A single-vendor, multiple-buyer coordinated supply chain model with unequal-sized batch shipments and cycle-dependent safety stocks

This paper investigates a single-vendor, multiple-buyer coordinated supply chain under stochastic demand. Each buyer implements a continuous review policy, and the shortage quantity is fully back-ordered. The lead time for the first shipment includes setup, production, and transportation times, while the lead time for the remaining shipments includes only transportation time. The production time is a function of production lot and production rate, which are both decision variables, and the setup and transportation times are controllable as well. For each buyer, the safety stock in the first replenishment cycle is not necessarily identical to that in the remaining cycles. Two successive shipments from the vendor to a buyer may have different sizes, and their ratio is a decision variable. The problem is finding the production and inventory replenishment policy, production rate, and lead times that minimise the long-run expected total cost per time unit. We demonstrate properties satisfied by the cost function and develop an optimisation algorithm, whose performance is compared to a benchmark algorithm based on a commercial solver in a numerical experiment. The experiments also investigate the benefits stemming from the proposed model when compared to models reproducing situations that leverage some or any of the controlled factors.


Introduction
For many decades, researchers have discussed how supply chains should be configured to achieve certain economic objectives (e.g.Dolgui and Proth 2013); in addition, there is an active discussion of how supply chain partnerships should be established (e.g.Chauhan, Nagi, and Proth 2004;Chauhan and Proth 2005).These topics remain valid in today's rapidly changing global business environment, in which it is becoming increasingly crucial for companies to leverage the benefits of cooperative value creation.This can be accomplished by formulating supply chain strategies and implementing channel-wide coordination mechanisms to align all supply chain members' decisions with a set of common goals (Ghasemi, Lehoux, and Rönnqvist 2022).Supply chain integration involves strategic, tactical, and operational tasks, with coordinated production and inventory replenishment decisions playing an important role because of their strong impact on supply chain costs (Glock and Grosse 2021).In the scientific literature, coordinated inventory CONTACT Davide Castellano davide.castellano@unimore.it "Enzo Ferrari" Department of Engineering, University of Modena and Reggio Emilia, Via P. Vivarelli 10, 41125 Modena, Italy Supplemental data for this article can be accessed online at https://doi.org/10.1080/00207543.2023.2240440.decisions within the supply chain are analysed by means of so-called joint economic lot size (JELS) models.
Originating with the pioneering work of Goyal (1976), this research stream has flourished in the last two decades.Interested readers should refer to Glock (2012a) for an in-depth review of the most important model categories and their corresponding features.According to Glock (2012a), JELS models can be classified along two dimensions: the scope dimension -that is, the number of cost parameters included in the model -and the depth dimension -that is, the number of stages considered in the supply chain.Regarding the second dimension, a further differentiation deals with the number of actors at each stage.In this paper, we consider a two-stage supply chain with a single vendor that supplies multiple buyers (we refer to this setup as an SVMB supply chain).Next, we provide an overview of the SVMB models.
Research on SVMB coordination originated with the work of Lal and Staelin (1984), Dada and Srikanth (1987), and Joglekar and Tharthare (1990).Chan and Kingsman (2007) showed that a significant total system cost reduction can be achieved if the buyers' ordering cycle lengths and lot sizes are coordinated with the vendor's production cycle.In the last two decades, research on SVMB supply chain coordination has been flourishing.Some authors have investigated how shipments should be made from the vendor to the buyers (e.g.Jha and Shanker 2013;Jia et al. 2016;Lee et al. 2016), while other authors have studied production and delivery patterns for cases in which multiple items are produced (e.g.Taleizadeh, Niaki, and Makui 2012;Uthayakumar and Ganesh Kumar 2019).Cases in which products deteriorate (e.g.Fauza et al. 2016;Chen 2018) or defective items may be produced (e.g.Mandal and Giri 2015) have also been investigated for SVMB supply chains.Beyond the classical, centralised inventory coordination scheme of JELS models, some researchers have investigated other forms of collaboration, such as consignment stock (CS) or vendor-managed inventory (VMI) (e.g.Zavanella and Zanoni 2009;Battini et al. 2010;Tarhini, Karam, and Jaber 2020).Recent studies have also examined the reverse flows of used materials in addition to the regular downstream flow of the finished product in an SVMB context (e.g.Ben-Daya, As'ad, and Nabi 2019; Chan et al. 2020).Table 1 provides an overview of the SVMB models that are most closely related to this paper.These models can be classified along several dimensions: • Demand can be deterministic (e.g.Chen 2018; Agrawal and Yadav 2020) or stochastic.In the latter case, researchers have often assumed a normally distributed demand or adopted the more general distribution-free approach (e.g.Huang and Song 2020;Castellano, Gallo, and Santillo 2021).• Especially in cases in which demand is stochastic, shortages can occur (e.g.Chan, Fang, and Langevin 2018;Castellano et al. 2019;Kurdhi, Yuliana, and Suratno 2021).Demand during the stockout period can be back-ordered or lost, or a mix of back orders and lost sales can occur (see, e.g.Chen 2018; Majumder, Jaggi, and Sarkar 2018;Castellano, Gallo, and Santillo 2021).• Inventory can be coordinated using classical continuous/periodic inventory review policies (e.g.Jha and Shanker 2014;Mandal and Giri 2015;Castellano, Gallo, and Santillo 2021), or a VMI or CS contract can be used (Chen 2018;Tarhini, Karam, and Jaber 2020).
Controllable lead times are another practical aspect that has been discussed in earlier research (e.g.Jha and Shanker 2013;Mandal and Giri 2015;Kurdhi, Yuliana, and Suratno 2021).Especially in uncertain demand settings, the ability to shorten lead times is crucial.Longer lead times increase the likelihood of shortages and the corresponding average cost (Glock 2012b), and lead-time length also influences demand in terms of back orders or lost sales (Shin et al. 2016).The benefits stemming from a reduced lead time are widely recognised in the supply chain literature (Hariga 2000;Glock 2012b).It is worth noting that lead time is usually assumed to consist of several independent components, such as order preparation time, order transit time, setup time, process time, and queue time (Tersine 2002).While production time can be controlled by using its relationship with lot size and production rate (Glock and Ries 2013), the remaining non-productive components can be shortened by paying a crashing cost (Glock 2012b).Originally proposed by Liao and Shyu (1991), this latter assumption about controlling lead time in inventory models has been adopted by several authors (e.g.Chuang, Ouyang, and Chuang 2004;Annadurai and Uthayakumar 2010;Hsu and Huang 2010).
