# Dynamics and Control on a Discrete Multi-Inventory System

Authors: Yuzhen Wang; Li Xu
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

In this paper, a discrete two item inventory model for deteriorating items with a linear stock dependent demand rate without regular supply is given first; then, regular supply is taken into account, which may be regarded as feedback control. An objective function is formed to calculate the net profit with respect to possible profits and possible loss. A necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the difference equations of the inventory is obtained. At last, an improved model is put forward.


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## Body

## 1. Introduction

Inventory management is one of the most important drivers for an effective enterprise. The mathematical modeling of inventory control for inventory management has caught up the mind of inventory researchers. Since the classical economic order quantity model which undoubtedly constitutes the cornerstone of inventory research [1], there are many models in the literature which can illustrate the behavior of inventory management, including not only the continuous-time framework (see [2–5], and the listed conference), but also discrete time one (for example, see [6, 7]), whose origins may be traced in the seminal work of Arrow [8].In recent years, there are strong calls for the development of multi-item deterioration inventory system management [9]. In general, deterioration is defined as decay, damage, spoilage, evaporation, obsolescence, pilferage, loss of utility, or loss of marginal value of a commodity that results in decrease of usefulness from the original one. Because of deterioration, a certain fraction of the items is either damaged or decayed which will not satisfy the future demand of customers better. Deterioration for such items is continuous and constant or time-dependent. A number of research papers have already been published on the above type of items by some authors, the details of which can be found in the articles [10–17].For example, some principles which occur in population dynamics can be applied to the multi -item deterioration inventory system in [17]. A two item competitive ordinary differential inventory model for deteriorating items with a linear stock dependent demand rate is given. By means of Pontryagin’s Maximum Principle, the author derived a necessary condition for the steady state optimal control problem in which objective function subjected to the constraints given by the competitive ordinary differential equations. The idea is interesting.In most models, it is assumed that the system events are monitored continuously and that the time axis is continuous. In fact, the inventory systems, in which events occur and decisions are made during fixed length intervals, are characterized, then discrete time inventory models are relatively more reasonable. And, there is a growing research interest in discrete time queues (see [11–14]), which are mainly motivated by their applications in the computer and communication systems. In such a discrete time model, events occur at evenly-spaced points in time t =0,1,2,  …N, where the integers represent any unit of time such as a day, a week, or a month. A discrete time system is a more natural manner to describe inventory dynamics.The model of discrete system control is both a simulation model of inventory dynamics and an optimization model which can give optimal control according to the defined performance criterion [18]. Here, the key idea is to construct a function called the Hamiltonian by introducing the adjoint function to attach the difference equation to the objective function, which can be dealt with by the discrete version of Pontryagin’s Maximum Principle [19]. For the objective functional subject to the state difference equation with initial condition, the Hamiltonian can be used to find the optimal control. So far, to our knowledge, there have been very few corresponding research works focusing on discrete version of Pontryagin’s Maximum Principle on discrete inventory model for deteriorating items. It will be interesting and meaningful.In this paper, corresponding to [17], a new discrete two item inventory model for deteriorating items with a linear stock dependent demand rate without regular supply is given first, then regular supply is taken into account. So this paper is organized as follows: in Section 2 we give some notations and assumptions. In Section 3, stability analysis is discussed for our discrete two item inventory model for deteriorating items without regular supply and with regular supply. In Section 4, an objective function is formed to calculate the net profit with respect to all possible profits and all possible loss. By means of the discrete version of Pontryagin’s Maximum Principle, a necessary condition for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the difference equations of the inventory is obtained. The next section is simulation discussion; a numerical example is used to illustrate how to find the optimal control and the corresponding state. Then an improved optimal problem with feedback control is put forward. The conclusion will be given in the last section.


## 2. Notations and Assumptions

For convenience, we use the similar assumptions and notation used in [17] which are as follows:


### 2.1. Notations

i: the index of products i=1,2 x i ( 0 ): economic order quantity which enters the inventory at time point n=0, xi0=xi(0) x i ( n ): inventory level at discrete time point n N: number of discrete time periods (inventory cycle length) k i: rate of supply to xi(n) which is inversely proportional to xi(n) and may be ki=ai/xi(n),  ai are scale parameter d i x i n: linear demand rate at the instantaneous level of inventory xi(n) at discrete time point n,di are scale parameter a i i: deterioration coefficient due to self-contact of xi(n) (self-deterioration coefficient) a i j ( i ≠ j ): deterioration coefficient of xin due to presence of unit of xj(n) at discrete time point n (cross-deterioration coefficient) K: order cost for one order c i k i: variable cost to maintain supply to xin at the rate kixin;ci are scale parameter p i: price per unit of xin θ i: the perishable rate of product xi(n)


### 2.2. Assumptions

( 1 ) A product may have a lifetime which ends until its utility reaches zero.( 2 ) Shortages are not allowed.( 3 ) The percentage of profit does not depend on the initial inventory.( 4 ) The rate of changing of holding cost is proportional to the square of xi; it is given by qixi2, qi are constants, i=1,2.


