Limit Distribution of Inventory Level of Perishable Inventory Model

Abstract

This paper studies a perishable inventory model, which assumes that each perishable item has finite lifetime, and only one item is consumed each time. The lifetimes of perishable items are independent random variables with the general distribution and so are the consumption internal. Under this assumption, by using backward equations and limit distribution of Markov skeleton processes, this paper obtains the existence conditions and the explicit expression of the limit distribution of the inventory level of perishable inventory model.

1. Introduction

Perishable goods are common in our daily life. In this paper, perishable goods refer to the items that have finite lifetime, like putrescible foods, easily-expired medicines, volatile liquids, and so on. Perishable inventory model can be widely used in blood banks, chemical and food industry.

In the past few decades, researchers have paid much attention to perishable inventory model. The inventory problem of perishable items was first studied by Whitin [1] who considered fashion goods perishing at the end of a prescribed storage period. Ghare and Schrader [2] proposed an inventory model, in which the rate of perishable is a constant, and the consumption internals have exponential distribution. Based on the inventory model proposed by Ghare and Schrader [2], series of studies are carried out (see Raafat [3], Goyal and Giri [4], and their references). Recently, Li et al. [5] considered some factors, like demand, deteriorating rate, price discount, allow shortage or not, inflation, time value of money, and so on, as important factors in the perishable inventory study, then they divided current perishable items inventory study literatures into two categories from the perspective of study scope and reviewed the literature for each category. Karmakar and Choudhury [6] focused on the modeling of perishable items with shortages and reviewed the corresponding inventory models. Other representative works can be seen in [7–15].

An interesting and important study of perishable inventory model is about the inventory level process. Ravichandran [16] obtained the explicit expression of the stationary distribution of the inventory level in operating the (𝑆,𝑠) policy, with positive lead time and poisson demand. Chiu [17] developed the expected inventory level to determine a best (𝑄,𝑟) ordering policy under a positive order lead time and fixed life perishability. Liu and Yang [18] analyzed an (𝑠,𝑆) continuous review model and obtains the matrix-geometric solutions for the steady-state probability distribution of the inventory level, with finite lifetimes and positive lead times. Sivakumar [19] obtained the joint probability distribution of the inventory level and the number of demands in the orbit, where the life time of each items is assumed to be exponential. Other related papers can be seen in [20–22] and so on.

In order to facilitate the mathematical treatment, most of these papers assume that the lifetime of item or consumption internal equals to constant or has exponential distribution, so that the inventory level of perishable inventory model can be reduced to a Markov process. However, in practice, the lifetime of item or consumption interval is not necessarily exponential, but a wide range of distribution. In this case, the inventory level of perishable inventory model is not and hardly been converted into a Markov process, which leads to a bottleneck on the mathematical treatment. To the best of our knowledge, no previous studies obtained the existence conditions of the limit distribution of the inventory level. Thus, we intend to work at it.

Markov skeleton process provides an effective solution to the problem. Markov skeleton processes which are proposed by Hou et al. [23] in 1997 are more extensive than Markov processes. Markov skeleton process has been in-depth studied (representative works, see [24–27]). This paper proves that the inventory level of perishable inventory model is a positive recurrent Doob skeleton process which is a special case of Markov skeleton processes. Hence, by applying backward equations and limit distribution of Markov skeleton processes, this paper obtains the existence conditions and the explicit expression of the limit distribution of the inventory level. Moreover, this paper obtains the probability of the inventory level greater than 0 and the probability of the inventory level less than or equal to 0, which can then be used for the evaluation of inventory system performance.

This paper is organized as follows. Section 2 introduces Markov skeleton processes and presents its backward equations and limit distribution. Section 3 introduces a perishable inventory model and applies Markov skeleton processes approach to study the limit distribution of inventory level process.

2. Markov Skeleton Processes

In this section, we introduce Markov skeleton processes and present its backward equations and limit distribution.