In many settings, production rates are under management control and can be varied within given limits to accelerate or decelerate the build-up of inventories as customer demand changes (Glock 2010).However, a variable production rate may influence various performance measures, such as setup, unit production, or energy costs.Glock and Grosse (2021) carried out an encompassing review of economic production quantity models that consider controllable production rates.Controllable production rates have also been considered in some of the reviewed models (e.g.Jha and Shanker 2014;Majumder, Jaggi, and Sarkar 2018;Sarkar et al. 2018).
Another important factor that distinguishes the reviewed SVMB models is the shipment strategy -that is, the way in which the lot is transported from the vendor to the buyers.If shipments can be made from a production lot before it is completed, these partial shipments to the buyer permit an earlier depletion of the vendor's inventory and an earlier start to the consumption cycle (Glock 2012a).In the case of partial shipping, equal -or unequal-sized batches can be considered for successive shipments, and both types of shipments have been considered in the research stream that is of interest to us (e.g.Chan, Fang, and Langevin 2018;Hoque and Bhattacharya 2020).Hsiao (2008) noted that in cases in which partial shipments are made, the lead time for the first shipment includes setup, production, and transportation times, while later shipments have a shorter lead time consisting of only the transportation time.To the best of the authors' knowledge, in existing SVMB models, different lead times for successive shipments have been taken into consideration only in Castellano et al. (2019), while this aspect has been addressed in the case of a  Glock (2012b).
If subsequent shipments to the buyers have different lead times, different safety stocks and reorder points can be established on the buyers' side.In the SVSB case, a few authors considered two different safety stocks for the first and the remaining shipments (Mou, Cheng, and Liao 2017;Sarkar and Giri 2022).To the best of the authors' knowledge, this aspect has not yet been investigated in the case of an SVMB model or in any other supply chain configuration.It is also necessary to observe that Mou, Cheng, and Liao (2017) and Sarkar and Giri (2022) did not correctly address the case of different safety stocks.In fact, they incorrectly linked the vendor's and the buyer's inventory, assuming that the expected time between two consecutive replenishments to the buyer is always equal to the ratio of the buyer's order quantity and the demand rate.This is incorrect, as we will demonstrate while developing the mathematical model.
This paper considers an SVMB coordinated supply chain under stochastic demand.Each buyer implements a continuous review policy, and the shortage quantity is fully back-ordered.We follow Hsiao (2008) and assume that the first shipment has a longer lead time than the remaining shipments.In particular, the lead time for the first shipment includes setup, production, and transportation times, while the lead time for the remaining shipments consists of only transportation time.The production time is a function of the production lot and the production rate, which are both decision variables, and the setup and transportation times are controllable by means of investments.For each buyer, the safety stock in the first replenishment cycle is not necessarily identical to that in the remaining inventory cycles.This feature adds operational flexibility to the inventory control policy, which, in turn, translates into better system performance.In other words, including a variable safety stock makes it possible to achieve a more accurate balance between inventory carrying charges and shortage costs.We further assume that two successive shipments from the vendor to the buyer may have different sizes and that their ratio is a decision variable.The problem is finding the production-inventory replenishment policy, the production rate, and the lead times that minimise the long-run expected total cost per time unit.
The main contributions of this paper can be summarised as follows: 1. We study an SVMB coordinated supply chain under stochastic demand considering the following main features: (i) different lead times and safety stocks between the first and the remaining replenishment cycles, (ii) unequal-sized batch shipments, and (iii) controllable lead times and production rate.These aspects have never been jointly considered in regard to any supply chain structure.Moreover, as observed in the literature review, the use of multiple safety stocks has received very little attention despite the additional flexibility it gives, and the studies known to us, which focus solely on an SVSB supply chain, do not correctly model this feature.Therefore, our work is the first to investigate this aspect (correctly) altogether.2. We develop an analytical treatment that correctly integrates the issue of multiple safety stocks with additional realistic features, such as unequal-sized batch shipments, and controllable lead times and production rate.We then present the long-run expected total system cost per time unit and formulate an optimisation problem.3. We prove several properties satisfied by the cost function, which permits us to develop an optimisation algorithm.4. Numerical experiments are carried out to analyse the performance of the proposed algorithm and to demonstrate the benefits of the model we propose.
The rest of the paper is organised as follows.Section 2 provides the problem description along with the adopted notation and assumptions.Section 3 develops the cost model, while Section 4 formulates the optimisation problem.Section 5 describes the optimisation method.Section 6 presents the numerical experiments, and Section 7 concludes the paper.

Problem description
We consider an SVMB coordinated supply chain.The vendor supplies one item type to the buyers, who face stochastic demand and use a continuous review, (Q, r) stock control policy to replenish their inventories.The product supplied by the vendor is delivered in m batches to the buyers, and subsequent batch shipments from a lot may have different sizes.Like Hsiao (2008), we assume that the lead time for the first shipment differs from that for the remaining shipments.For the first batch, some kind of setup would be required to bring the production facility into a ready-to-produce mode, leading to a setup time that has to be considered a component of the first batch's lead time along with the time to produce the first batch.Given that the remaining batches are continuously produced until the lot has been completed, no further setups and production times are required, resulting in a shorter lead time for these batches.Some time to deliver the batch to the buyer is needed in every replenishment cycle.In a situation in which the buyer establishes a safety stock to protect itself against stockouts during the lead time, the buyer can decide to maintain a lower safety stock for batches 2, . . ., m.Assuming in addition that the production rate and the setup and transportation times are controllable, the system has various options to protect itself against stockouts and to minimise its long-run expected total cost rate.In the rest of this section, we introduce the notation and the assumptions adopted in developing the mathematical model.

