Container management in a singlevendormultiplebuyer supply chain
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Abstract
This paper studies a supply chain consisting of a single vendor and multiple retailers that uses returnable transport items, such as containers or crates, to facilitate shipping products from the vendor to the retailers. The paper considers two different strategies for transporting finished products from the supplier to the retailers: In case of early shipments, deliveries to a retailer can be made while the production process at the supplier is still in progress, while in the case of late shipments, the supplier has to wait until the entire production lot has been finished before shipments can be made from the lot. The paper develops mathematical models for both strategies and derives optimal solutions for the cycle time, the container size, the individual order quantities of the retailers and the shipment sequence with the intention to minimize the average total costs of the system. The behavior of the models is analyzed with the help of numerical examples.
Keywords
Returnable transport item Joint economic lot size RTI Container management Singlevendormultiplebuyers1 Introduction
Governmental regulations and changes in customer preferences have induced companies to reduce the environmental impact of their operations. Among the various areas which have been identified as drivers of sustainable supply chain management, the distribution of products has often been characterized as a key driver of sustainability [2, 39, 51].
In transforming traditional distribution systems into more environmentally friendly ones, the transportation equipment and the packaging materials used are of major importance. More than a decade ago, Hekkert et al. [23] hypothesized that using reusable instead of oneway packaging material has the potential to reduce global CO_{2} emissions from production and transportation by up to 16 %. Other studies suggested that employing reusable packaging material may reduce the gross energy requirement and waste generation of transportation significantly [39]. Socalled returnable transport items (RTIs), which represent a specific type of reusable packaging material, such as pallets, crates, railcars or (maritime) containers, are used in a variety of industries today, e.g., in the automotive or consumer goods industries or the grocery sector [19, 29, 44, 46, 47].
It is clear that the use of reusable packaging material has to be adequately coordinated to fully realize the benefits this type of packaging material offers. This is especially important in case highvalue or highvolume packaging material is used, whose mismanagement could lead to a significant increase in transportation cost. Management actions associated with the use of reusable packaging material include the initial purchase and the replacement of damaged or lost units, the collection and return of used items as well as the organization of cleaning and repair processes [29].
In recent years, researchers have developed models that assist decision makers in planning the purchase, distribution and return of reusable packaging material. However, as will be shown in the next section, publications in this area had a focus on conceptual models of RTI management, on the use of radiofrequency identification (RFID) in the management of RTIs and on container routing and repositioning models. Models that consider the different logistical costs associated with the use of RTIs as well as the interdependencies that arise between the management of RTIs and the distribution of finished products, in contrast, have not been proposed thus far. To close this gap, this paper studies a supplier who uses returnable packaging material for supplying a product to multiple customers. The objective is to develop a mathematical model that helps decision makers in coordinating the flow of finished products and RTIs along the supply chain and to minimize the total logistics cost of the chain.
The remainder of the paper is structured as follows. The next two sections give an overview of related works and outline the assumptions and definitions that will be used in the remaining parts of this paper. Sections 4 and 5 develop models of a supply chain with a single vendor and multiple buyers that use returnable transport items, and Sect. 6 contains numerical examples. Section 7 concludes the paper and provides suggestions for future research.
2 Literature review
Two streams of research are of special importance to this paper, namely works that focus on the management of RTIs in supply chains and works that study the coordination of replenishment decisions in singlevendormultibuyer supply chains. Both research streams will be discussed briefly in the following.
2.1 RTI management