2.1. Definition of Markov Skeleton Processes

Definition 2.1 (see [26]). A stochastic process 𝑋={𝑋(𝑡,𝜔),0≤𝑡<∞)} which takes values on a polish space (𝐸,ℰ) is called a Markov skeleton process if there exists a sequence of optional stopping times {𝜏_𝑛}_𝑛≥0, satisfying (i)𝜏_𝑛↑+∞ with 𝜏₀=0, and for each 𝑛≥0, 𝜏_𝑛<∞; (ii)for all 𝑛=0,1,…, 𝜏_𝑛+1=𝜏_𝑛+𝜃_𝜏𝑛⋅𝜏₁; (iii)for every 𝜏_𝑛 and any bounded ℰ^[0,∞)-measurable function 𝑓 defined on 𝐸^[0,∞)𝐸𝑓𝑋𝜏_𝑛+⋅∣ℱ^𝑋_𝜏𝑛=𝐸𝑓𝑋𝜏_𝑛+⋅∣𝑋𝜏_𝑛P-a.s., (2.1) where Ω_𝜏𝑛=(𝜔∶𝜏_𝑛(𝜔)<∞), and ℱ^𝑋_𝜏𝑛={𝐴∶∀𝑡≥0,𝐴∩(𝜔∶𝜏_𝑛≤𝑡)∈ℱ^𝑋_𝑡} is the 𝜎-algebra on Ω_𝜏𝑛. {𝜏_𝑛}^∞_𝑛=0 is called skeleton time sequence of the Markov skeleton process 𝑋. Furthermore, if on Ω_𝜏𝑛𝐸𝑓𝑋𝜏_𝑛+⋅∣ℱ^𝑋_𝜏𝑛=𝐸𝑓𝑋𝜏_𝑛+⋅∣𝑋𝜏_𝑛=𝐸_{𝑋(𝜏𝑛)}𝑓(𝑋(⋅)) (2.2) P-a.s. holds, where 𝐸_𝑥(⋅) denotes the expectation corresponding to 𝑃(⋅∣𝑋(0)=𝑥), then 𝑋 is called a time homogeneous Markov skeleton process.

Definition 2.2 (see [26]). A time homogeneous Markov skeleton process 𝑋={𝑋(𝑡,𝜔),≤𝑡<∞}→{𝑋(𝑡,𝜔),0≤𝑡<∞} is called normal, if there exists a function ℎ(𝑥,𝑡,𝐴) on 𝐸×𝐑⁺×𝜀, such that (i)for fixed 𝑥 and 𝑡, ℎ(𝑥,𝑡,⋅) is a finite measure on 𝜀, (ii)for fixed 𝐴∈𝜀,ℎ(⋅,⋅,𝐴) is 𝜀×ℬ(𝐑⁺) measurable function on 𝐸×𝐑⁺, (iii)for any 𝑡≥0,𝐴∈𝜀,ℎ𝑋𝜏_𝑛,𝑡,𝐴=𝑃𝑋𝜏_𝑛+𝑡∈𝐴,𝜏_𝑛+1−𝜏_𝑛>𝑡∣𝑋𝜏_𝑛P-a.s. (2.3)

2.2. Backward Equations of Markov Skeleton Processes

Theorem 2.3 (see [26]). Suppose that 𝑋={𝑋(𝑡);𝑡≥0} is a normal Markov skeleton process with {𝜏_𝑛}^∞_𝑛=0 as its skeleton time sequence, then for any 𝑥∈𝐸,𝑡≥0,𝐴∈𝜀, 𝑝(𝑥,𝑡,𝐴)=ℎ(𝑥,𝑡,𝐴)+_𝐸^𝑡₀^∞_𝑛=1𝑞^(𝑛)𝑥,𝑑𝑠,𝑑𝑦ℎ𝑦,𝑡−𝑠,𝐴. (2.4) Thus, 𝑝(𝑥,𝑡,𝐴) is a minimal nonnegative solution to the following nonnegative equation system: ∀𝑥∈𝐸,𝑡≥0,𝐴∈𝜀, 𝑝(𝑥,𝑡,𝐴)=ℎ(𝑥,𝑡,𝐴)+_𝐸^𝑡₀𝑞𝑥,𝑑𝑠,𝑑𝑦𝑝𝑦,𝑡−𝑠,𝐴. (2.5) Formula (2.5) is called the backward equations of Markov skeleton processes.

