Optimal manufacturer's cost sharing ratio, shipping policy and production rate: A two-echelon supply chain

Herbon, Avi; David, Israel Article Optimal manufacturer's cost sharing ratio, shipping policy and production rate: A two-echelon supply chain Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Herbon, Avi; David, Israel (2023) : Optimal manufacturer's cost sharing ratio, shipping policy and production rate: A two-echelon supply chain, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 10, pp. 1-14, https://doi.org/10.1016/j.orp.2022.100264 This Version is available at: https://hdl.handle.net/10419/325751 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Optimal manufacturer’s cost sharing ratio, shipping policy and production rate – A two-echelon supply chain Avi Herbon * , Israel David Department of Management, Bar-Ilan University, Ramat-Gan 52900, Israel ARTICLE INFO Keywords: Integrated inventory model Shipping policy Sharing shipment cost Partial coordination ABSTRACT We analyze an integrated inventory supply chain and seek the optimal production lot, optimal production rate, and optimal (integer) number of shipments per production lot. An increasing need for higher operational efficiency, as well as growing competition among multiple products for a limited storage capacity, is driving retailers to require more frequent shipping. This imposes pressure on suppliers to share the shipping cost with retailers. The sharing ratio of the shipment cost has not previously been considered within the context of an integrated supply chain. Therefore, we contribute to the literature by investigating this entirely new parameter, assuming that the shipment cost is shared between a manufacturer and a retailer. We also consider a distributed supply chain in which each party optimizes its own cost. We analyze the problem of finding the optimal sharing ratio of the shipment cost for such a supply chain and show that there exists a specific choice of shipment cost-sharing ratio (set by the manufacturer) that results in total costs similar to those obtained in the integrated inventory model. We develop deterministic models that provides basic insights into the investigated problem. Through mathematical analysis of a nested-designs model, we provide intermediate results (which are of interest in their own right) as well as optimal analytical solutions. We show, through numerical examples, that in the scenario where each party optimizes its own cost, the manufacturer’s shipment cost is a central control variable in the sense that it affects the costs of both parties. 1. Introduction 1.1. Motivation and research objective As industrial environments have become increasingly competitive, effective supply-chain management has become essential [1]. Coordination between supply-chain members is often suggested as a means of facing some of the challenges that arise within the operational management of supplier-retailer chains. The partnership between suppliers and retailers along a supply chain has a substantial impact on supply-chain success. If retailers’ expectations are not met on a day-to-day basis, this could result in unsatisfied consumers, which could affect the profits of not only the retailers but also the suppliers. The ability to reduce cycle times can constitute a powerful competitive advantage, and manufacturers invest considerable efforts in pursuing this goal. In many industries, an increasing need for higher operational efficiency, as well as growing competition among multiple products for a limited storage capacity, are factors that drive retailers to seek more frequent shipping. Accordingly, a quick response to customer demand is one of the prerequisites for manufacturers to sustain their global ‘competitive advantage’ [2]. Most manufacturers sell to resellers and distributors, who in turn sell to consumers. The ability to manufacture a product on a tighter schedule enhances the manufacturer’s perceived reliability and improves relationships with potential distributors, thus increasing the manufacturer’s market share. Table 1 in Taifa and Vhora [3] presents an overview of the various methods deployed for cycle time reduction (e.g., efficient maintenance, process step reduction, and fine tuning). Many real-life industries (e.g., the manufacture of semi-conductors in Taiwan, the automobile industry in India, and the production of agricultural and construction machinery in Korea) have utilized techniques for cycle time reduction (see Table 2 in Ref. [3]). However, once shorter cycles are achieved, the manufacturer is confronted with the operational problem of how to adapt to the new situation. Our model suggests an optimal solution to this problem. In particular, it determines the optimal production lot, optimal production rate, and optimal (integer) number of shipments per production lot. When attempting to improve supply-chain management through * Corresponding author. E-mail address: [email protected] (A. Herbon). Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp https://doi.org/10.1016/j.orp.2022.100264 Received 11 July 2022; Received in revised form 24 December 2022; Accepted 25 December 2022 Operations Research Perspectives 10 (2023) 100264 2 greater coordination between members, it needs to be borne in mind that the perspective of each party is intrinsically different and that optimizing each member’s own performance does not necessarily achieve optimal performance of the entire supply chain. For example, it might by more costly to hold a unit of a product for a unit of time on the retailer’s shelf than at the supplier’s warehouse [4]. The most common assumption in the existing literature (as well as in our paper) is that the manufacturer-buyer chain operates under full cooperation. An example of supply-chain cooperation, and the focus of the current paper, is shared shipment costs between the manufacturer and the buyer. Unfortunately, this is not the case in many real-life supply chains, for numerous reasons. These include: unwillingness of one of the parties to accept external intervention regarding organizational decisions; unwillingness of the manufacturer to share the cost of shipping; unwillingness of the retailer to accept a different shipping policy from that originally planned; and inability to reach an agreement contract that adequately divides the total profit. Within the scope of an integrated supply chain, the ratio between the supplier’s and the retailer’s share of the shipment cost has no effect on supply-chain performance. However, in a non-cooperative supply chain, lack of willingness on the part of the supplier to share the shipment cost results in a situation where the retailer incurs higher costs. This could decrease the retailer’s incentive to work with such suppliers. Thus, we contribute to the literature by investigating an important parameter that has not previously been studied: the ratio between the share of the shipment cost paid by the manufacturer and that paid by the retailer. None of the aforementioned studies, nor, the vast body of literature on supply-chain coordination has not considered the case where the production rate acts as a decision variable, the optimal value of which has the potential to make a positive contribution to the total performance. This paper analyzes a two-echelon supply chain consisting of a manufacturer who converts raw materials into a final single product, stores the product at his warehouse, and then ships it to a retailer who sells the item to a market with a known fixed demand rate. The manufacturer incurs costs when capacity is limited by slow production, especially when the production-run absorbs almost the entire cycle length. This may result in costs due to, for example, reduced time for preventive maintenance, workers being required to work overtime, and longer usage of manufacturing tools. A production rate that is too high, on the other hand, also incurs costs, due to factors such as faster accumulation of inventory (resulting in greater holding costs) and the need to purchase more expensive equipment. 1.2. The current study and its contribution In today’s hypercompetitive and highly complex business environment, companies are constantly searching for ways of gaining a competitive advantage by improving the speed, efficiency, and quality of the goods and services they deliver to customers through their supply chains. The key to sustained success in supply-chain management is the ability to make optimized business decisions by finding the best possible solution to the company’s planning and scheduling problems. Optimization algorithms that generate optimal solutions outperform their heuristic counterparts and enable businesses to maximize both their profit and their operational efficiency. Explicit solutions provide managers with a deeper understanding and suggest practical insights. Such solutions involve determining the best way of synchronizing supply and demand across the supply-chain network, so as to boost customer satisfaction and enhance bottom-line results. The current research involves mathematical analysis of a nesteddesigns model. The analysis provides intermediate results that are of interest in their own right as well as explicit solutions. More specifically, our research contributes to the existing literature as follows: •Most previous studies on the subject of integrated supply chains have analyzed the behavior of the proposed model (or extensions of the model) in an extensive numerical experiment implemented in a commercial solver. In contrast to these studies, the solution approach developed in the current paper is purely analytical. •Through mathematical analysis of a nested designs model, we provide intermediate results that are of interest in their own right and explicit solutions. •We generalize the two-echelon supply-chain problem