Notation
The notation adopted in the mathematical model is as follows: Ratio between the second and the first shipments to any buyer within a batch production run -that is, μ = q 2n /q 1n , for n = 1, 2, . . ., N λ Ratio between shipments i + 1 and i, with i = 2, 3, . . ., m, to any buyer within a batch production run -that is, λ = q i+1n /q in , for i = 2, 3, . . ., m and n = 1, 2, . .Positive real numbers Additional notation will be introduced as needed.

Assumptions
The mathematical model is developed subject to the following assumptions: 1.One vendor supplies one item type to N buyers.2. The vendor and the buyers act cooperatively -that is, decisions within the supply chain are centralised.3. Buyer n, for n = 1, 2, . . ., N, orders a lot of size m i=1 q in and the vendor manufactures n=1 q in , with a finite production rate P in one setup.The production rate is such that P ∈ [P min , P max ] and Then, in the ith replenishment cycle, with i = 1, 2, . . ., m, the vendor ships quantity Q i to meet the demand of all buyers, who have the same average ordering cycle time -that is, The assumption of identical average ordering cycle times implies that 4. Each buyer faces stochastic demand, and stockouts are permitted at the buyers' side.Moreover, the shortage quantity is completely back-ordered.5.Each buyer implements the classical continuous review, lot size -reorder point inventory control policy.For n = 1, 2, . . ., N, the reorder point of the first replenishment cycle is , where L 1n = Q 1 /P + t S + t n , with Q 1 /P being the production time of the lot Q 1 , t S the setup time, and t n the transportation time from the vendor to buyer n.The reorder point of replenishment cycles 2, 3, . . ., m is , where L 2n = t n .The quantities S 1n = z 1n σ n √ L 1n and S 2n = z 2n σ n √ L 2n are, respectively, the safety stock in the first replenishment cycle and the safety stock in replenishment cycles 2, 3, . . ., m.The lead time for the first replenishment cycle and the lead time for replenishment cycles 2, 3, . . ., m differ, as explained above.However, in contrast to Hsiao (2008), the safety stocks S 1n and S 2n can differ.6.For n = 1, 2, . . ., N, the lead-time demand of buyer n for the first replenishment cycle, X 1n , is a Gaussian random variable with mean D n L 1n and standard deviation σ n √ L 1n .The lead-time demand of buyer n for replenishment cycles 2, 3, . . ., m, X 2n , is a Gaussian random variable with mean D n L 2n and standard deviation σ n √ L 2n .7. The average consumption time of the safety stock is identical for all buyers -that is, D N .8. The vendor incurs unit production cost as a function of the production rate, P. The unit production cost is supposed to have the following functional form (see, e.g.Khouja and Mehrez 1994): ( 1 ) 9. The setup time, t S , is controllable according to the procedure originally proposed by Liao and Shyu (1991).In particular, it is assumed that t S consists of R mutually independent, deterministic, and constant components, which are arranged in series.The generic pth component has a minimum duration w p , a normal duration W p , and a unit crashing cost c p , with The components are crashed one at a time starting with the component with the lowest c p and so on.If t Sp is the setup time with components 1, 2, . . ., p crashed to their minimum duration, we have where t S0 = R j=1 W j is the maximum duration of the setup time.The setup time crashing cost, T S , is thus given by The setup time crashing cost is defined in the interval [t SR , t S0 ], where t SR is the minimum duration of the setup time, obtained by crashing all of its components to their respective minimum duration.10.For n = 1, 2, . . ., N, the transportation time of shipments from the vendor to buyer n, t n , is controllable with a formulation similar to t S .The transportation time t n is supposed to be made up of M n mutually independent, deterministic, and constant components, which are arranged in series.The generic gth component has a minimum duration u ng , a normal duration U ng , and a unit crashing cost The components are crashed one at a time starting with the component with the lowest v ng and so on.If t ng is the transportation time of the nth buyer with components 1, 2, . . ., g crashed to their minimum duration, we have where t n0 = M n j=1 U nj is the maximum duration of the transportation time.The transportation time crashing cost, T n , is thus given by The transportation time crashing cost T n is defined in the interval [t nM n , t n0 ], where t nM n is the minimum duration of the transportation time, obtained by crashing all of its components to their respective minimum duration.11.The mathematical model is developed considering an infinite time horizon.

Cost model development
In this section, the long-run expected total system cost rate is developed.In the first subsection, the cost function of buyer n is analysed, while the cost function of the vendor is studied in the second subsection.The last subsection presents the total system cost.