2.3. Limit Distribution of Markov Skeleton Processes

Definition 2.4 (see [27]). Suppose that 𝑋(𝑡) is a normal Markov skeleton process with {𝜏_𝑛}^∞_𝑛=0 as its skeleton time sequence. If there exists probability measure 𝜋(⋅) on (𝐸,ℰ), such that for any 𝐴∈ℰ, 𝑃𝑋𝜏₁∈𝐴∣𝑋(0)=𝑥,𝜏₁=𝑠=𝑃𝑋𝜏₁∈𝐴=𝜋(𝐴), (2.6) then 𝑋(𝑡) is called a Doob skeleton process, 𝜋(⋅) is called the characteristic measure of 𝑋(𝑡), and {𝜏_𝑛,𝑛=1,2,…} is the Doob skeleton time sequence of 𝑋(𝑡).
For any 𝑛∈𝑁,𝑡≥0,𝐴∈ℰ, 𝑞^(𝑛)𝑋𝜏_𝑚,𝑡,𝐴=𝑃𝑋𝜏_𝑚+𝑛∈𝐴,𝜏_𝑚+𝑛≤𝑡∣𝑋𝜏_𝑚 (2.7) and 𝑞⁽¹⁾(𝑥,𝑡,𝐴) is abbreviated to 𝑞(𝑥,𝑡,𝐴), 𝑝(){𝑋(𝑡)(0)},𝐹()𝜏𝑥,𝑡,𝐴=𝑃∈𝐴∣𝑋=𝑥𝑥,𝑡=𝑃₁(0)𝐹(𝑡)=≤𝑡∣𝑋=𝑥,∀𝑥∈𝐸,𝑡≥0,_𝐸𝐹()𝜋()ℎ()=𝑥,𝑡𝑑𝑥,𝑡≥0,𝑡,𝐴_𝐸ℎ(𝑥,𝑡,𝐴)𝜋(𝑑𝑥),𝑚=^∞₀𝑡𝑑𝐹(𝑡). (2.8)

Definition 2.5 (see [27]). Suppose that 𝑋(𝑡) is a Doob skeleton processes. If 𝑚<∞ and for any 𝑥∈𝐸, 𝐹(𝑥,0)≡0, 𝐹(𝑥,∞)≡1, then 𝑋(𝑡) is called a positive recurrent Doob skeleton process.

Theorem 2.6 (see [27]). Suppose that 𝑋(𝑡) is a positive recurrent Doob skeleton process. If 𝐹(𝑡) is not lattice distribution, then for ∀𝐴∈ℰ, the limit distribution 𝑝(⋅) of 𝑋(𝑡) exists, 𝑝(𝐴)=lim_𝑡→∞𝑝()=∫𝑥,𝑡,𝐴^∞0ℎ(𝑡,𝐴)𝑑𝑡𝑚, (2.9) and 𝑝(⋅) is a probability distribution in (𝐸,ℰ).

3. Limit Distribution of Inventory Level of Perishable Inventory Model

The perishable inventory model studied in this paper has been proposed and investigated in [26], which obtained the backward equations of the inventory level of this model. Different from [26], this paper study the limit distribution of the inventory level.

3.1. Perishable Inventory Model

First, we present the details of the perishable inventory model as follows (see [26]). (1)Assume that lifetimes of inventory commodities are i.i.d random variables, with a common distribution function 𝐹(𝑡), where 𝐹(𝑡) is continuous and satisfies ∫^∞₀𝑡𝑑𝐹(𝑡)=𝜇₁.(2)Sell one item each time, and the sale times of each item are i.i.d random variables, with a common distribution function 𝐺(𝑡), where 𝐺(𝑡) is continuous and satisfies ∫^∞₀𝑡𝑑𝐺(𝑡)=𝜇₂. Assume that the sale times are also independent of the commodities' lifetimes. (3)The maximum capacity of the warehouse is a fixed value 𝑆_max. When the inventory level becomes 𝑆_min(𝑆_min<0) (i.e., the quantity of out of stock arrives 𝑆_min), new commodities are replenished to increase the inventory level until it reaches 𝑆_max.

