replenishment rates(𝛾=1.4,𝛽=1.3,𝛼=4,𝑝=0.85).

Table 2 

Optimal values corresponding to various demand rates and perishable rate (𝛽=1.3,𝛼=4,𝜇=4,𝑝=0.85).

Table 3 

Optimal values corresponding to various demand rates and the vacation time rates(𝛾=1.4,𝛼=4,𝜇=4,𝑝=0.85).

Table 4 

Effect of primary and postponed demand rates on the optimal values(𝛾=1.4,𝛽=1.3,𝜇=4,𝑝=0.85).

Table 5 

Optimal values corresponding to various perishable rates and the replenishment rates (𝜆=14,𝛽=1.3,𝛼=4,𝑝=0.85).

Table 6 

Influence of the postponed demand rate and replenishment rate on the optimal values(𝜆=14,𝛾=1.4,𝛽=1.3,𝑝=0.85).

Table 7 

Influence of the vacation time rate and the replenishment rate on the optimal values(𝜆=14,𝛾=1.4,𝛼=4,𝑝=0.85).

Table 8 

Optimal values corresponding to various perishable rates and vacation time rates (𝜆=14,𝛼=4,𝜇=4,𝑝=0.85).

Table 9 

Influence of the postponed demand rate and vacation time rate on the optimal values(𝜆=14,𝛾=1.4,𝜇=4,𝑝=0.85).

Example 1. In the first example, we study the impact of the demand rate 𝜆, the lead time rate 𝜇, the perishable rate 𝛾, the selection rate 𝛼_𝑖, and the vacation time parameter 𝛽 on the optimal values 𝑠^∗,𝑆^∗,𝑁^∗, and TC^∗. Towards this end, we first fix 𝛼_𝑖=𝑖𝛼,𝑖=1,2,…,𝑁, and the cost values as 𝑐_ℎ=0.3,𝑐_𝑠=15,𝑐_pr=0.3,𝑐_sh=5, and 𝑐_𝑝=3. We observe the following from Table 1 to 10.(1)The optimal values 𝑠^∗,𝑆^∗, and TC^∗ increase monotonically with increase in 𝜆. This is to be expected since, when the arrivals occurs closer and closer, the optimal reorder level and the maximum inventory level increase to prevent more waiting time of the demands in the pool. We also note that, as 𝜆 increases, 𝑁^∗ increases except for the case in which 𝛾 increases.(2)As is to be expected, the optimal cost rate TC^∗ decreases monotonically, when 𝜇,𝛼, and 𝛽 increase and monotonically increases when 𝛾 increases.(3)When 𝜇 increases, 𝑠^∗ and 𝑆^∗ decrease and 𝑁^∗ increases. This can be explained intuitively as follows. If the replenishments occur further and further apart, the inventory level becomes zero frequently. During the stock-out period, a part of arriving demands enter into the pool. To prevent the loss of demand, we have necessity to increase the pool size. For the same reason as 𝛽 increases, 𝑠^∗ and 𝑆^∗ decrease and 𝑁^∗ increases.(4)We note that, as 𝛾 increases, 𝑠^∗ and 𝑁^∗ decrease monotonically. We also note that except for the combination of increasing 𝛾, 𝑆^∗ decreases when 𝛼 increases.(5)As 𝛼 increases, 𝑁^∗ increases and 𝑠^∗ decreases. We also note that, except for the combination of increasing 𝜇, 𝑆^∗ increases when 𝛼 increases.

Table 10 

Optimal values corresponding to various perishable rates and postponed demand rates (𝜆=14,𝛽=1.3,𝜇=4,𝑝=0.85).