by determining both the optimal production rate, P, and the optimal (integer) number of shipments to the buyer, n, for each production lot Q. Furthermore, we consider bounds on the cycle length (at the manufacturer’s warehouse) and on the production rate. •As Aldurgam et al. [5] remark, variable production rates in the context of an integrated inventory model have been studied very infrequently. We contribute to the scarce literature that considers an integrated inventory combined with a variable production rate. •In addition to the integrated supply chain, we analyze a partiallycoordinated supply chain in which coordination takes place by means of the shared shipment cost, but each party optimizes its own objective. We show that there is a specific value of the shipment costsharing ratio (selected by the manufacturer) that results in total costs of the supply chain that are nearly the same as those obtained in the integrated inventory model. The main result (Theorem 1) pertains to the non-convexity of the optimum cost as a discrete-argument function of n, at the point n =2. In particular, we demonstrate that the speed of production is unimportant (i.e., remains optimal) in the case where the lot size is split into two shipments. 2. Literature review 2.1. Integrated approach In common with most of the literature on supply-chain inventory management, we assume in this paper that a central decision-maker sets a policy that aims to optimize the total supply-chain performance. This so-called “integrated” approach between the vendor and the buyer (or buyers) for improving performance has been studied for many years (see [6–14]). Utama et al. [15] presented a systematic review of integrated Table 1 Cases for Theorem 1. Case Description I R(1) ≤R(2) II R(1) >R(2), R(n∗ 2) ≥ 0 III R(1) >R(2) Table 2 Optimal policy for several values of the bound on the cycle length, T. T nact min nT max Case in Theorem 2 C∗ T n∗Q∗P∗ 1 3.873 1 f 6025.0 4 200 266.67 2 7.746 1 c 3775.0 8 400 266.67 3 11.619 4 c 3191.7 12 600 266.67 4 15.492 11 c 3025.0 15 800 266.67 5 19.365 26 b 3010.8 17 890.14 266.67 6 23.238 50 b 3010.8 17 890.14 266.67 A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 3 procurement production. Kim and Ha [16] developed an integrated inventory model with a JIT (just in time) approach and a small lot size with the objective of minimizing the joint total cost. Chou [17] showed that an integrated two-stage inventory model for deteriorating items results in higher profits than a non-integrated approach. Similarly, Klein et al. [18] found that performance improves when parties share strategic information and customize IT. A review of integrated inventory models is provided by Glock [19]. Gharaei et al. [20] employed the null-space method (NSM) for solving a nonlinear programming (NLP) formulation representing an integrated lot-sizing model of a multi-level supply chain. Dolai et al. [21] developed an integrated profit function where the demand function depends on the product’s green degree, the advertisement frequency, and the retailer’s selling price. Panja and Mondal [22] suggested a game theoretical approach for the case where demand depends not only on the product’s green degree and the retailer’s selling price but also on the credit period offered by the retailer to customers. Khara et al. [23] investigated a supply chain of used products consisting of a supplier, a manufacturer, a retailer and a “collector” (who collects the used product from consumers). In particular, their model seeks the optimal number of deliveries from supplier to manufacturer, from manufacturer to retailer, and from collector to manufacturer that together maximize the total integrated profit. 2.2. Variable production rate as an important managerial decision Numerous studies that aim to determine inventory and production policies allow for the production rate P to be a decision variable. This can affect the problem in a number of ways, as P can influence the rate of defects, the stochastic time period until failure or repair, inventory levels, and ultimately, the total cost. Glock [24] studied the impact of production rates on the buildup of inventory in a two-stage production system and showed that deviating from the ‘design production rate’ of the system may reduce the system’s total costs. Khouja [25], Giri et al. [26], Larsen [27], Sana [28], Manna et al. [29], Dolai and Mondal [30], and Askari et al. [31] all chose to study the well-known EPQ model – the EOQ model with production (see, e.g., [32,33]). Thus, they investigated the scenario in which a manufacturer optimizes the batch size given the parameters of the EOQ model and an additional parameter P – the production rate. A comprehensive and systematic overview of EPQ-type lot-sizing models that consider controllable production rates is given in Glock and Grosse [34]. Relatively few studies incorporate both an integrated inventory model and