Let 𝑆(𝑡) denote inventory level at time 𝑡. When 𝐹(𝑡) and 𝐺(𝑡) are not exponential distributions, {𝑆(𝑡),𝑡≥0} is not a Markov process. In this case, we introduce supplementary variables as follows: 𝜃(𝑡) denotes the lifetime of the item in stock at time 𝑡, and 𝜃(𝑡) denotes the time interval between the last sale before 𝑡 and time 𝑡.

As one item is consumed and the other item perishes at the same time is a rare event, so we don't consider this case and suppose 𝐹(𝑡) and 𝐺(𝑡) are continuous distribution. Let 𝜏_𝑛 denote the 𝑛th discontinuous point of (𝑆(𝑡),𝜃(𝑡),𝜃(𝑡)), that is, one item is consumed or perishes at 𝜏_𝑛. At 𝜏_𝑛, (𝑆(𝑡),𝜃(𝑡),𝜃(𝑡)) has Markov property, so by Definition 2.1, (𝑆(𝑡),𝜃(𝑡),𝜃(𝑡)) is a Markov skeleton process with 𝜏_𝑛 as its Markov skeleton time sequence.

3.2. Limit Distribution of Inventory Level

In this subsection, we obtain the limit distribution of inventory level.

Suppose that 𝑇₀=0, and 𝑇_𝑛 denotes the 𝑛th times when the process (𝑆(𝑡),𝜃(𝑡),𝜃(𝑡)) returns to state (𝑆_max,0,0). Let 𝑇_𝑛+1=𝑇_𝑛+𝜃_𝑇𝑛⋅𝑇₁𝑌,𝑛=1,2,…,₀=𝑇₁,𝑌_𝑖=𝑇_𝑖+1−𝑇_𝑖,𝑖≥1, (3.1) then 𝑌_𝑖 is the replenishment interval. By Definition 2.2, ℎ_𝑆max,𝑗(𝑡)=𝑃𝑆(𝑡)=𝑗,𝑡≤𝑇₁∣𝑆(0)=𝑆_max,𝜃(0)=𝜃(0)=0. (3.2) Let 𝑀(𝑡) denote the distribution function of 𝑌_𝑖, and 𝑀 denote the expectation of 𝑌_𝑖, then, 𝑀=𝐸𝑇₁∣𝑆(0)=𝑆_max(0)=𝜃(0)=,𝜃=0_∞0𝑃𝑇₁≥𝑡∣𝑆(0)=𝑆_max(0)=𝜃(0)=,𝜃=0𝑑𝑡_{∞0𝑆max𝑗=𝑆min+1}ℎ_𝑗(𝑡)𝑑𝑡. (3.3)

Theorem 3.1. If 𝜇₁<+∞, 𝜇₂<+∞, 𝑆(𝑡) is a positive recurrent Doob skeleton process with {𝑇_𝑛} as its Doob skeleton time sequence.

Proof. As 𝑇_𝑛 denotes the beginning of the 𝑛th replenishment and 𝑇_𝑛− denotes the moment before the 𝑛th replenishment, we have 𝑆(𝑇_𝑛−)=𝑆_min, 𝑆(𝑇_𝑛)=𝑆_max, then 𝑆(𝑡) satisfies Markov property at 𝑇_𝑛, which assures that 𝑆(𝑡) is a Markov skeleton process. At the beginning of every replenishment, 𝑆(𝑇_𝑛)≡𝑆_max, so we have 𝑃𝑆𝑇₁=𝑗∣𝑆(0)=𝑖,𝑇₁=𝑠=𝛿₀𝑗−𝑆_max. (3.4) Thus, 𝑆(𝑡) is Doob skeleton process by Definition 2.4. If 𝜇₁<+∞, 𝜇₂<+∞, we obtain 𝑀=𝐸[𝑌_𝑖]<+∞. Therefore, 𝑆(𝑡) is a positive recurrent Doob skeleton process with {𝑇_𝑛} as its Doob skeleton time sequence.