## 2.1. Notations

i: the index of products i=1,2 x i ( 0 ): economic order quantity which enters the inventory at time point n=0, xi0=xi(0) x i ( n ): inventory level at discrete time point n N: number of discrete time periods (inventory cycle length) k i: rate of supply to xi(n) which is inversely proportional to xi(n) and may be ki=ai/xi(n),  ai are scale parameter d i x i n: linear demand rate at the instantaneous level of inventory xi(n) at discrete time point n,di are scale parameter a i i: deterioration coefficient due to self-contact of xi(n) (self-deterioration coefficient) a i j ( i ≠ j ): deterioration coefficient of xin due to presence of unit of xj(n) at discrete time point n (cross-deterioration coefficient) K: order cost for one order c i k i: variable cost to maintain supply to xin at the rate kixin;ci are scale parameter p i: price per unit of xin θ i: the perishable rate of product xi(n)


## 2.2. Assumptions

( 1 ) A product may have a lifetime which ends until its utility reaches zero.( 2 ) Shortages are not allowed.( 3 ) The percentage of profit does not depend on the initial inventory.( 4 ) The rate of changing of holding cost is proportional to the square of xi; it is given by qixi2, qi are constants, i=1,2.


## 3. Model and Stability Analysis

In this section, we will show the discrete model which can express the multi-item deterioration inventory. It can also be seen as the discrete version of the model in [17], which can be explained by some principles occurring in population dynamics. Furthermore, stability conditions will be discussed for our discrete two item inventory model without regular supply and with regular supply.Case 1 (without regular supply).
We give the following two item discrete inventory model(1)x1n+1=x1n1-d1-θ1-a11x1n-a12x2nx2n+1=x2n1-d2-θ2-a21x1n-a22x2nwhere d1,d2,θ1,θ2,a11,a12,a21,a22 are nonnegative, and the above system has four fixed points (0,0), (0,-d2+θ2/a22), (-d1+θ1/a11,0) and (-d1+θ1a22-d2+θ2a12/a11a22-a12a21, -d1+θ1a21-d2+θ2a11/a11a22-a12a21), (0,0) is the only nonnegative one. The linearization of (1) about (0,0) has the Jacobian matrix(2)J=1-d1-θ1001-d2-θ2,which has two eigenvalues(3)λi=di+θi-1,i=1,2.
ifλi<1 or di+θi<2, then steady state (0,0) is locally asymptotically stable; this means that the inventory is empty at last. Then we discuss the case with supply.Case 2 (with regular supply).
If the regular supplykixi is taken into account, the following model can be given(4)x1n+1=x1n1-d1-θ1-a11x1n-a12x2n+k1x1nx2n+1=x2n1-d2-θ2-a21x1n-a22x2n+k2x2n
Clearly, system (4) has four possible steady states, i.e., E0=0,0, exclusion points E1=0,k2-(d2+θ2)/a22, E2=k1-(d1+θ1)/a11,0, and nontrivial coexistence point E3=x1∗,x2∗, where(5)x1∗=k1-d1+θ1a22-k2-d2+θ2a12a11a22-a12a21x2∗=k2-d2+θ2a11-k1-d1+θ1a21a11a22-a12a21and ki-di+θi>0,i=1,2,a11a22>a21a12,a11/a22>k1-d1-θ1/k2-d2-θ2>a12/a22 or a11a22<a21a12,a11/a22<k1-d1-θ1/k2-d2-θ2<a12/a22.
At the fixed pointE0, the Jacobian matrix has the form(6)JE1=k1+1-d1-θ100k2+1-d2-θ2,and the corresponding eigenvalues of (6) are(7)λ1=d1+θ1-1-k1,λ2=d2+θ2-1-k2.
Ifλi<1 or 0<di+θi-ki<2,i=1,2, then steady state E0 is locally asymptotically stable. That is to say, if rate of supply exceeds the sum of the natural deterioration rate and linear demand rate at the instantaneous level of inventory, the inventory will not be empty.
The linearization of (4) about E1 has the Jacobian matrix and the corresponding eigenvalues are(8)λ1=1-d1-θ1+k1-a12k2-d2+θ2a22λ2=1-d2-θ2+k2-a22k2-d2+θ2a22.
Ifλi<1 or d2+θ2<k2<d2+θ2+1 and d1+θ1+a12/a22(k2-d2-θ2)-2<k2<d1+θ1+a12/a22(k2-d2-θ2), then steady state E1 is locally asymptotically stable. From this, we can conclude that if the above conditions are met, the number of x1 will go to zero, and the number of x2 will go to k2-(d2+θ2)/a22.
At the fixed pointE2, the corresponding stable conditions can be given as follows:(9)d1+θ1<k1<d1+θ1+1d12+θ2+a21a11k1-d1-θ1-2<k2<d12+θ2+a21a11k1-d1-θ1
Similarly, the number ofx2 will go to zero, and the number of x1 will go to k1-(d1+θ1)/a11.For the fixed pointE3, the Jacobian matrix has the form(10)JE3=1-r1a11a22-r2a11a12a11a22-a12a21-r1a12a22-r2a122a11a22-a12a21-r2a11a21-r1a212a11a22-a12a211-r2a11a22-r1a21a22a11a22-a12a21.where r1=1-d1-θ1+k1,r2=1-d2-θ2+k2.The characteristic equation ofJE3 is given by(11)λ2-trJλ+det⁡J=0,where(12)trJ=2-r1+r2a11a22-r2a11a12-r1a21a22a11a22-a12a21det⁡J=1+r1r2-r1+r2a11a22+r22-r2a11a12+r12-r1a21a22a11a22-a12a21.According to the Jury’s criteria [20], E3 is asymptotically stable if the following conditions hold:(13)F-1=1-trJ∗-1+det⁡J>0F1=1-trJ∗1+det⁡J>01-det⁡J>0,then we have the following conclusion that if r1r2+2G>(r1+r2)a11a22+(r22-r2)a11a12+(r12-r1)a21a22>r1r2G,G=a11a22-a12a21 holds, the fixed point E3 is asymptotically stable. We can say that x1 and x2 can coexist.