a variable production rate. Glock [35] considered a single-vendor single-buyer JELS (Joint Economic Lot Size) model with stochastic demand and variable lead-time. In this model, lead-time can be shortened in a number of ways: by reducing the lot size, by increasing the production rate, or by crashing a constant delay time. Jauhari and Pujawan [36], in analyzing the JELS model, employed an iterative procedure to determine simultaneously the following parameters: the safety factor, delivery lot size, delivery frequency, production batch, raw material lot size, and production rate. The objective was to minimize the total cost under stochastic demand. Following the work of Glock [35], Aldurgam et al. [37] analyzed a generalization that considers shipment constraints and raw material purchases. They considered a two-echelon single-vendor single-manufacturer supply chain where a vendor produces an intermediate product that is shipped to a manufacturer. The manufacturer transforms the intermediate product into a final product and has the opportunity to vary its production rate, which in turn affects the speed of completion of the lot size. The unit production cost Cp(P) is parametrically dependent on the (expected) cycle length, which is influenced by P, a decision variable. In Aldurgam et al. [37], the manufacturer is the buyer, while in our model, the manufacturer is the vendor. In spite of the relatively general setting, they did not provide analytical solutions. Kim and Glock [38] assumed that the production rates of the manufacturer’s machines may be varied within given limits. This allows the manufacturer to adjust his total production capacity according to the demand and to exploit different cost structures of the available machines. The authors analyzed the behavior of the proposed model in an extensive numerical experiment where the model was implemented using a commercial solver. In recent work by Jauhari et al. [39], in addition to utilizing regular production as a production base, the vendor also uses green production, which is costlier than regular production, but generates lower emissions. In a follow-up paper [40], this model was generalized by assuming imperfect production. In particular, the authors assumed that the production rate is adjustable and that it influences both the production cost and the emissions that result from the production and reworking processes. In contrast to the abovementioned studies, the solution approach used in the current research is purely analytical. We further contribute to the existing literature by assuming that the shipment cost is shared between the manufacturer and the retailer, and then investigating the optimal value of this cost-sharing ratio. 2.3. Closely related studies In this subsection, we describe the previous work that is most relevant to the current study and we identify the gap in the literature addressed by this paper. Firstly, we note that, in common with previous publications that examine JELS models (e.g., [41–43]), we assume that the shipments are all of equal size. Our study extends the approach of David and Eben-Haim [44], who analyzed an integrated inventory system in which the production rate is fixed and the cycle length is unrestricted. Very long cycle lengths inevitably mean very large lots, which, in turn, give rise to a large and inefficient warehouse. In addition, very long cycle lengths indirectly imply a long production-cycle time. To prevent these unwelcome scenarios, we restrict the cycle length and denote the upper bound on it by T. The present paper studies a similar problem setting to that investigated by Herbon [45]. Yet, there are several key differences between the two studies, as described in the following, which are crucial for understanding our current contribution: (a) The problem analyzed in Herbon [45] addresses the issue of bounding the production cycle length, while the current manuscript assumes a bound on the entire cycle length. Although these two parameters are related, they remain fundamentally different. (b) A fundamental difference is that Herbon [45] develops a heuristic to solve the complex problem, which is classified as an integer programming problem, while in the current paper, we develop a mathematical method to optimally solve the general problem. (c) Under small bounds on the production cycle length (see first line in Table 3 in Ref. [45]), non-feasible solutions (i.e., production rates) are obtained. Our current approach, on the other hand, rules out such scenarios by definition and is therefore applicable to all given bounds. (d) The current study analyzes the problem of finding the optimal sharing ratio of the shipment cost. This was not addressed in Herbon [45], nor has it been tackled by any previous literature on the topic of manufacturer-retailer chains. Furthermore, although both articles analyze an integrated inventory supply chain (i.e., cooperative