Theorem 3.2. If 𝜇₁<+∞, 𝜇₂<+∞, and 𝐹(𝑥), 𝐺(𝑥) are not lattice distribution, then, for 𝑗∈{𝑆_min+1,𝑆_min+2,…,𝑆_max}, 𝑝_𝑗(𝑡)=lim_𝑡→∞𝑝𝑆(𝑡)(0)=∫=𝑗∣𝑆=𝑖^∞0ℎ_𝑗(𝑡)𝑑𝑡𝑀=∫^∞₀ℎ_𝑗(𝑡)𝑑𝑡∫^∞₀∑^𝑆max_{𝑗=𝑆min+1}ℎ_𝑗(𝑡)𝑑𝑡; (3.5) for 𝐴={1,2,…,𝑆_max}, 𝑝(𝐴)=lim_𝑡→∞𝑝(𝑆(𝑡)(0))=∫∈𝐴∣𝑆=𝑖^∞0ℎ(𝑡,𝐴)𝑑𝑡𝑀=∫^∞₀∑^𝑆max_𝑗=1ℎ_𝑗(𝑡)𝑑𝑡∫^∞₀∑^𝑆max_{𝑗=𝑆min+1}ℎ_𝑗(𝑡)𝑑𝑡; (3.6) for 𝐵={𝑆_min+1,𝑆_min+2,…,0}, 𝑝(𝐵)=lim_𝑡→∞𝑝(𝑆(𝑡)(0))=∫∈𝐵∣𝑆=𝑖^∞0ℎ(𝑡,𝐵)𝑑𝑡𝑀=∫^∞₀∑⁰_{𝑗=𝑆min+1}ℎ_𝑗(𝑡)𝑑𝑡∫^∞₀∑^𝑆max_{𝑗=𝑆min+1}ℎ_𝑗(𝑡)𝑑𝑡, (3.7) and 𝑝(⋅) is a probability distribution in (𝐸,ℰ).

Proof. If 𝐹(𝑥), 𝐺(𝑥) are not lattice distribution, then 𝑀(𝑡) is not lattice distribution. According to Theorems 2.6, 3.1, and formula (3.3), we get formulas (3.5)–(3.7). Thus, the proof of the theorem is completed.

3.3. The Explicit Expression of ℎ_𝑗(𝑡)

Next, we intend to give the explicit expression of ℎ_𝑗(𝑡) by applying backward equations of Markov skeleton processes.

By formula (3.4), we have 𝜋(𝑑𝑥)=𝛿₀(𝑥−𝑆_max). Then, ℎ_𝑗(𝑡)=^∞_𝑖=0ℎ_𝑖𝑗(𝑡)𝛿₀𝑖−𝑆_max=ℎ_𝑆max,𝑗(𝑡), (3.8) where ℎ_𝑆max,𝑗(𝑡) is defined in (3.2).

Let 𝜏₀=0, 𝜏_𝑛=𝜏_𝑛∧𝑇₁,𝑛=1,2,…, where 𝜏_𝑛 denotes the 𝑛th discontinuous point of (𝑆(𝑡),𝜃₁(𝑡),𝜃₂(𝑡)), then 𝜏_𝑛↑𝑇₁,𝑛↑+∞. (3.9)

According to Theorem 3.1, (𝑆(𝑡),𝜃₁(𝑡),𝜃₂(𝑡),𝑡<𝑇₁) is a Markov skeleton process with 𝜏_𝑛 as its skeleton time sequence.

For 𝑖>0, let ℎ𝑖,𝜃,𝜃,𝑗,𝐴,𝐴,𝑡=𝑃𝑆(𝑡)=𝑗,𝜃(𝑡)∈𝐴,𝜃(𝑡)∈𝐴,𝑡<𝜏₁(0)(0)𝜃(0)=𝜃;𝑞𝐴𝑆()()𝜃()∈(0)(0)𝜃(0)=𝜃;𝑝𝑆(𝑡)(𝑡)𝜃(𝑡)∈∣𝑆=𝑖,𝜃=𝜃,𝑖,𝜃,𝜃,𝑑𝑠,𝑗,𝐴,=𝑃𝑑𝑠=𝑗,𝜃𝑑𝑠∈𝐴,𝑑𝑠𝐴∣𝑆=𝑖,𝜃=𝜃,𝑖,𝜃,𝜃,𝑗,𝐴,𝐴,𝑡=𝑃=𝑗,𝜃∈𝐴,𝐴,𝑡<𝑇₁∣𝑆(0)=𝑖,𝜃(0)=𝜃,𝜃(0)=𝜃. (3.10) Thus, ℎ_𝑆max,𝑗(𝑡) can be expressed as follows: ℎ_𝑆max,𝑗(𝑡)=𝑝𝑆_max,0,0,𝑗,𝑡. (3.11)