Long-run expected cost rate for buyer n
We adopt the arguments that Hadley and Whitin (1963) followed in developing their well-known heuristic lot size -reorder point inventory model.The authors considered a non-negative reorder point, which is an assumption that is frequently met in practice, as companies seldom wait until a stockout occurs to issue a replenishment order (Nahmias 2013;Silver, Pyke, and Thomas 2017).As a result, this assumption has frequently been made in the inventory management literature (e.g.Hariga and Ben-Daya 1999;Atan, Snyder, and Wilson 2018;Pinçe, Gürler, and Berk 2008).Figure 1 gives an illustrative example considering a system with a single vendor and two buyers, assuming m = 3, and with r 11 > r 21 , r 12 > r 22 , S 21 > S 11 , and S 22 > S 12 .Because for both buyers, the safety stock of the first replenishment is different from the safety stock of the other replenishments 2, 3, . . ., m, it is apparent that, even in the case of identical shipment sizes, the expected cycle lengths of the first and last replenishment differ from the value they would take in the 'traditional' case of identical safety stocks.This peculiarity is clearly visible in Figure 1.The models of Mou, Cheng, and Liao (2017) and Sarkar and Giri (2022), which are the only two works known to us that consider different safety stocks, did not correctly capture this aspect (they also neglected the presence of unequal-sized batch shipments).
Let Z 1n and Z 2n be the expected cycle length of the first replenishment and the last replenishment, respectively, for the nth buyer.We can then write and The expected cycle length for the remaining m − 2 cycles is q in /D n , for i = 2, 3, . . ., m − 1.As a result, the expected total duration of all m cycles, τ n , is given by To establish the expected average inventory level of buyer n, we first calculate the expected area below the inventory curve considering all m cycles, Ūn , which is determined as follows: The expected average inventory of buyer n, Īn , is obtained by dividing Eq. ( 9) by Eq. ( 8): Under the assumption of a Gaussian lead-time demand, the expected shortage in the first replenishment cycle is while the expected shortage in replenishment cycle i, with ψ being the standard normal (first-order) loss function, defined as ψ(x) = φ(x) + x( (x) − 1).Hence, the expected shortage per time unit is The transportation time crashing cost, T n , and the transportation cost per shipment, C n , are paid m times every τ n time units (in compliance with the literature, the transportation time crashing cost is transferred to the buyer; see, e.g.Glock 2012b).The ordering cost per order, A n , is paid once every τ n time units.Hence, the long-run expected cost per time unit for buyer n, K n , is given by where and q 1 = (q 11 , q 12 , . . ., q 1N ) and q n = (q 1n , q 2n , . . ., q mn ).
In Eq. ( 15), we recall that Q 1 = N n=1 q 1n .The fact that Eq. ( 15) includes the quantity Q 1 P follows from the assumption that the expected time between two consecutive replenishments is the same for all buyers.Now, let μ = q 2n /q 1n and λ = q in /q i−1n , for i = 3, . . ., m.If we let q 1n = q n , we can then write q 2n = μq n , q 3n = λq 2n = λμq n , q 4n = λq 3n = λ 2 μq n , etc. -that is, q in = λ i−2 μq n -for i = 2, 3, . . ., m.The cost function can thus be rewritten as follows: where q = (q 1 , q 2 , . . ., q N ), and S 1n is given by Eq. ( 15), in which Q 1 is replaced by Q = N n=1 q n , with q n being the order lot of the first shipment for buyer n.Using the equalities m i=2 Eq. ( 17) becomes for each n 20)

Long-run expected cost rate for the vendor
We briefly recall some of the assumptions introduced in Section 2.2.The expected time between two consecutive replenishments is the same for all buyers: We also assumed that and that From Eqs. ( 22) and ( 23), we have The production lot of the vendor is n=1 q in is the total lot shipped to the buyers in replenishment cycle i.The batches q i1 , q i2 , . . ., q iN are shipped consecutively to each buyer in the ith replenishment cycle.The production-replenishment policy implemented by the vendor is governed by the relations in Eqs. ( 21)-( 24). Figure 1 depicts the time plot of the vendor's inventory for the adopted productionreplenishment policy.
According to our assumptions, the expected inventory cycle of the vendor, τ V , is identical to the expected total duration of all m cycles of any buyer -that is, τ V = τ 1 = τ 2 = . . .= τ N .Considering the nth buyer, for which τ n is given by Eq. ( 8), we can write The following relation can be obtained from Eq. ( 21) after some algebraic manipulations: where D = N n=1 D n .Combining Eqs. ( 25) and ( 26), we get The expected average inventory of the vendor, ĪV , can be calculated from Figure 1, for example, by using the method proposed by Joglekar (1988), as follows: Equation ( 28) was written exploiting Eqs. ( 24) and ( 26) and the following two relations, which are derived from Eqs. ( 22) and ( 23), respectively: and where S 1 = N n=1 S 1n and S 2 = N n=1 S 2n .If we assume q 1n = q n and q in = λ i−2 μq n , for i = 2, 3, . . ., m and n = 1, 2, . . ., N, then we can write for i = 2, 3, . . ., m, where we put Q = N n=1 q n .As a result, we have: where we put Note that = 1 when m = 1.Using Eqs. ( 31) and (32), Eq. ( 28) becomes Using Eq. ( 18) with m − 1 in place of m and the equality which is demonstrated in Appendix A in the online supplement, Eq. ( 34) becomes, after some algebraic manipulations, where is given by Eq. ( 33).
The setup time crashing cost, T S , is fully borne by the vendor because in practice, any initiative aimed at reducing the setup time is often launched by this actor.T S and the unit setup cost, C S , are paid once every τ time units.Considering the production cost per time unit, which is expressed as we have the following long-run expected cost per time unit for the vendor: where . ., t N ), z 1 = (z 11 , z 12 , . . ., z 1N ), z 2 = (z 21 , z 22 , . . ., z 2N ), and is given by Eq. (33).

Long-run expected total system cost rate
We assumed q 1n = q n , for each n.Hence, Eq. ( 26) becomes with Q = N n=1 q n .Using Eqs. ( 33) and ( 39), the longrun expected cost rate for buyer n (see Eq. ( 20)) can be rewritten as follows: The long-run expected total system cost per time unit, K, includes the costs of vendor and buyers.We thus have