mode), in the current manuscript, we also mathematically analyze two further scenarios. The first is a non-cooperative game between the manufacturer and the retailer in which each party optimizes its own cost. We analytically determine the best response of each party given the opponent’s decisions. Second, we analyze a partially-coordinated supply chain, in which coordination takes place by means of the shared shipment cost, but each party optimizes its own objective. In Section 3, the integrated inventory supply chain is described and the problem is formulated. In Section 4, we fully solve the mixed integer program for an unrestricted cycle length, while in Section 5, the A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 4 generally constrained problem is addressed iteratively (namely, P given Q and n, then Q given n using optimal P, and finally optimizing over n). Section 6 analyzes the case of partial cooperation between the parties, where each party optimizes its own costs. Section 7 concludes. 3. Supply-chain description and problem formulation In this section we introduce the two-echelon supply chain analyzed in this paper. We present the notations, formulate the mathematical model of the integrated inventory supply chain, and state the key assumptions. 3.1. Supply chain description A single retailer (hereafter, buyer), who does not permit shortages, observes a market demand rate of D for a single product. All items are purchased from a single manufacturer (hereafter, vendor) who supplies them periodically. We assume that the manufacturer and the retailer fully cooperate and that the relevant information is available to both parties. The manufacturer produces the product with rate P≥D (to be determined) and, in common with the buyer, does not permit shortages. Each commencement of a production cycle (which results in a production lot of size Q) is associated with a setup cost of K. The lot is delivered in several shipments, n (to be determined), each of size q (i.e., q=Q/n), from the manufacturer’s storage facility to the retailer’s warehouse. Fig. 1, adapted from Herbon [45], presents the integrated inventory level, the buyers’ inventory level, and the vendor’s inventory level. At this juncture, it is useful to define the demand-to-production rate ratio r =D/P, such that r replaces P as a decision variable. Interestingly, the combined inventory never empties, except in the special case where the production rate is infinite, P=∞ (i.e., purchasing). For the common scenario in whichP>D, i.e., the scenario presented in Fig. 1, the vendor’s inventory increases step-wise over time for the entire duration of production, after which it decreases step-wise to 0. The sudden "drops" in inventory level along the cycle length represent shipments to the buyer. In Figs. 2a–2c, we illustrate inventory levels for three special cases, where we denote the maximal production rate by U and the cycle length (for the manufacturer) byL=Q/D. Setting P=D, as presented in Fig. 2a, indicates that the manufacturer never stops production. This, of course, is impracticable. At the other extreme, Fig. 2b presents the case whereP=U=∞. This indicates that the manufacturer purchases raw materials, but does not produce the item. Fig. 2c presents a more common case in which there are exactly two equal shipments per lot. In this figure, we denote half the vendor’s idle time in a cycle byΔ=Δq,P,D= q/D−q/P. To complete the set of notations, we let hV and hB be the vendor’s holding cost and the buyer’s holding cost (respectively), for each unit per unit time. Similarly, we let kV and kB denote the shipment cost associated with the vendor (i.e., manufacturer) and the buyer (i.e., retailer), respectively. 3.2. Notations and assumptions Parameters D – demand rate for a single product (expressed in units per unit time) r – demand-to-production rate ratio, r =D/P (dimensionless) Q – production lot size (units) U – maximal production rate (units per unit time) hV– vendor’s holding cost for each unit per unit time hB– buyer’s holding cost for each unit per unit time kV– shipment cost ($) associated with the vendor (i.e., manufacturer) kB– shipment cost ($) associated with the buyer (i.e., retailer) Fig. 1. Inventory levels at the vendor and the buyer, along with the total inventory level A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 5 K – setup cost ($) associated with the commencement of a production cycle (i.e., a production lot of size Q) rmax– upper bound on r (dimensionless) Decision variables n – number of shipments (integer) per lot q – shipment size, i.e., q=Q/n (units) P – production rate (units per unit time) Variables Ct(q,r,n)– total cost per unit time ($ per unit time) The key assumptions of the proposed model are as follows: A1. The manufacturer and the retailer cooperate through an integrated inventory system. A2. Market and operational information (i.e., information about demand and