## 4. Necessary Conditions for Dynamic Optimality

From Case1 in Section 2, it is a fact that the inventory will be empty at last if di+θi<2. On the other hand, it is unreasonable if di+θi is too large. The adjustment for the system is necessary which will result in Case 2. From the control theory, the supply action may be regarded as a feedback control one. In other words, one may wish to move the undesired trajectory towards the desired orbit. Here we will alter the positions of positive equilibrium, but to obtain its stability. This motivates us to adjust the model form as follows.(14)x1n+1=x1n1-d1-θ1-a11x1n-a12x2n+k1x1n=x1n1-d1-θ1-a11x1n-a12x2n+b1u1nx2n+1=x2n1-d2-θ2-a21x1n-a22x2n+k2x2n=x2n1-d2-θ2-a21x1n-a22x2n+b2u2nwhere k1=b1u1(n) and k2=b2u2(n), u1 and u2 are control variables.Under the condition the inventory will not be empty; we will calculate the net profit of the inventory. The objective function for the inventory problem should first be formed. Note that the percentage of profit does not depend on the initial inventory and the nonnegativity constraints hold for inventory variables, and percentage of profit maybe is found to be more for a lower initial stock than that for a higher initial stock [17]. Naturally we can formulate the optimization problem to maximize the net profit (performing index) G taking the form(15)G=-K+∑i=12∑n=0N-1pidixin-piaii+qixi2n+∑i=12∑n=0N-1pi-cibiuinxinxin-di-θi-aiixin-aijxjn+biuins . t (16) x 1 n + 1 = x 1 n 1 - d 1 - θ 1 - a 11 x 1 n - a 12 x 2 n + b 1 u 1 n x 2 n + 1 = x 2 n 1 - d 2 - θ 2 - a 21 x 1 n - a 22 x 2 n + b 2 u 2 n .and(17)r1r2+2G>r1+r2a11a22+r22-r2a11a12+r12-r1a21a22>r1r2G,where G=a11a22-a12a21,r1=1-d1-θ1+b1u1,r2=1-d2-θ2+b2u2.For simplicity, we noteS={(d1,d2,θ1,θ2,b1,b2,a11,a12,a21,a22)|r1r2+2G>(r1+r2)a11a22+(r22-r2)a11a12+(r12-r1)a21a22>r1r2G}.We can easily obtain the next results.Theorem 1.
There exists an optimal controlu∗=(u1∗,u2∗) with corresponding states from (14) such that G(u∗)=min⁡G(u).Now we consider the Hamiltonian(18)Hn=-KN+∑i=12pidixin-piaii+qixin2-∑i=12pi-cibiuinxinxin1-di-θi-aiixin-aijxjn+biuin+λ1n+1x1n1-d1+θ1-a11x1n-a12x2n+b1u1n+λ2n+1x2n1-d1+θ1-a21x1n-a22x2n+b2u2n,λi are the costate variables, then, from discrete Pontryagin’s Maximum Principle [19], we can derive the necessary conditions in the next theorem.Theorem 2.
Given an optimal controlu∗=(u1∗,u2∗) and corresponding states x1∗ and x2∗ from (14), there exist adjoint functions λ1 and λ2 satisfying(19)λ1n=p1d1-2p1a11+q1x1n+p1x1-c1b1u1a11-p1x1n+1+λ1n+11-d1+θ1-2a11x1n-a12x2nλ2n=p2d2-2p2a22+q2x2n+p2x2-c2b2u2a22-p2x2n+1+λ2n+11-d2+θ2-a21x1n-2a22x2nλ1N=λ2N=0,and(20)λin+1=pixin-cibiuin-cixin+1xinatui∗n.where n=0,1,2,…,N-1.Proof.
Using the discrete version of Pontryagin’s Maximum Principle(21)λ1n=∂Hn∂x1n=p1d1-2p1a11+q1x1n+p1x1-c1b1u1a11-p1x1n+1+λ1n+11-d1+θ1-2a11x1n-a12x2nλ2n=∂Hn∂x2n=p2d2-2p2a22+q2x2n+p2x2-c2b2u2a22-p2x2n+1+λ2n+11-d2+θ2-a21x1n-2a22x2n,
Additionally, from the adjoint final conditions, we can get(22)λ1N=λ2N=0.Furthermore,(23)∂Hn∂ui=ci1-di-θi-aiixin-aijxjn+biuin-pixin-cibiui+λin+1xi=cixin+1-pixin-cibiuin+λin+1xin=0,atui∗nxi≠0,from which we can get λi(n+1)=pixi(n)-cibiui(n)-cixi(n+1)/xi(n).


