OPTIMAL CONTROL FOR PRODUCTION INVENTORY SYSTEM WITH VARIOUS COST CRITERION SUBRATA GOLUI CHANDAN PAL MANIKANDAN R. AND ABHAY SOBHANAN

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OPTIMAL CONTROL FOR PRODUCTION INVENTORY SYSTEM

WITH VARIOUS COST CRITERION

SUBRATA GOLUI, CHANDAN PAL, MANIKANDAN, R., AND ABHAY SOBHANAN

Abstract. In this article, we investigate a dynamic control problem of a production-

inventory system. Here, demands arrive at the production unit according to a Poisson

process and are processed in an FCFS manner. The processing time of the customer’s

demand is exponentially distributed. The production manufacturers produce the items

on a make-to-order basis to meet customer demands. The production is run until the

inventory level becomes suﬃciently large. We assume that the production time of an

item follows exponential distribution and the amount of time for the produced item to

reach the retail shop is negligible. Also, we assume that no new customer joins the queue

when there is a void inventory. This yields an explicit product-form solution for the

steady-state probability vector of the system. The optimal policy that minimizes the

discounted/average/pathwise average total cost per production is derived using a Markov

decision process approach. We ﬁnd optimal policy using value/policy iteration algorithms.

Numerical examples are discussed to verify the proposed algorithms.

Key Words: Production inventory system, controlled Markov chain, cost criterion, value

iteration algorithm, policy iteration algorithm.

Mathematics Subject Classiﬁcation: Primary 93E20, Secondary 49L20, 60J27.

1. Introduction

Inventory theory has useful applications in various day-to-day real-life scenarios. One

such application is production control, in which decision-makers focus on controlling costs

while satisfying customer demands and maintaining their goodwill. Over the last decade

research on complex integrated production-inventory systems or service-inventory systems

has found much attention, often in connection with the research on integrated supply chain

management, see He et al. (2002); He and Jewkes (2000); Helmes et al. (2015); Kr-

ishnamoorthy et al. (2015); Krishnamoorthy and Narayanan (2013); Malini and Shajin

(2020); Pal et al. (2012); Sarkar (2012); Veatch and Wein (1994). In these articles the

authors considered (s, S)/(s, Q)-type policy to study their inventory models.

Sigman and Simchi-Levi (1992) and Melikov and Molchanov (1992) introduced the

integrated queueing-inventory models. Whereas the article by Sigman and Simchi-Levi

arXiv:2210.15251v1 [math.OC] 27 Oct 2022

2 SUBRATA GOLUI, CHANDAN PAL, MANIKANDAN, R., AND ABHAY SOBHANAN

(1992), considered the Poisson arrival of demands, arbitrarily distributed service time, and

exponentially distributed replenishment lead time. Also, they showed that the resulting

queueing-inventory system is stable if and only if the service rate is higher than the customer

arrival rate. The authors considered that the customers may join the system even when

the inventory level is zero and discussed the case of non-exponential lead-time distribution.

Berman et al. (1993) followed them with deterministic service times and formulated the

model as a dynamic programming problem. For more inventory models with positive service

times, see Berman and Kim (1999), (2004); Arivarignan et al. (2002); Krishnamoorthy et

al. (2006a), (2006b); for a recent extensive survey of literature, we refer in Krishnamoorthy

et al. (2021), it provides the summary of work done until 2019.

We recall the remarkable work by Schwarz et al. (2006). They propose product form

solutions for the system state distribution under the assumption that customers do not join

when the inventory level is zero, where the service/lead time is exponentially distributed

and demands follow a Poisson distribution. Krishnamoorthy and Narayanan (2013) reduced

the Schwarz et al. (2006) model to a production inventory system with single-batch bulk

production of the quantum of inventory required. The production inventory with service

time and protection for a few of the ﬁnal phases of production and service is discussed in

Sajeev (2012).

Saﬀari et al. (2011) considered an M/M/1 queue with inventoried items for service,

where the control policy followed is (s, Q) and the lead time is a mixed exponential distribu-

tion. They assumed that when inventory stock is empty, fresh arrivals are lost to the system,

and thus, they obtain a product form solution for the system state probability. Inventory

system with queueing networks was studied by Schwarz et al. (2007). The authors assumed

that at each service station, an order for replenishment is made when the inventory level at

that station drops to its reorder level; hence, no customer is lost to the system. Zhao and

Lian (2011) used dynamic programming to obtain the necessary and suﬃcient conditions

for a priority queueing inventory system to be stable.