information about costs) is fully shared. A3. Both parties prohibit shortages. A4. The supply-chain profit in the integrated inventory system is divided between the two parties according to a pre-arranged agreement between them. Fig. 2. a. P=D (n=∞). b. "lot-for-lot" (n=1). c. Any P(n=2) Fig. 3. Total cost as a function of the production rate, relative to the optimal cost solution Fig. 4. Peak inventory versus production rate, relative to the peak inventory for the optimal cost solution A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 6 A5. Once production has started, it is uninterrupted until the lot is completed (see Ref. [8]). A6. Production setup is required only in the case where there is a finite interval of time during which an inventory level of 0 is observed at the manufacturer (see Fig. 1). 3.3. Problem formulation The average total inventory level over time is It=rq +(1−r)nq /2 and the average inventory level for the buyer is IB=Q /(2n). Thus, after excluding fixed costs associated with purchasing (in the case of the buyer) and production (in the case of the manufacturer), the total cost per unit time is Ct(Q,P,n) = DK Q+kVD⋅n Q+hV(D P Q n+Q 2(1−D P)−Q 2n)+kBD⋅n Q+hB Q 2n. (1) The suggested formulation considers two additional factors that are likely to apply in real-life systems. Firstly, the formulation prohibits scenarios in which the production rate approaches the demand rate, that is, scenarios in which production takes place almost without breaks. Thus, we denote by rmaxan upper bound on r, and accordingly, there is a minimal production rate P=Pmin. Secondly, we avoid scenarios in which the cycle length (at the manufacturer’s warehouse) is unbounded. The problem we address in this paper is formally defined as Problem (P1): min q,r,nCt(q,r,n) s.t D/U≤r≤rmax nq ≤DT q>0 n≥1,integer (2) Ct(q,r,n), or equivalentlyCt(Q,P,n), is given in (1). Model formulation (2) is classified as a mixed integer non-linear optimization problem and the analysis begins in the following section. 4. Optimization for a given integer n – unbounded cycle length This section addresses a reduced version of the problem in formulation (2). This solution strategy enables us to obtain intermediate insights, while the optimal solution acts as a lower bound on the original problem. The reduced problem that we address in this section is formally defined as Problem (P2): min q,r,nCt(q,r,n) s.t D/U≤r≤rmax q>0 n≥1,integer (3) 4.1. Optimization of the production ratePgiven n and q We begin by stating the following insightful conclusion: Proposition 1.For a given number of shipments n (integer) and shipment quantity q, the optimal production rate is P∗(q,n) = P∗(n) = ⎧ ⎨ ⎩ U n =1 P∈ [Pmin,U]n=2 Pmin n≥3 .(4) Proof. Considering that q=Q/n and r =D/P, cost function (1) is identical to Ct(q,r,n) = DK nq +D(kB+kV) q+hV(rq +(n−1)q 2−rnq 2)+hB q 2. By grouping terms in the above expression, it can be seen that the cost function is linear in r, with slope hVq(1−n 2). Therefore, when seeking the minimum cost under the condition thatD/U≤r≤rmax, we obtain the following three outcomes: for n=1, the optimum r is the endpoint r=D/U, which means that P=U; for n=2 (zero slope), r is arbitrary; and for n≥3, r=rmax, or P=Pmin. Since solution (4) is feasible under the first constraint of problem (P2), it is also optimal for the general problem (P1) for any feasible set (q, n), i.e., for the problem that assumes a cycle-length bound. In particular, it is optimal for the optimal set (q∗,n∗)of problem (P2). The optimal value of P is independent of lot size q. Interestingly, unless a production run consists of exactly two equal shipments, the optimal strategy is either to produce at the maximum rate (if the whole batch is shipped at once) or to produce at the minimal allowable rate (if there are at least three shipments). Similar adequate production regimes (e.g., n=2) are also found to be optimal for controlling simple dynamic production systems with a single machine (see Maimon et al. [46]). Intuitively, if n=1, then the manufacturer ships the entire lot immediately upon completion, and has the opportunity to completely eliminate all inventory holding costs by making P=∞ (if feasible). This is the case in Fig. 2b. If, on the other hand, n≥2, the vendor (manufacturer) necessarily holds some inventory, but the system can minimize the holding costs by producing at the demand rate while incurring no production setup costs and only shipping costs. This is the case in Fig. 2a. In the special case wheren=2, the vendor’s average inventory level over time is shown to be equal to q/2, regardless of P (see Fig. 2c). The average inventory of the buyer is also q/2. Since P enters the cost expressions only through the vendor’s holding cost, the optimal total cost per unit time does not depend on the chosen production rate in the case where it is optimal to ship exactly twice per lot. 4.2. Optimizing the shipment size q given n Using (1), setting Q=qn, and defining k=kB+kV, Ct(q,r,n) = KD nq +kD q+hB q 2+hV((1−r)nq 2+rq −q 2).