Works that study returnable transport items can roughly be differentiated into three different streams of research. Papers that fall along the first stream develop mathematical models that assist decision makers in determining deployment quantities and dates for RTIs as well as the timing and quantity of replacement orders of RTIs. One example is the work of Kelle and Silver [35], who proposed different methods to forecast the expected demand and the expected returns of containers. A related paper is the one of Goh and Varaprasad [22], which developed a method for estimating the container return distribution by assuming that containers are subject to damage and loss. Buchanan and Abad [9] studied the inventory control problem for containers and considered the returns in a given period as a stochastic function of the number of containers in the field. The authors used dynamic programming to derive the optimal inventory control policy for the system.
The second stream of research studies the repositioning of containers in a logistics network. This problem is common in several scenarios, such as in international maritime shipping, the distribution of empty freight cars in a railway network or the allocation of vehicles to depots of a transport service provider. The intention of works in this area usually is to match customer demand and supply of RTIs and to minimize the number and distance of empty RTI shipments. Crainic et al. [15], for example, studied the empty container allocation problem and considered uncertain supply and demand data and several specific operational characteristics, such as the substitution of container types or product imports and exports. Del Castillo and Cochran [17] studied a manufacturer who operates several plants, which, in turn, serve multiple depots of a customer. In this scenario, the question arises which depot should return which container to which plant. The authors formulated this problem as a linear program and derived results with the help of simulation. Dang et al. [16] studied the positioning of empty containers in a port area with multiple depots. Customer demand and returning containers in depots were assumed to be serially correlated and dependent random variables. Three alternatives were considered for repositioning containers: positioning from other oversea ports, inland positioning between depots, and leasing. Related papers are those of Jordan and Turnquist [33], Choong et al. [14] and Di Francesco et al. [18], among others. A review of works that studied the empty container repositioning problem at a regional level can finally be found in [8].
The third research stream studies the use of RFID to simplify tracking and handling of RTIs. Equipping RTIs with RFID tags increases asset visibility and helps to identify which member of the supply chain holds which RTI in inventory. Hellström [24], for example, conducted a series of case studies to identify best practices in the implementation of RFID for managing and controlling RTIs. Hellström and Johansson [25] examined the impact of different control strategies on the management of returnable transport items. The results of a simulation study indicated that RTI shrinkage can be reduced by either using tracking systems or by implementing an appropriate control strategy. Related works are the ones of Johansson and Hellström [32] and Ilic et al. [29], who analyzed the impact of RFID on asset visibility and the efficiency of an RTI supply network.
2.2 Singlevendormultibuyer models
In recent years, research on inventory management experienced a gradual shift in perspective: while classical inventory models had a focus on individual companies, research nowadays more and more considers entire supply chains and tries to minimize inventoryrelated costs on the supply chain level. Early works along this stream of thought had a focus on twostage supply chains with a single actor on each echelon, while newer works studied more complex supply networks (see [21] for a recent review).
One important issue in coordinating a singlevendormultibuyer supply chain is the scheduling of the buyers’ replenishment cycles and the vendor’s production cycle. If the buyers order their individually optimal order quantities, which may result from applying the economic order quantity model, for example, discrete and unequally spaced depletions of the vendor’s inventory may occur. This makes it very difficult to calculate the average inventory at the vendor and to determine how much inventory should be kept to avoid shortages. To avoid this problem, Banerjee and Burton [3] suggested to implement a common replenishment cycle for each buyer and assumed that the buyers are only allowed to order at the beginning of each cycle. The production cycle of the vendor, in turn, was assumed to be an integer multiple of the replenishment cycle to allow the vendor to economize on setup costs. This paper was extended by Viswanathan and Piplani [48], who assumed that the vendor offers a price discount to the buyers to induce them to deviate from their individually optimal order quantities and to participate in the cooperation. Another extension was proposed by Siajadi et al. [45], who assumed that the production cycle of the vendor and the replenishment cycles of the buyers are of equal sizes. In addition, they assumed that the product is delivered in equalsized batches to the buyers and that the number of shipments may differ from buyer to buyer. Hoque [26] extended the work of Siajadi et al. [45] by studying three alternative shipment policies: two with equalsized batches and one with unequalsized ones. Kim et al. [37] developed a model that considered the procurement of raw material as well as the production and delivery of multiple items to multiple retailers. The authors considered the production sequence as a key decision factor for effectively coordinating procurement, production and delivery activities.