In all the papers quoted above, customers are provided with an item from the inven-

tory after their completion of service. In Krishnamoorthy et al. (2015), customers may not

get an inventory after their completion of service. They studied the optimization problem

and obtained the optimal pairs (s, S) and (s, Q) corresponding to the expected minimum

costs.

OPTIMAL CONTROL FOR PRODUCTION INVENTORY SYSTEM WITH VARIOUS COST CRITERION 3

In this study, we do not use any common inventory control policies such as (s, S)/(s, Q)-

type. We consider the problem of ﬁnding the optimal production rates for a discounted/long-

run average/pathwise average cost criterion of the production inventory system. Here, we

consider an M/M/1/∞production inventory system with positive service time. Customers’

demands arrive one at a time according to a Poisson processes. Service and production times

follow an exponential distribution. Each production is 1 units, and the production process

is run until the inventory level becomes suﬃciently large (inﬁnity). It is assumed that the

amount of time for the item produced to reach the retail shop is negligible. We assume

that no customer joins a queue when the inventory level is zero. This assumption leads

to an explicit product-form solution for the steady-state probability vector using a sim-

ple approach. In this paper, we have applied matrix analytic methods for ﬁnding system

steady-state equations. Readers are referred to Neuts (1989), (1994), Chakravarthy and

Alfa (1986) and Chakravarthy (2022a and 2022b).

In this paper, we ﬁnd an optimal stationary policy by policy/value iteration algo-

rithm. We see that there are many studies on inventory production control theory on

continuous-time controlled Markov decision processes (CTCMDPs) for discounted/ aver-

age/ pathwise average cost criteria (see Federgruen and Zipkin (1986), (1986); Helmes et

al. (2015)). However, the articles discussed on algorithms for ﬁnding an optimal stationary

policy are, Federgruen and Zhang (1992); He et al. (2002); He and Jewkes (2000). The

ﬁxed costs of ordering items or setting up a production process arise in many real-life sce-

narios. In their presence, the most widely used ordering policy in the stochastic inventory

literature is the (s, S) policy. In this context, we mention two important survey papers for

discrete/continuous-time regarding (s, S) replenishment policy: Perera and Sethi (2022a,

2022b). They comprehensively surveyed the vast literature accumulated over seven decades

in these two papers for the discounted/average cost criterion on discrete/continuous-time.

The motivation for studying discounted problems comes mainly from economics. For

instance, if δdenotes a rate of discount, then (1 + δ)Lwould be the amount of money one

would have to pay to obtain a loan of Ldollars over a single period. Similarly, the value of

a note promising to pay Ldollars ttime steps into the future would have a present value of

(1+δ)t=αtL, where α:= (1 + δ)−1denotes the discount factor. This is the case for ﬁnite-

horizon problems. But in some cases, for instance, processes of capital accumulation for an

economy, or some problems on inventory or portfolio management, do not necessarily have

a natural stopping time in the deﬁnable future, see Hern´andez-Lerma and Lasserre (1996);

Puterman (1994). Now when decisions are made frequently, so that the discount rate is

very close to 1, or when performance criterion cannot easily be described in economic terms,

4 SUBRATA GOLUI, CHANDAN PAL, MANIKANDAN, R., AND ABHAY SOBHANAN

the decision maker may prefer to compare policies on the basis of their average expected

reward instead of their expected total discounted reward, see Piunovsky and Zhang (2020).

The ergodic problem for controlled Markov processes refers to the problem of mini-

mizing the time-average cost over an inﬁnite time horizon. Hence, the cost over any ﬁnite

initial time segment does not aﬀect ergodic cost. This makes the analytical analysis of

ergodic problems more diﬃcult. However, the sample-path cost r(·,·,·), deﬁned by (3.19),

corresponding to an average-cost optimal policy that minimizes the expected average cost

may ﬂuctuate from its expected value. To take these ﬂuctuations into account, we next

consider the pathwise average-reward (PAC) criterion. In this study, we investigate the

production inventory control problem for the discounted/average/pathwise average cost

criterion. We ﬁnd the optimal production rate through a value/policy iteration algorithm.