Example 2. In this example, we study the impact of the setup cost 𝑐_𝑠, holding cost 𝑐_ℎ, perishable cost 𝑐_pr, shortage cost 𝑐_sh, and the waiting cost 𝑐_𝑝 on the optimal values 𝑠^∗,𝑆^∗,𝑁^∗, and TC^∗. Towards this end, we first fix the parameter values as 𝜆=14,𝛾=1.4,𝛽=1.3,𝛼_𝑖=𝑖𝛼,𝑖=1,2,…,𝑁,𝛼=4,𝜇=4, and 𝑝=0.85. We observe the following from Table 11 to 20:(1)The optimal cost increases, when 𝑐_ℎ,𝑐_𝑠,𝑐_pr,𝑐_sh, and 𝑐_𝑝 increase. The optimal cost is more sensitive to 𝑐_ℎ than to 𝑐_𝑠,𝑐_pr,𝑐_sh, and 𝑐_𝑝.(2)As 𝑐_ℎ increases, as is to be expected, the optimal values 𝑠^∗,𝑆^∗, and 𝑁^∗ decrease monotonically. This is to be expected since the holding cost increases, we resort to maintain low stock in the inventory.(3)When 𝑐_sh increases, as is to be expected, the optimal values 𝑠^∗,𝑆^∗, and 𝑁^∗ increase monotonically. This is to be expected since for the high shortage cost to reduce the number of customer to be lost, we have to increase the pool size, maintain high inventory, and place order in the higher level.(4)If the waiting cost 𝑐_𝑝 increases, the optimal values of 𝑠^∗ increase and 𝑁^∗ decreases monotonically. We also note that, except for the combination of increasing the holding cost 𝑐_ℎ, 𝑆^∗ decreases when 𝑐_𝑝 increases.(5)If the setup cost 𝑐_𝑠 increases, it is a common decision that we have to maintain more stock to avoid frequent ordering. This fact is also observed in our model.(6)We note that, when the perishable cost 𝑐_pr increases 𝑠^∗,𝑆^∗, and 𝑁^∗ decrease monotonically.

Table 11 

Variation in optimal values for different values of 𝑐_ℎ and 𝑐_𝑝(𝑐_𝑠=15,𝑐_sh=5,𝑐_pr=0.3).

Table 12 

Optimal values corresponding to various holding and perishable costs (𝑐_𝑠=15,𝑐_sh=5,𝑐_𝑝=3).

Table 13 

Optimal values corresponding to various setup and holding costs (𝑐_sh=5,𝑐_𝑝=3,𝑐_pr=0.3).

Table 14 

Variation in optimal values for different values of 𝑐_ℎ and 𝑐_sh(𝑐_𝑠=15,𝑐_𝑝=3,𝑐_pr=0.3).

Table 15 

Effect of 𝑐_𝑠 and 𝑐_𝑝 on the optimal cost rate (𝑐_ℎ=0.3,𝑐_sh=5,𝑐_pr=0.3).

Table 16 

Effect of 𝑐_𝑠 and 𝑐_pr on the optimal cost rate(𝑐_ℎ=0.3,𝑐_sh=5,𝑐_𝑝=3).

Table 17 

Sensitivity of optimal values to variation in 𝑐_𝑠 and 𝑐_sh(𝑐_ℎ=0.3,𝑐_𝑝=3,𝑐_pr=0.3).

Table 18 

Variation in optimal values for different values of 𝑐_𝑝 and 𝑐_sh(𝑐_𝑠=15,𝑐_ℎ=0.3,𝑐_pr=0.3).

Table 19 

Effect of 𝑐_sh and 𝑐_pr on the optimal cost rate (𝑐_𝑠=15,𝑐_ℎ=0.3,𝑐_𝑝=3).

Table 20 

Variation in Optimal values for different values of 𝑐_𝑝 and 𝑐_pr(𝑐_𝑠=15,𝑐_ℎ=0.3,𝑐_sh=5).