Implementing a single replenishment cycle for all buyers may be very restrictive in certain scenarios, for example, when there is a significant difference between the cost parameters of the buyers. To provide a more flexible solution for such situations, AbdulJalbar et al. [1] relaxed the assumption of a common replenishment cycle and assumed that the order interval of each buyer is an integer multiple of the order interval of the buyer who is scheduled before this buyer in the sequence of deliveries. Thus, by sorting the buyers and by calculating different replenishment intervals, the total costs of the system can be reduced. Chan and Kingsman [10] suggested a different coordination mechanism and implemented a basic cycle approach, where the replenishment interval of each buyer is restricted to integer multiples of the basic cycle. This coordination mechanism was adopted by Chan et al. [11] and Chan and Lee [12], among others.
Other authors extended these models by including further stages of the supply chain in their study, for example, distributors or raw material suppliers. Banerjee et al. [4], for instance, considered a supply chain consisting of a single manufacturer, multiple suppliers and multiple buyers. The production cycle of the manufacturer was assumed to be an integer multiple of the replenishment cycle of the buyers. BenDaya and AlNasser [6] developed a model of a threestage supply chain with multiple actors on each stage and assumed that the cycle time of each stage is an integer multiple of the cycle time of the adjacent downstream stage. Related works are the ones of Khouja [36], Wee and Yang [49], Jaber and Goyal [30] and Sarker and Diponegoro [43], among others. Other extensions included consignment stock policies [5, 7], deteriorating items [28, 52], learning effects in production [40], order cost reduction [50] and lead time reduction [27, 31].
2.3 Synthesis of both research streams
The literature review illustrates that research on returnable transport items and on supply chain inventory models has thus far been conducted widely independently of each other. Considering the management of reusable packaging material in an integrated inventory model, however, may lead to many benefits. If the production and distribution of finished products and the distribution and return of RTIs are adequately coordinated, this may lead to lower levels of RTI inventory in the supply chain and fewer stockout situations. Both aspects contribute to lower total system costs. The only work we are aware of that studied RTIs in an integrated inventory model is the one of Kim et al. [38], which, however, considered only a single vendor and a single retailer in modelling their supply chain. It is obvious that the planning problem changes if multiple retailers are considered, which requires that the sequence of deliveries to the retailers (and the sequence of returns from the retailers) is determined in addition. This paper extends the existing literature by developing a model that considers the coordination of finished products and RTIs in a supply chain consisting of a single vendor and multiple retailers.
3 Problem description
 1.
Endcustomer demand at the retailers is deterministic and constant over time.
 2.
The supplier uses one type of RTI for shipping products to the retailers.
 3.
After a shipment arrives at one of the retailers, the retailer empties the RTIs and subjects them to a cleaning and repair process. After a lead time of l_{ i } units of time, which is required for preparing the RTIs for their next usage, the empty RTIs arrive at the supplier.
 4.
The paper considers two types of production processes. If the first one is implemented, the supplier has to wait until the entire lot has been finished before shipments can be made from the lot. If the second one is used, the supplier can make shipments to the buyers while the production process is still in progress. Both production processes are representative for different scenarios and have often been studied in the literature. We refer to the first production process as the case of “late shipments” and to the second one as the case of “early shipments.”
 5.
To coordinate production and consumption, the supplier implements a common replenishment cycle for all retailers and replenishes each retailer exactly once per cycle.
Parameters
 n