However, there may be some issues in obtaining an optimal policy. Hence, in this study,

we also examine an ε-optimal policy. Finally, numerical examples are included to verify the

proposed algorithms.

The remainder of this paper is organized as follows. First, we deﬁne the production

control problem in section 2. In Section 3, we discuss the steady-state analysis of this model

and describe the evaluation of the control system. In addition, we deﬁne our cost criterion

and assumptions required to obtain an optimal policy. Section 4 discusses the discount

cost criterion. Here, we ﬁnd a solution for the optimality equation corresponding to the

discounted cost criterion, and provide its value/policy iteration algorithms. In the next

section, we deal with the optimality equation and policy iteration algorithm corresponding

to the average cost criterion. We perform the same analysis in Section 6 for the pathwise

average cost criterion, as in Section 5. Finally, in Section 7, we provide concluding remarks

and highlight the directions for future research.

Notations:

N(t): number of customers in the system at time t.

I(t): inventory level in the system at time t.

e: (1,1,· · · ,1,· · · ) a column vector of 10sof appropriate order.

(LI)QBD: (Level independent) Quasi birth and death process.

N0=N∪ {0}, where Nis set of all natural numbers.

Cb(N0×N0) is the collection of all bounded functions on N0×N0.

OPTIMAL CONTROL FOR PRODUCTION INVENTORY SYSTEM WITH VARIOUS COST CRITERION 5

2. Problem Description

2.1. Production inventory model. We consider an M/M/1/∞production inventory

system with positive service time. Demands by customers for the item occur according

to a Poisson process of rate λ. Processing of the customer request requires a random

amount of time, which is exponentially distributed with parameter µ. Each production is

of 1 unit and the production process is keep run until inventory level becomes suﬃciently

large (inﬁnity). To produce an item it takes an amount of time which is exponentially

distributed with parameter β. We assume that no customer is allowed to join the queue

when the inventory level is zero; such demands are considered as lost. It is assumed that

the amount of time for the item produced to reach the retail shop is negligible. Thus the

system is a continuous-time Markov chain (CTMC) {X (t); t≥0}={(N(t),I(t)) ; t≥0}

with state space Ω=

∞

n=0

L(n),where L(n) is called the nth level of the CTMC, is given by,

{(n, i); i∈N0}.

Now the transition rates in the CTMC are:

•(n, i)→(n+ 1, i) : rate is λ,n∈N0,i∈N

•(n, i)→(n−1, i −1) : rate is µ,n, i ∈N

•(n, i)→(n, i + 1) : rate is β,n, i ∈N0.

•All other transition rates are zero.

Write,

P{N(t) = n, I(t) = i}=Pn,i(t).

These satisﬁes the system of diﬀerence-diﬀerential equations:

n,0(t) = −βPn,0(t) + µPn+1,1(t), n ∈N0(2.1)

n,i(t) = −(λ+β+µ)Pn,i(t) + µPn+1,i+1(t) + λPn−1,i−1(t) + βPn,i+1, n, i ∈N.(2.2)

The steady-state time derivative is equated to zero under the condition for its existence is

λ<µ, which will be proved in the subsequent Lemma 3.1.

Write,

lim

t→∞ Pn,i(t) = Pn,i, n, i ∈N.

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OPTIMALCONTROLFORPRODUCTIONINVENTORYSYSTEMWITHVARIOUSCOSTCRITERIONSUBRATAGOLUI,CHANDANPAL,MANIKANDAN,R.,ANDABHAYSOBHANANAbstract.Inthisarticle,weinvestigateadynamiccontrolproblemofaproduction-inventorysystem.Here,demandsarriveattheproductionunitaccordingtoaPoissonprocessandareprocessedinanFCFSmanner...

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OPTIMAL CONTROL FOR PRODUCTION INVENTORY SYSTEM WITH VARIOUS COST CRITERION SUBRATA GOLUI CHANDAN PAL MANIKANDAN R. AND ABHAY SOBHANAN

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