Optimisation problem
The model developed in the previous section includes the two lot multipliers λ and μ.Two different multipliers are needed to account for the different safety stocks in the first replenishment cycle and the remaining ones.In the deterministic demand case, there is a single multiplier, which can take any value in the interval [1, P/D] (see, e.g.Hill 1997).We argue that the same observation can be made in the stochastic demand case when S 1 = S 2 .In the general case in which S 1 and S 2 are not necessarily identical, the lower limit of λ and μ is reasonably identical and equal to 1 as in the deterministic case (see, e.g.Hill 1997).The upper limit is not necessarily the same (this is the main motivation for introducing two multipliers).To understand this point, the following arguments should be considered.
The first batch produced by the vendor, within the production lot Q, is Q 1 = Q.According to the assumptions of the model developed in the previous section, the second lot is shipped to the buyers on average Q−(S 2 −S 1 ) D time units after the shipment of the first lot.As a result, the maximum size of the second lot produced by the vendor, by assumption, the value of μ corresponding to the lot Evidently, the maximum size of the ith lot shipped by the vendor, Q i , with i = 3, . . ., m, is PQ i−1 /D.Because Q i = λQ i−1 by assumption, for i = 3, . . ., m, the value of λ corresponding to the lot Equations ( 42) and (43) provide the upper limit of the multipliers μ and λ, respectively.Note that μ max is a function not only of P and D but also of Q, S 1 , and S 2 .The upper limit of λ is identical to the deterministic case, or to the stochastic demand case with S 1 = S 2 .We can finally observe that μ max = λ max for S 1 = S 2 .
3. For fixed (z 1 , z 2 , Q, P, m, t S , μ, λ), with m ≥ 2, K is strictly concave in t n , for each n, if and only if the following condition is satisfied for each n: where 4. For fixed (t, Q, P, m, t S ), with m = 1, K is strictly convex in z 1 .5. For fixed (t, Q, P, m, t S , μ, λ), with m ≥ 2, K is strictly convex in (z 1 , z 2 ). 6.For each n, K → +∞ as z 1n → +∞, and Proof.See Appendix B in the online supplement.The following corollary readily follows from the third point of Proposition 1: From point No. 1 of Proposition 1, we know that K is concave in t S , which implies that with t S ∈ [t Sp , t Sp−1 ], the minimum of K in t S lies on either t Sp or t Sp−1 .In general, we can thus affirm that the t S -component of the optimal solution is in the set S = {t Sp : p = 0, 1, . . ., R}.
Corollary 1 and point No. 3 of Proposition 1 assure that K is concave in t n for each n.This means that with t n ∈ [t ng , t ng−1 ], the minimum of K in t n coincides with either t ng or t ng−1 .Hence, more generally, the t-component of the optimal solution belongs to the set T = {(t 1g , t 2g , . . ., t ng , . . ., t Ng ) : g = 0, 1, . . ., M n ∀n}.
We observe that the (t, t S )-component of the optimal solution has to be searched over all elements of T × S.
If we take the first-order optimality condition in z 1n and z 2n , we get , for n = 1, 2, . . ., N, (60) respectively (recall that z 2n applies to shipments 2, 3, . . ., m, for each n).Note that Eq. ( 59) admits a solution in z 1n , which we denote by ẑ1n , if and only if while Eq. ( 60) has a solution in z 2n , which we denote by ẑ2n , if and only if the following two relations are jointly verified: If inequalities ( 61)-( 63) are not satisfied, then K does not admit any finite minimiser in z 1n and z 2n for each n.It is thus required that these three conditions be verified for a solution to problem (44) under constraints ( 45)-( 53) to exist.Note that conditions (61) and ( 63) are satisfied when back orders are significantly costlier than keeping stock in inventory (see both sides of inequalities ( 61) and ( 63)).This is a situation that is often met in practice (Nahmias 2013;Silver, Pyke, and Thomas 2017;Hadley and Whitin 1963).The optimisation procedure will thus assume that inequalities (61)-( 63) hold.
We observe that K is not, in general, convex in Q.However, according to point No. 7 of Proposition 1, we deduce that the Q-component of the minimum of K necessarily lies on a stationary point.Based on arguments similar to, for example, Glock's (2012b), we also note that K is not, in general, convex in P and that the minimum in P does not necessarily lie on a stationary point.From the first-order optimality condition in Q and P, we get, respectively, where = Note that the first-order optimality condition in P may not admit a solution (in P) depending on the sign of , which may be negative in a given problem instance.According to Glock (2012b), we use the following rule: (69) For a fixed (t, m, t S , μ, λ), the (z 1 , z 2 , Q, P)-component of the optimal solution can be determined solving Eqs. ( 59), ( 60), (64), and ( 69) with an iterative method that is well known in the literature (see, e.g.Glock 2012b; Hadley and Whitin 1963).We denote by (ẑ 1 , ẑ2 , Q, P) the optimal solution for a fixed (t, m, t S , μ, λ), where ẑ1 = (ẑ 11 , ẑ12 , . . ., ẑ1N ) and ẑ2 = (ẑ 21 , ẑ22 , . . ., ẑ2N ).
Concerning m, μ, and λ, their optimisation is tackled by means of some heuristic arguments.In particular, the search over m is carried out starting from the value m = 1 and proceeding with the m = 2, 3, . .as long as a smaller cost can be achieved.This procedure is widespread in the inventory management literature (see, e.g.Castellano et al. 2019;Glock 2012b;Jha and Shanker 2014).Similar to Hill's (1997) method, the optimisation of μ and λ is instead performed with a line search over 1, P D 1 + S 1 −S 2 Q and 1, P D , respectively, starting from the left endpoint of these intervals and considering the following observations: • For m = 1, neither μ nor λ affect the productioninventory replenishment policy.• For m = 2, only μ affects the production-inventory replenishment policy.• For m ≥ 3, both μ and λ affect the productioninventory replenishment policy.
The previous arguments are included in Algorithm 1 (please refer to Appendix C in the online supplement), which is the heuristic optimisation algorithm we propose to approach problem (44) under constraints ( 45)-( 53).

Numerical experiments
This section presents the numerical experiments with a twofold objective.First, we analyse the quality of the solution obtained by Algorithm 1 by comparing it with a benchmark solver.Second, we investigate the benefits that can be achieved through controlling setup and transportation times, adopting different safety stocks, and implementing unequal-sized batch shipments.
Experiments are carried out using a platform developed in MATLAB ® R2021b on a machine with an Intel ® Core TM i7-8750H CPU at 2.20 GHz and 16GB RAM memory.The search step, ε, in Algorithm 1 is set to 0.05.
Throughout the experiments, we assume that setup and transportation times are made up of three components -that is, R = 3 and M n = 3 -for each n.Moreover, the following operating conditions, assumed to be identical for each supply chain actor, are considered: 8 h/day, 7 days/week, and 46 weeks/year.

