The critical points of production rate under vendor–buyer coordination: explicit solutions

• Author: Israel David,Avi Herbon
• Date: 01 May 2021
• Published date: 01 May 2021
• DOI: http://doi.org/10.1111/itor.12840

Intl. Trans. in Op. Res. 28 (2021) 1330–1346

DOI: 10.1111/itor.12840

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The critical points of production rate under vendor–buyer

coordination: explicit solutions

Israel David and Avi Herbon∗

Department of Management, Bar-Ilan University,R amat-Gan 52900, Israel

E-mail: idavid@bgu.ac.il [David]; avher@bezeqint.net [Herbon]

Received 5 June2019; received in revised form 18 January 2020; accepted 16 May 2020

Abstract

This paper revisits the deterministic joint vendor–buyer production-inventory problem and assumes that

the production rate can be controlled. To guard against unwarranted extreme solutions, such as too abrupt

production or, on the other hand, everlasting production aligned with demand, we place lower and upper

bounds on the production rate. In addition, we incorporate an independent upper bound on the overall cycle

of producing and remaining idle. Through identifying several critical points of the production rate, we solve

the resulting problem for the optimal triplet (P,Q,n), where Pis the constant production rate,a key decision

variable, Qis the entire lot size produced in a cycle, and nis the number of equal successive shipments of

this lot to the buyer. Our owntreatment is purely analytical, which adds value from a theoretical perspective.

Worst values of the production rate, when the other decision values are optimal for P, are found. We prove

that only two production rates, the lowest and the highest, can yield minimal joint costs, and we identify

which of the two is optimal under given relative positions of the defined critical points. Numerical illustration

indicates that the joint cost sharply increases for small values of the bound on the overall cycle length. This

study highlights the importance of solving variants of the suggested model and of developing managerial

alternatives that relax this constraint as much as possible.

Keywords:vendor–buyer coordination; optimization; production and ordering policy; lot sizing

1. Introduction and background literature

Variable production rates, which serve as a key decision variable in our model, may prevent the

rapid accumulation of inventories and the costs and extra handling burden associated with them

(Kimemia and Gershwin, 1983; Akella and Kumar, 1986; Liberopoulos and Caramanis, 1994;

Schweitzer and Seidmann, 1991). Researchers such as Khouja and Mehrez (1994), Eiamkan-

chanalai and Banerjee (1999), Giri et al. (2005), and Sana (2010) have studied the cost–benefit

of variable production speed within the simple prototypical economic production quantity (EPQ)

∗Corresponding author.

© 2020 The Authors.

International Transactionsin Operational Research © 2020 International Federation of OperationalResearch Societies

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I. David and A. Herbon / Intl. Trans. in Op. Res. 28 (2021) 1330–1346 1331

model (the economic order quantity [EOQ] with production). Thus, the scenario they investigate is

that of a single item, which is produced continuously during a certain production-runlength (PRL).

The manufacturer needs to optimize the batch size (and hence the PRL) given the parameters of the

basic EOQ model plus a decision variable P—the productionrate. Moon and Cha (2005) developed

an inventory model under which, at the time of entering into a contract with the manufacturer, the

retailer can negotiate the lead time by considering the production rate of the manufacturer who

usually has the option of increasing his regular production rate up to the maximum (designed) pro-

duction capacity. Herbon and Khmelnitsky (2009) developed a model thatseeks an optimal control

function (i.e., transient production rate), which minimizes a performance measure along the plan-

ning horizon. Optimal control modeling and process solving, which consider aspects of decision

making with a limited forecast, are exemplified by a single-part type production system. It is only

one step forward from this literature on the basic EPQ with variable Pto incorporate such a vari-

able Pinto the celebrated paradigm of joint vendor–buyer (integrated) inventory production. The

possibility of Pbeing a decision variable within the joint paradigm potentially reduces overall costs,

controls inventory accumulation, and allows for the planning and utilization of idle times.

However, as is rightfully stressed in Jauhari and Pujawan (2014), and in Jauhari et al. (2016), not

enough attention has been paid in the joint vendor–buyer (integrated) inventory-production litera-

ture to variable production rates, and most authors have assumed that Pis a constant parameter

rather than a decision variable (see Kelle et al., 2003, 2009). In the joint vendor–buyer paradigm,

the buyer is normally perceived as the stronger party of the two, frequently a just-in-time (JIT)

vendor in its own right. The “vendor” is perceived as a JIT supplier to the “buyer.” We will use

the terms “vendor,” “supplier,” and “manufacturer” synonymously throughout this short paper. A

review of this literature first appeared in Goyal and Gupta (1989), and no less than three additional

reviews have followed it to date (Sarmah et al., 2006; Ben-Daya et al., 2008; Glock, 2012).

This study revisits the deterministic joint vendor–buyer inventory-production problem. The goal

is to determine the optimal triplet (P, Q , n ), where Qis the lot size produced in a cycle and nis

the number of equal successive shipments of this lot to the buyer, given the problem parameters.

The objective is to minimize the joint production, inventory carrying, and shipment costs (i.e., of

the buyer and vendor combined) to reflect the partnership between the two parties. Our research is

especially important for companies facing high in-process inventories, since varying the production

rate gives production planners flexibilityin smoothing material flows and in avoiding the accumula-

tion of inventory at bottleneck stations(Glock, 2011). Production systems such as the one described

in this paper can be found in a variety of application areas, for example, in the automotive indus-

try, electronics industry, and wood furniture industry. David and Eben-Chaime (2003) highlight the

degree of independence between the parties under the joint vendor–buyer (integrated) inventory-

production paradigm; namely, the manufacturer can independently select the production lot size Q

(as long as P>D, the demand rate), and once the delivery quantity is agreed upon, the (approxi-

mate) ratio of these two quantities is the desired shipment frequency. In contrast to these variables,

the demand-to-production rate ratio turns out to be very significant in the coordination as a whole.

It has been well established in the literature that adopting joint vendor–buyer (integrated) inventory-

production results in considerable savings in comparison to dictating the shipment frequency and

lot size unilaterally.

In the studies that do allow Pto be variable, including the present one, such advantages

have been clearly shown. Zavanella and Zanoni (2009) analyzed the advantage of a certain

© 2020 The Authors.

International Transactionsin Operational Research © 2020 International Federation of OperationalResearch Societies