Lemma 3.3. When 𝐹(𝑡)and𝐺(𝑡) are continuous, we have ℎ𝑆_max,0,0,𝑗,𝑡=0,𝑗≠𝑆_max;(1−𝐹(𝑡))^𝑆max(1−𝐺(𝑡)),𝑗=𝑆_max,𝑞𝑆_max=𝐶,0,0,𝑑𝑠,𝑗^1𝑆max(1−𝐹(𝑠))^𝑆max−1(1−𝐺(𝑠))𝐹(𝑑𝑠),𝑗=𝑆_max𝐶−1,oneitemperishes,andnoitemisconsumed;_1𝑆max(1−𝐹(𝑠))^𝑆max𝐺(𝑑𝑠),𝑗=𝑆_max−1,oneitemisconsumed,andnoitemperishes;0,𝑗≠𝑆_max−1. (3.12)

Proof. By the definition of ℎ(𝑆_max,0,0,𝑗,𝑡), there is no state transition of 𝑆(𝑡) up to 𝑡. If 𝑗≠𝑆_max, ℎ(𝑆_max,0,0,𝑗,𝑡)=0. If 𝑗=𝑆_max, which means that no item is consumed or perishes up to 𝑡, then ℎ(𝑆_max,0,0,𝑗,𝑡)=(1−𝐹(𝑡))^𝑆max(1−𝐺(𝑡)).
By the definition of 𝑞(𝑆_max,0,0,𝑑𝑠,𝑗), 𝑆(𝑡) will transfer from state 𝑆_max to state 𝑗 at 𝑑𝑠. As 𝐹(𝑡),𝐺(𝑡) are continuous, when 𝑗≠𝑆_max−1, 𝑞(𝑆_max,0,0,𝑑𝑠,𝑗)=0.
If one item perishes at time 𝑑𝑠, and no item is consumed up to 𝑠, then 𝑗=𝑆_max−1, 𝑞(𝑆_max,0,0,𝑑𝑠,𝑗)=𝐶¹_𝑆max(1−𝐹(𝑠))^𝑆max−1(1−𝐺(𝑠))𝐹(𝑑𝑠).
If one item is consumed at time 𝑑𝑠, and no item perishes up to 𝑠, then 𝑗=𝑆_max−1, 𝑞(𝑆_max,0,0,𝑑𝑠,𝑗)=𝐶¹_𝑆max(1−𝐹(𝑠))^𝑆max𝐺(𝑑𝑠).

According to Theorem 2.3 and Lemma 3.3, we have the following.

Theorem 3.4. 𝑝(𝑆_max,0,0,𝑗,𝑡) is the minimal nonnegative solution to the following nonnegative linear equation, 𝑝𝑆_max,0,0,𝑗,𝑡=𝛿_𝑆max,𝑗(1−𝐹(𝑡))^𝑆max((𝑡+1−𝐺))_𝑡0𝐶¹_𝑆max(1−𝐹(𝑠))^𝑆max−1(1−𝐺(𝑠))𝐹(𝑑𝑠)𝑝𝑆_max+−1,𝑠,𝑠,𝑗,𝑡−𝑠_𝑡0𝐶¹_𝑆max(1−𝐹(𝑠))^𝑆max𝐺(𝑑𝑠)𝑝𝑆_max−1,𝑠,0,𝑗,𝑡−𝑠. (3.13)

Thus, combining formulas (3.8), (3.11), and Theorem 3.4, the explicit expression of ℎ_𝑗(𝑡) is obtained; ℎ_𝑗(𝑡)=𝑝(𝑆_max,0,0,𝑗,𝑡) is the minimal nonnegative solution to formula (3.13).

Acknowledgments

This work was supported by the National Natural Science Foundation for Young Scholars of China (no. 11001179). The authors thank the editor and the anonymous referees for their constructive comments and suggestions for improving the quality of the paper.