Solution quality of Algorithm 1
The solution obtained by Algorithm 1 is compared with the one returned by an algorithm exploiting the 'fmincon' MATLAB ® solver in Optimization Toolbox TM .The MATLAB ® solver is used, specifically, to obtain the minimum of K in (t, z 1 , z 2 , Q, P, t S , μ, λ) for fixed m under constraints ( 45)-( 49) and ( 51)-((53).The minimum over m is determined similarly to Algorithm 1: the minimisation in (t, z 1 , z 2 , Q, P, t S , μ, λ) for fixed m is performed by increasing m by one unit at a time, starting from m = 1, until the minimum cost for fixed m increases, and then we take the solution in ω that achieved the minimum cost.We denote the benchmark algorithm based on the MATLAB ® solver by M-ALG.Note that we do not know in advance how well M-ALG will perform.We use M-ALG as a benchmark algorithm as we rely on the fact that Optimization Toolbox TM in MATLAB ® includes well-established commercial solvers.Of course, our numerical experiments also permit us to assess the performance of a MATLAB ® solver on the problem we posed.
Algorithm 1 and M-ALG are compared on the dataset given in Appendix D in the online supplement.The results obtained with Algorithm 1 and M-ALG are shown in Tables 2 and 3, respectively.Considering the instances we generated, the performances of Algorithm 1 and M-ALG are comparable, as Algorithm 1 achieved the best solution in 50% of cases, and M-ALG in the remaining ones.Moreover, the worst solution (for both algorithms) is always less than 0.5% worse than the best solution.Also note that the relative performance does not seem to change as the number of buyers increases.

Model performance
This section demonstrates the benefits that can be achieved through the model we developed, compared to situations in which the buyers receive equal-sized shipments, safety stocks are identical, and setup and transportation times are not controllable.Equal-sized shipments are simply obtained by setting λ = μ = 1.If setup and transportation times are not controllable, this means that t S = t S0 and t n = t n0 for each n.Safety stocks are identical if for each n, we take which gives z 2n = z 1n √ L 1n /L 2n .Four simpler models are considered in the comparison, whose features are given in Table 4.This table also summarises the features of the model developed in this paper, which is referred to as model M.Moreover, this analysis is conducted in the setting corresponding to the previous instance P1 (please refer to Appendix D).
Table 5 compares the proposed model (i.e.model M) with the simpler models shown in Table 4 (i.e.models M1 to M4).The comparison is carried out by considering the optimal solution achieved in the setting corresponding to the instance P1 (Table 5 shows only the optimal values of m, Q, P, μ, and λ; note that the optimal value of a decision variable is not given for those models that do not include that specific decision variable) and with reference to the following performance measures evaluated in correspondence to the optimal solution: • The long-run expected total system cost per time unit, K. • The inventory cycle of the vendor, τ V , as given by Eq. ( 27).• The overall amount of safety stock in the system (SS).
In case of models M and M4, which consider different safety stocks, the value of SS reported in Table 5 is the average value over the m shipments.• The total stockholding cost rate of the system (THC).
• The total shortage cost rate of the system (TSC).
First, we note that model M leads to the lowest value of K with a maximum percentage decrease of 5.44 in comparison with model M1.This confirms that leveraging setup and transportation times, batch sizes, and safety stocks in a joint manner is beneficial to the overall system cost performance.The fact that model M4 performs second best in terms of the minimum cost, with a percentage difference of 0.57, suggests that the batch size of successive shipments has the least impact on the total cost performance.On the contrary, the value of K characterising model M3 shows that the cycle-dependent safety stock is the factor that most impacts the total cost performance.Moreover, the value of K for model M2 suggests that controlling setup and transportation times leads to some cost improvement compared to the simplest condition in terms of decision variables (i.e.model M1).Still, by comparing the cost performance of M2 and M3, we can say that the batch size of successive shipments does not make much of a difference when controlling setup and transportation times.
To enable a deeper understanding of the relative performance of model M with respect to each model in Table 4, we also consider the optimal solution achieved in terms of m, Q, P, μ, and λ, along with the performance measures reported above.We immediately observe that for the considered instance, all the models present the same value of P.
While all alternative models entail an increase in the optimal number of shipments compared to model M, it is worth noting that controlling setup and transportation times has a substantial effect only when combined with different safety stocks (model M3) or unequal-sized shipments (model M4).
Controlling setup and transportation times (model M2) increases the optimal solution for Q by about 40%.However, in the presence of a controllable lead time, the adoption of unequal-sized shipments (model M3) has the overall effect of reducing Q, while a varying safety stock (model M4) has the opposite effect of further increasing Q by more than 60%.In the case of model M3, equal safety stocks in successive shipments are compensated with a smaller value of μ and a small adjustment of λ.
Only controlling lead time (model M2) does not have much of an effect on getting the inventory cycle of the vendor, τ V , close to that of model M.However, the combination of a controllable lead time with cycle-dependent safety stock (model M4) improves this performance measure (only a 19% increase with respect to model M).At the same time, controlling lead time and successive shipment sizes (model M3) slightly reduces the value of τ V with respect to model M.
Regarding SS, the major effect of controlling setup and transportation times occurs when it is combined with varying safety stock.In fact, model M4 produces an increase in SS of only 7% compared to model M. It is worth observing that controlling successive shipment sizes in addition to lead time (model M3) even deteriorates the system performance in terms of SS, compared to the case in which only lead time is controlled (model M2).
THC shows the lowest average percentage variation once K has been excluded.All the models show a percentage increase in THC compared to model M, with models M2 and M3 having very close values.This latter situation confirms that the combined effect of controlling lead time and unequal-sized shipments is negligible.The least percentage increase pertains to model M4, which further illustrates the superior ability of varying safety stocks in limiting the increase in THC.
As far as TSC is concerned, some interesting observations can be made.Models M1, M2, and M3 present a sizeable percentage decrease in TSC compared to model M (up to 88%).In absolute terms, these models are able to almost nullify shortage costs.On the contrary, model M4 delivers a performance in terms of TSC comparable to M, even with a percentage increase of 6%.Hence, controlling lead time in combination with unequal-sized shipments does not have a large effect in terms of further decreasing TSC.Moreover, controlling lead time only in combination with cycle-dependent safety stocks further increases shortage costs.The performance delivered by the various models in terms of TSC are to be considered in combination with those in terms of SS, THC and K. Specifically, the ability of models M1, M2, and M3 to achieve a very low shortage cost (compared to models M and M4) is the direct consequence of maintaining higher safety stocks and, therefore, implies higher stockholding and total system costs.