Example 3. In this example, we look to the impact of the demand rate 𝜆, the reorder rate 𝜇, the selection rate 𝛼, and the mean vacation time 1/𝛽 on the fraction, 𝜂_𝐹, of time the server is on vacation for the combination of various values of postponed demand rate 𝛼^(𝑛)(𝑛=1,2,3), 𝛼^(𝑛) is the increasing, constant, and decreasing functions, where 𝛼⁽¹⁾=5+𝑖∗0.1, 𝛼⁽²⁾=5, and 𝛼⁽³⁾=5+𝑖∗(−0.1), 𝑖=1,…,𝑁. From Figures 2 to 5, we observe the following.(i)As is to be expected as 𝜆 increases, 𝜂_𝐹 increases. This is because, if the demands rate increases, the stockout occurs frequently, and hence the server takes vacation frequently. Hence, 𝜂_𝐹 increases. Similarly the same thing is observed in the case 𝛾.(ii)As 𝜇 increases, 𝜂_𝐹 decreases. This happens because, if 𝜇 increases, the ordered items are received at higher rate. Hence, fewer stock-out periods occur. Hence the server returns to the system more often. Hence, the intervacation time increases. Therefore 𝜂_𝐹 decreases. Similarly, the same thing is observed in the case 𝛽.(iii)In these cases, we observe 𝛼⁽¹⁾>𝛼⁽²⁾>𝛼⁽³⁾.

Figure 2 

Fraction of time server is on vacation—𝜆 versus 𝛼(𝑛) (𝑆=90,   𝑠=15,   𝑁=25,   𝛽=1.4,   𝛾=4,   𝜇=0.8,   𝑝=0.75.)

Figure 3 

Fraction of time server is on vacation—𝛽 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛾=4,𝜇=0.8,𝑝=0.75.)

Figure 4 

Fraction of time server is on vacation—𝜇 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝛽=1.4,𝛾=4,𝜆=8,𝑝=0.75.)

Figure 5 

Fraction of time server is on vacation—𝛾 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝛽=1.4,𝜆=8,𝜇=0.8,𝑝=0.75.)

Example 4. In this example, we now study the impact of 𝑝 on the performance measures, expected Inventory level 𝜂_𝐼, expected reorder rate 𝜂_𝑅, expected perishable rate 𝜂_PR, mean shortage rate 𝜂_SR, expected number of customers in the pool 𝜂_𝑃, fraction of time the server is on vacation 𝜂_𝐹 and the total expected cost TC(𝑠,𝑆,𝑁), for the combination of various values of postponed demand rate 𝛼^(𝑛)(𝑛=1,2,3), 𝛼^(𝑛) is the increasing, constant, and decreasing functions, where 𝛼⁽¹⁾=5+𝑖∗0.1, 𝛼⁽²⁾=5 and 𝛼⁽³⁾=5+𝑖∗(−0.1), 𝑖=1,…,𝑁. From Figure 6 to 12, we observe the following.(i)As TC(𝑠,𝑆,𝑁), 𝜂_SR and 𝜂_𝑃 are monotonically increasing, when 𝑝 increases. In these cases, we observe 𝛼⁽¹⁾<𝛼⁽²⁾<𝛼⁽³⁾.(ii)As 𝜂_𝑅 and 𝜂_𝐹 are monotonically increasing, when 𝑝 increases. In these cases, we observe 𝛼⁽¹⁾>𝛼⁽²⁾>𝛼⁽³⁾.(iii)As 𝜂_𝐼 and 𝜂_PR are monotonically decreasing, when 𝑝 increases. In these cases, we observe 𝛼⁽¹⁾<𝛼⁽²⁾<𝛼⁽³⁾.

Figure 6 

The total expected cost rate—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜆=8.)

Figure 7 

Expected Inventory Level—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Figure 8 

Expected reorder rate—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Figure 9 

Expected number of customers in the pool—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Figure 10 

Expected perishable rate—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Figure 11 

Mean loss of demand rate—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Figure 12 

Fraction of time server is on vacation—𝑝 versus 𝛼(𝑛)(𝑆=90,𝑠=15,𝑁=25,𝜆=8,𝛽=1.4,𝛾=0.7,𝜇=0.8.)

Acknowledgments

The authors wish to thank the anonymous referee for the comments, which significantly improved the presentation of this paper. G. Arivarignan would like to thank the University Grants Commission (UGC), India, for their financial support (F. no. 32-170/2006(SR)). B. Sivakumar would like to thank the Council of Scientific and Industrial Research (CSIR)—India for their financial support (no. 25(0813)/10/EMR-II).