(5) For the sake of brevity, assume herein that P∗(2) = D in (4), so we may re-write (4) as r∗(q,n) = r∗(n) = {D/U n =1 rmax n≥2.(6) Consequently, r∗(n)is applied; thus, Ct(q,n) ≡ Ct(q,r∗(n),n) =DK nq +Dk q+hV(r∗(n)⋅q+(n−1)q 2−r∗(n)⋅n⋅q 2)+hB q 2.(7) Noting that min q,r,nCt(q,r,n)may be written asmin n{min q>0Ct(q,n)}and defining epar n=D⋅K+k⋅D⋅n, the FOC for the problem min q>0Ct(q,n), in the case where n =1, is hVD 2U−epar 1 q2+hB 2=0(8) thus, q2(1) = 2UD⋅(K+k) hVD+hBU. For n≥2, the FOC takes on the form hVrmax 2−n 2n+hV n−1 2n−epar n n2q2+hB 2n=0. A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 7 Thus, q∗(n) =  2epar n hV(n(n−1)− rmaxn(n−2))+hBn √(9) and Q∗(n) =  2nepar n hV(n−1−rmax(n−2))+hB √.(10) Note that q∗(n)holds for any set of parameters where hV +hB >0. In particular, q∗(2) =  epar 2 hV+hB √and Q∗(2) =  4epar 2 hV+hB √(regardless of rmax, as shown geometrically in Fig. 2c). In addition, q∗ ∞≡limn→∞q∗(n) = ⎧ ⎪ ⎨ ⎪ ⎩  2Dk hV+hB √rmax =1 0rmax <1 .(11) Again, regardless of rmax,Q∗(n)→∞ as n tends to infinity. If rmax =1, this limiting-infinite batch is comprised of an infinite number of equal shipments of finite, positive size q∗ ∞ (see Fig. 2a). Intuitively, this case is warranted if the setup cost is high (this argument is expounded upon in the following section). If, however, rmax <1, the limiting-infinite batch is comprised of an infinite number of shipments of infinitesimal size. However, this scenario seems to lack any practical significance. We summarize the above results in the following proposition: Proposition 2.The problem min q>0Ct(q,n)has a minimal solution q∗(n)for any n ≥1. This shipment size is given by q∗(n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  2UD(K+k) hVD+hBU √n=1  2epar n hV(n(n−1)− rmaxn(n−2))+hBn √n≥2 .(12) 4.3. A numerical illustration of the effect of production rate Defining Ct(n) = Ct(r∗(n),q∗(n),n)(q∗(n)is inserted into (7)), we are left with the problem min n≥1,integerCt(n),(13) which solves Problem (P2) – the version that disregards the cycle length. In this case, Ct(n) = epar n nq∗(n)+hVq∗(n) 2⋅(n− (n−2)r∗(n))+(hB−hV)q∗(n) 2.(14) A general conclusion from (14) is that P clearly does not affect the buyer’s costs. This reduces the inter-dependency between the two parties, thus increasing the likelihood that they will cooperate. Applying (6) and (12), we get Ct(1) =  2(hVD/U+hB)epar 1 √,forn=1 Ct(n) =  2D(K/n+k)(hV(n−1)− hVrmax(n−2)+hB) √forn≥2.(15) For the given problem setting (see Fig. 1), the maximal total inventory is given by Imax t(n) = Imax t(r∗(n),q∗(n),n) = r∗(n)q∗(n) + (1− r)nq∗(n). Let us pause here to illustrate the results so far by means of a numerical example. Example 1.The effect of production rate To highlight the effect of the production rate on the integrated inventory system and on the maximal mutual inventory, Fig. 3 depicts the ratio between Ct(n)(see (15)) and Ct(q∗(r),r,n)(see (14), where r*(n) is replaced by the given r), as a function of the production rate P and the number of shipments n, for the following choice of parameter values: D=200 (daily), U=500 (daily), K=5000, k V =50, k B =50, h V =10, h B =10, and r max =0.75. Figs. 3 and 4 illustrate the case of a fixed unit-production cost of c V =100 (D⋅cVis added to Ct(Q,P,n)). Four curves are shown, for four different n values (1, 2, 5 and 10). Similarly, Fig. 4 depicts the ratio of Imax t(q∗(r),r,n)to Imax t(n)as a function of production rate. A comparison of Figs. 3 and 4 reveals that the peak inventory is strongly sensitive to the production rate, while the total cost shows much lower sensitivity. In both cases, the sensitivity increases with n, n≥2. The peak inventory is particularly sensitive to a deviation from the optimal P. For high values of n, the lion’s share of the total system inventory is allocated to the vendor. Interestingly, exceptional behavior is observed for the case of a single shipment (n=1), namely, a decrease in the C t ratio with production rate. This behavior originates from (4) and Fig. 5. Lot-for-lot inventory patterns for the vendor (above) and buyer (below) Fig. 6. The ratio between the total optimal costs of the partially-coordinated supply chain and the integrated supply chain, as a function of the level of coordination, where hV=hB. Fig. 7. The ratio between the optimal costs of the partially-coordinated supply chain and the integrated supply chain, as a function of the level of coordination, when hV>hB. A. Herbon and I. David Operations Research Perspectives 10 (2023) 100264 8 (6), under which the form of the optimal solution differs for a single shipment (n=1). For such a case, the vendor holds a total of Q 2 /2P inventory units per cycle (see Fig. 5 below), according to which the cost is minimized by P =U, regardless of Q. 4.4. Overall optimization We now seek the optimal number of shipments, n∗(an integer). Let us denote by [x] either the left-hand side integer neighbor of a positive real x, or its right-hand side integer neighbor, whichever gives the smaller value of C t (n) (see (15)). If x is an integer, the left-hand side neighbor is x. We shall also need the abbreviation n∗ 2={[n∗(rmax)]Ctn∗(rmax) ≥ 2 2n∗(rmax)<2,(16) where, n*(rmax) is the real quantity that minimizes the second row in (15)). Let us further define R h =h B /h V and R k =k/K. Writing down the inequality Ct(n)<Ct(1)for n≥2, we see (by (15)) that it is equivalent to R(n) <0, where R(n) is the quadratic an2+bn +d and a=Rk(1−rmax) b=1−Rh− (rmin +rmax)− (1−2rmax +rmin)Rk d=Rh− (1−2rmax) .