Sensitivity analysis
This section analyses the impact of modifications in the model parameters on the optimal solution and on the performance measures considered in Section 6.2 with reference to model M. The analysis is carried out by varying each of the considered parameters by −50%, −25%, + 25%, and +50%, starting from the base values defined in the setting corresponding to instance P1 (please refer to Appendix D).The parameter values are modified one at a time while keeping the others fixed at the base value.
The results of this analysis are reported in Table 6.Please note that changes in σ n are obtained by varying the coefficient of variation ρ n = σ n /D n while keeping the value of D n fixed.
Appendix E contains the results of an additional sensitivity analysis carried out on models M1-M4 with respect to model M. The percentages in Tables E.1-E.4 are percentage variations with respect to the value pertaining to model M under the same percentage change in parameters.The sensitivity analysis of the performance of models M1-M4 with respect to model M aims to demonstrate that the superiority of model M is preserved under changes in the parameter values.It can be observed that the maximum percentage cost reduction achieved by model M is more than 7%.Moreover, the sensitivity analysis permits us to identify under which conditions the cost advantage provided by model M increases.In particular, even if model M is always preferable to models M1-M4, we observe that the cost performance of model M, compared to that of the other models, improves as C S , h V , and A n decline and a 1 , C n , h n , π n , and ρ n increase.
From Table 6, it becomes apparent that the model is sensitive to changes in each parameter, even if it turns out to be more dependent on some parameters than others.Specifically, the model appears to be practically insensitive to C S ; very sensitive to h n and ρ n (with an overall average absolute percentage variation of 14.7% and 16.0%, respectively); moderately sensitive to h V , a 1 , and C n (with an overall average absolute percentage variation of 7.6%, 8.6%, and 9.0%, respectively); and slightly sensitive to A n and π n (with an overall average absolute percentage variation of 6.0% and 6.3%, respectively).Moreover, we observe that the optimal solution remains unchanged regarding P * and λ * for any change in parameters.The results can be summarised as follows: • The optimal solution is influenced by an increase in C S only as far as Q * is concerned even if the corresponding increase in Q * is almost negligible.With reference to performance measures, an increase in C S produces only negligible increases in τ V and THC.• An increase in h V reduces m * , while any variation in h V is accompanied by a decrease of Q * (on average, -16.3%) and an increase of μ * (on average, + 18.6%).With reference to performance measures, an increase in h V produces a decrease in τ V and only negligible increases in SS, THC, and K * , while any variation of h V produces an average reduction in TSC of -3.5%.• Any variation in a 1 increases m * (on average, + 9.2%) and μ * (on average, + 18.6%), while it decreases Q * (on average, -16.6%).As far as the performance measures are concerned, any variation in a 1 increases τ V (on average, + 7.7%) and decreases TSC (on average, -3.6%), while it slightly affects SS and THC with an average decrease of -0.5%.K * is the performance parameter which is most affected by variations ina 1 with an overall absolute percentage variation > 60%: an increase in a 1 produces a reduction in K * .• Any variation in A n reduces Q * (on average, -16.1%) and increases μ * (on average, + 17.9%), while an increase in A n leads to an increase in m * .With reference to performance measures, while SS, THC, and K * are negligibly affected by an increase in A n , τ V increases with A n , and any variation of this parameter produces an average reduction in TSC of -3.5%.).The performance measures show smaller variations, especially in K * and THC, which both increase with π n .While TSC decreases with π n , any variation of this parameter induces an increase in τ V (on average, + 7.5%).Only in correspondence of a -50% variation in π n , we get a reduction of -3.3% in SS, while any other variation in the parameter implies an average increase in SS of +2.9%.• As observed above, ρ n is one of the parameters which produce the highest changes in both the optimal solution and the performance measures.Regarding the former, we can observe that μ * increases with ρ n with an absolute range of variation of about 97%.Any variation in ρ n produces an increase in m * (on average, + 9.2%).Only in correspondence of a -50% variation in ρ n we observe a growth of 12.3% in Q * , while any other variation of the parameter implies an average decrease in Q * of -25.2%.Regarding the performance measures, they all increase with ρ n , with the only exception being τ V .Specifically, SS and TSC share the same absolute range of variation of 81.4%, while THC and K * show analogous values of 48.3% and 13.5%, respectively.Any variation in ρ n produces an increase in τ V (on average, + 11.6%), with the only exception being the -50% variation, which induces a slight reduction in τ V of -1.4%.