## 5. Simulation and Discussion

In this section, we will show a simple example in order to intuitively illustrate how to find the optimal control firstly. We fix some of the constants in the parameter regionS, and let d1=0.4,d2=0.3,p1=1,p2=2,θ1=0.0002,θ2=0.0003,b1=1,b2=1,a11=0.001,a12=0.02,a21=0.03,a22=0.002. The initial values will be set to x10=10,x20=15, and let N=10, by means of forward-backward sweep, the optimal solutions and optimal control can be displayed in Figures 1 and 2.Figure 1
Optimal control and state forx1 and u1 with 10 time steps.Figure 2
Optimal control and state forx2 and u2 with 10 time steps.According to the theoretical analysis in Sections3 and 4, stability analysis at the steady state of the discrete two item inventory model for deteriorating items with regular supply is done to ensure the warehouse is not empty and keep at an equilibrium level. With the view of control theory, the supply action can also be regarded as a feedback control. And we have altered the positions of positive equilibrium to obtain its stability. In fact, we can rewrite the discrete two item inventory model, in which the control variables satisfy a certain difference equation [21].(24)x1n+1=x1n1-d1-θ1-a11x1n-a12x2n+b1u1nx2n+1=x2n1-d2-θ2-a21x1n-a22x2n+b2u2nu1n+1=1-e1u1n+d1x1nu2n+1=1-e2u2n+d2x2n.where the parameters d1,d2,e1, and e2 are positive constants.Similarly, an objective function is formed to calculate the net profit with respect to all possible profits and all possible loss for the system with the above state equation. We can obtain an optimal control problem as follows:(25)max⁡Gx,us . t (26) x 1 n + 1 = x 1 n 1 - d 1 - θ 1 - a 11 x 1 n - a 12 x 2 n + b 1 u 1 n x 2 n + 1 = x 2 n 1 - d 2 - θ 2 - a 21 x 1 n - a 22 x 2 n + b 2 u 2 n u 1 n + 1 = 1 - e 1 u 1 n + d 1 x 1 n u 2 n + 1 = 1 - e 2 u 2 n + d 2 x 2 n .Corresponding to system (15)-(16), applying the discrete version of Pontryagin’s Maximum Principle, a necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the difference equations (24) may be different. Perhaps the problem is worth discussing.


## 6. Conclusion

In this paper, we first get a new discrete two item inventory model for deteriorating items with a linear stock dependent demand rate without regular supply, but steady state(0,0) is locally asymptotically stable which means that the inventory is empty at last, so regular supply is the fact. To maximize net profits, an objective function is formed with the difference equation constraints with feedback control. Then there is a necessary condition for the steady state optimal control problem by the discrete Pontryagin’s Maximum Principle; the numerical can also verify the validity of theoretical analysis. At last, a revised optimal control problem is put forward.



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