(17) Substituting and cancelling terms, we find that R(1) = (rmax − rmin)(1+Rk), which is a positive number. We are now ready to summarize the state of affairs when the constraint on L (the cycle length for the manufacturer) is ignored: Theorem 1.Consider the classification of three mutually exclusive possible cases, as shown in Table 1. If hB≥hV(1−2rmax)then n∗={1CaseIandCaseII [m]CtCaseIII ,(18) where, m=max{n|Rh≥ (1−2rmax)+n2k K⋅(1−rmax)}(19) Proof. See Appendix A. Note that if rmax =1, we formally get n*(r max ) =∞. Thus, it follows from Theorem 1 that n* is either 1 or infinity (Fig. 2a or 2b). Example 1.(continued). Optimal solutions for various parameter settings For the parameters in Example 1, Case III (see Table 1) applies. The optimal solution is characterized by n∗=17,Q∗=890.4,P∗=266.67, and q∗=52.36 q*=52.36, with I∗max T=261.80 and a minimal cost of C∗ t =3010.8$. If we change some of the parameters, specifically, if we set rmax =0.5, k V =1250, and k B =1250, Case III still applies, but with an optimal solution of n* =2 (such that Fig. 2c applies). Further, we find that the minimal cost is C∗ t=6324.6$ and the optimal lot size is Q∗= 632.45. Thus, for such a combination of parameter values, it is optimal to produce Q∗units per lot, irrespective of the production rate, and to split them into two equal shipments. A solution with n* =1 would only hold for a small setup cost (relative to the delivery cost) and a high vendor-to-buyer ratio for the holding cost parameter, as can be gleaned from (14). Theorem 1 specifies this qualification: if, for instance, k V =k B =4000, andrmax =0.5, then Case I applies with an optimal solution of n* =1, Q* =609.45, and C∗ t=8532.29$ (by (15)). Fig. 5 shows a schematic representation of such a lot-for-lot inventory pattern. In this case, (Q∗ 1)2→2D(K+k)/hB as U→∞ . U=∞ may be interpreted as outside procurement (with zero lead-time). The picture is then one of immediate outsourcing, shipping the quantity q= 2D(K+k)/hB √to the buyer and then resting for the entire duration of the q/D cycle length. Example 1 and Figs. 3 and 4 effectively represent the scenario where there is no constraint on the cycle length. 5. An exact, complete solution of Problem (P1) We return to the original problem (P1), including the constraint on the cycle length L≤T, or Q≤DT, alongside the existing constraint on the proportional production time r≤rmax. 5.1. The maximal possible number of shipments for each production lot The optimal production rate (6) is not affected by the constraint on the cycle length; thus, r∗ L(n) = {D/U n =1 rmax n≥2,(20) where, r∗ L(n)is the doubly constrained minimum. Since objective (5) is strictly convex in q, q∗ L(n) = min{DT /n,q∗(n)} (21) For n≥1, where q∗(n)is given in (9) and q∗ L(n)is the doubly constrained minimum, given n. For n≥2, Q∗(n)in (10) increases in n. Denote by nT max the maximal possible number of shipments for each production lot when n≥2, that is nT max =max{integern,n≥2 2nepar n hV(n−1)− hVrmax(n−2)+hB √≤DT}. (22) If Q∗(2)>DT, that is, if  4epar 2/(hV+hB) √>DT, we let nT max =1. The following proposition summarizes the conclusion that the maximal possible number of shipments for each production lot is either nT max =1 or, when nT max ≥2, it is given by the "integer part" (i.e., rounding down) of the positive root of a quadratic equation in n (denoted below by p(n) =0). Proposition 3.Let p(n)be the convex parabola p(n) = An2+Bn+C, where A=2Dk,B=2DK −hV(1−rmax)D2T2,and C =D2T2(hV(1−2rmax)− hB) (23) If nT max ≥2 then nT max =⌊−B+ B2−4AC √ 2A⌋(24) where, ⌊x⌋ signifies the "integer part" (i.e., the rounding down) of x, and A, B and C are given in (23). Otherwise,nT max =1. 5.2. The optimal number of shipments for each production lot The rest of this section is devoted to finding the optimal number of shipments in a single lot, n∗, when we consider two key intermediate variables, nT max (the maximal possible number of shipments per production lot) and nact min (defined below). Below, in (25), is the expression for the total cost. Consider the case where the constraint representing an upper bound on the cycle length becomes active (i.e., Q=DT is inserted into (7)). Using the superscript act to denote this case, we obtain: A. Herbon and I. David