Discussion and outlook
This paper studied an SVMB coordinated supply chain under stochastic demand, in which each buyer adopts a continuous review policy and demand during the stockout period is fully back-ordered.The lead time for the first shipment was expressed as the sum of setup, production, and transportation times, while the lead time for the remaining shipments includes only transportation time.
The production time was assumed to be a function of the production lot and the production rate, which were considered decision variables, and the setup and transportation times were controllable according to a crashing cost function.For each buyer, we assumed that the safety stock in the first replenishment cycle is not necessarily identical to that in the remaining inventory cycles.Moreover, the ratio between two successive shipments from the vendor to a generic buyer was supposed to be a decision variable.The problem was finding the production and inventory replenishment policy, production rate, and lead times, minimising the long-run expected total cost per time unit.We derived many properties satisfied by the cost function, and we proposed an optimisation algorithm (Algorithm 1), whose performance was evaluated by means of a comparison with a benchmark algorithm (M-ALG) based on a MATLAB ® solver in several problem instances.The performance analysis showed that for the considered instances, the performances of Algorithm 1 and M-ALG are comparable, as Algorithm 1 achieved the best solution in 50% of cases.Moreover, the worst solution (for both algorithms) was always less than 0.5% worse than the best solution.The relative performance was observed to be almost insensitive to changes in the number of buyers.
Beyond comparing the relative performance of Algorithm 1 and M-ALG, the numerical experiments investigated the benefits stemming from the proposed model (M) when compared to other models (i.e.models M1-M4), reproducing situations that leverage some or none of the controlled factors (controllable setup and transportation times, equal/unequal-sized shipments, and variable safety stocks for successive shipments).The results showed that controlling the above-mentioned factors in a joint manner delivers the highest positive impact on economic system performance.The experiments also reveal that when controlling lead time, cycle-dependent safety stocks prove to be more effective than unequal shipments in limiting the total amount of safety stock and the total stockholding cost of the system.However, this reduced overall safety stock in the system comes with higher shortage costs compared to the situation in which none of the above factors is controlled.
A sensitivity analysis permitted us to observe that the cost advantage provided by the proposed model also dominates the other models for alternative model parameters.The results obtained in the sensitivity analysis are particularly significant.In fact, we demonstrated that our model performs even better when the system variability increases and when the unit ordering cost, the unit stockholding cost rate, and the unit shortage cost increase.In other words, the model we presented enables companies to respond to uncertainty more efficiently, providing the best balance, compared to simpler (traditional) models, between inventory carrying charges and shortage costs.
The experiments we conducted thus demonstrated the superiority of our model compared to models that consider only a subset of all the decisions included in ours.However, besides the advantages provided, an important question that should be analysed is the practical applicability of the model.First, we observe that lead time can be controlled in several ways in practice.For example, the setup time component may be reduced by investing in new tools and equipment or in the implementation of efficient methods -for instance, the single-minute exchange of die system that originated from lean production.Concerning the transportation time component, it may be shortened through investments in more efficient vehicles or in new equipment that may speed up the truckloading activity.This lead-time component may also be shortened by investing in software devoted to optimising routings or by developing procedures aimed at increasing the efficiency of tasks involved with transportation, such as the application of lean thinking (see, e.g.Villarreal, Garza-Reyes, and Kumar 2016).Other activities that may be carried out to reduce lead time are discussed by, for example, Liao and Shyu (1991).Evidently, any activity aimed at reducing lead time is worthy of implementation, provided that overall system performance increases.
A key feature of our model is the presence of a cycledependent (i.e.variable) safety stock, which is appealing because it adds flexibility to the management of inventories in the presence of uncertainty.As we demonstrated, from an economic viewpoint, this flexibility leads to better system performance, resulting in a better balance between inventory carrying charges and shortage costs.Our variable safety stock policy is especially interesting in situations in which the product in question is expensive.In this case, a lot of capital is bound in the safety stock, which may also provide an incentive to select a lower safety stock level than required.Our policy enables companies to select appropriate safety stock levels to achieve a high service level, and it can reduce inventory carrying costs, as the safety stock is dynamically adjusted where possible.In terms of implementation, this very much depends on the warehouse management system (WMS) that is in use.If stock control is partially automated and supported by a WMS, the company could automatically lower the stock level and consume additional units when the safety stock should be decreased, and when it should be increased again, additional units from the last order are blocked for consumption.Evidently, some implementation effort would be needed to include such a rule in the WMS, but afterwards, any operational problems should be excluded.
Future research may be devoted to overcoming some limitations of the model we developed.For example, we may drop the assumption of a normally distributed demand at the buyers and extend the model to the distribution-free case.We may also consider a generalised production and shipment policy, as in Hill (1999).Another interesting topic to investigate would be a scenario in which items deteriorate such that the safety stock has to be replaced from time to time.In such situations, varying the size of the safety stock over time may be interesting, as it offers the opportunity to partially consume the safety stock on a regular basis, which may lower the spoilage of items.Moreover, the model may be extended to include multiple vendors with supplier selection.Future investigations may also include GHG emissions and the possibility of implementing transshipments in a similar supply chain structure.An additional interesting topic to consider in future research is delivery performance -that is, the effect of early or late deliveries.Finally, it may be interesting to compare the model we presented with its counterpart developed under a VMI -CS agreement to evaluate the respective performance and draw insights into, for example, when one model is preferable to the other.

Figure 1 .
Figure 1.Inventory -time plots for a single-vendor, two-buyer system, considering generalised, increasing batch shipments and two different safety stocks at the buyers.The figure illustrates the average net inventory at the buyers (image on the left) and possible random dynamics (image on the right).

Table 1 .
Comparison of the reviewed single-vendor, multi-buyer coordinated supply chain models.

Table 2 .
Solution obtained by Algorithm 1 in each instance.

Table 3 .
Solution obtained by M-ALG in each instance.

Table 4 .
Features of the models considered in the comparison.

Table 5 .
Comparison of models.Within brackets, the relative performance with respect to model M is reported.
SS: Total safety stock in the system.THC: Total stockholding cost rate.TSC: Total shortage cost rate.

•
The optimal solution is influenced by an increase in C n ; in fact, it reduces m * and increases Q * .Moreover, any variation in C n increases the value of μ * (on average, + 25.0%).Additionally, performance measures are affected by variations in C n .Specifically, as C n increases, τ V , THC, and K * increase, while SS reduces its value.It is worth noting that any variation in C n produces a negative percentage variation in TSC with an overall average value of -1.6%.•As mentioned earlier, the model is highly sensitive to h n : as this parameter increases, m * increases and λ *

Table 6 .
Sensitivity analysis of model M. .These percentage variations are higher, in absolute terms, when percentage variations in h n are positive.With reference to performance parameters, they are all increasing, with the exception of SS, as h n increases with TSC, showing the highest range of variation.•π n variations influence, even if only slightly, both the optimal solution and the performance measures.Any variation in π n produces an increase in m * )ρ n denotes the coefficient of variation σ n /D n decreases.Any variation in h n seems to reduce Q * and increase μ * * (on average, + 10.5%) and μ * (on average, + 17.2%) and a reduction of Q * (on average, -16.7%