Number of retailers
 p

Production rate of the supplier in units per year
 S

Setup cost in dollars per setup
 h _{ R }

Inventory holding cost for RTIs in dollars per unit per year
 h _{ F }

Inventory holding cost for finished goods in dollars per unit per year
 c _{ α }

Annual cost of managing a container, including depreciation and repair, per unit container capacity
 s

Scale factor for container capacity affecting the annual cost of managing a container
 d

Demand rate at the supplier in units per year, where $d={\sum}_{i=1}^{n}{d}_{i}$
 d _{ i }

Demand rate at retailer i in units per year
 h _{ i }

Inventory holding cost for finished goods at retailer i in dollars per unit per year
 A _{ i }

Ordering cost of retailer i in dollars per order
 l _{ i }

RTI return lead time of retailer i in years
Decision variables
 T

Cycle length in years
 r _{ i }

Amount of RTIs required for a single delivery to retailer i, where ${r}_{i}=\u2308\frac{{d}_{i}T}{\mathit{\alpha}}\u2309$
 Z

Shipment sequence
 Α

Container capacity in units, where α ∈ [α_{min}, α_{max}]
Other symbols
 d _{max}

Maximum demand rate at the retailers in units per year, i.e., ${d}_{\mathrm{max}}={\mathrm{max}}_{1\le i\le n}\left\{{d}_{i}\right\}$
 q _{ i }

Shipment quantity per delivery for retailer i in units, where q_{ i } = d_{ i }T
 Q

Production lot size in units, where $Q={\sum}_{i=1}^{n}{q}_{i}$
 l

Sum of RTI return lead times in years, i.e., $l={\sum}_{i=1}^{n}{l}_{i}$
 r _{max}

Maximum amount of available RTIs at the manufacturer, where r_{max} = max_{1≤i≤n}{r_{ i }}
 T _{max}

Maximum possible cycle time in years
 T _{min}

Minimum possible cycle time in years
4 Model 1: Late shipments
4.1 The objective function
The sequence of events is as follows: At the beginning of a cycle, the supplier starts producing the final product and finishes production after ${\sum}_{i=1}^{n}{q}_{i}/p$ units of time. Subsequently, a shipment to retailer i is made, who receives the batch and subjects the RTIs to a screening and repair process. After l_{ i } units of time, the empty RTIs arrive at the supplier, who despatches a shipment to one of the retailers who has not yet been served in this cycle (if any). Note that the sequence in which the retailers are served influences the total costs of the system, first by determining how many finished products are removed from the inventory of the supplier (this favors that large shipments are made early in the sequence) and second by determining how long the supplier has to wait before the next shipment can be made (this favors that shipments to retailers with short return lead times are made early in the sequence).
s denotes a scale parameter that determines how the container capacity impacts the cost of managing a container. If s > 1, then managing one capacity unit of a large container leads to higher costs than managing one capacity unit of a small container, which could be the case if large containers are more difficult to repair than small ones, for example. If s < 1, in contrast, then scale effects occur, and the cost of managing one capacity unit is reduced as the size of the container increases.
4.2 Solution of the model
If the shipment schedule is established according to Eq. (2), then we can assume that the shipment sequence has been established prior to assigning values to the other decision variables. By definition, the number of required RTIs should be an integer value. Due to this restriction, it is difficult to find a solution for the objective function analytically. Therefore, we relax the integral condition defined above, i.e., ${r}_{i}=\u2308\frac{{d}_{i}T}{\mathit{\alpha}}\u2309,\forall i$, in the following and treat the number of required RTIs as a continuous variable, i.e., ${r}_{i}=\frac{{d}_{i}T}{\mathit{\alpha}},\forall i$. Similarly, the value of r_{max} is relaxed as $\frac{{d}_{\mathrm{max}}T}{\mathit{\alpha}}$, where d_{max} = max_{1≤i≤n}{d_{ i }}.
To generate a feasible schedule, the cycle length should satisfy the condition $(\frac{Q}{p}+{\sum}_{i=1}^{n}{l}_{i})\le T$. The lower bound of the cycle length could thus be set to ${T}_{min}=\frac{{\sum}_{i=1}^{n}{l}_{i}}{(1\frac{d}{p})}$.
To find an optimal solution to the problem, it would be necessary to establish the Karush–Kuhn–Tucker conditions of Eq. (3) and the constraints on T and α. Due to the complexity of the objective function, this is not possible. We are, however, able to show that the objective function (3) is either convex or concave in T and α for given values of the respective other decision variable, depending on the parameter values. This enables us to find a solution for T and α that is at least locally optimal.
Optimal container capacity for different values of T and s
Case  Condition for T  Condition for s  $\frac{\mathit{\partial}TRC{\left(\mathit{\alpha},TZ\right)}_{{M}_{1}}}{\mathit{\partial}\mathit{\alpha}}$  $\frac{{\mathit{\partial}}^{2}TRC{\left(\mathit{\alpha},TZ\right)}_{{M}_{1}}}{\mathit{\partial}{\mathit{\alpha}}^{2}}$  α ^{*} 

1.1  $T>\frac{{\sum}_{i=1}^{n}{d}_{i}{l}_{i}}{{d}_{\mathrm{max}}}$  s ≤ 1  Always negative  Always positive  α^{*} = α_{max} 
1.2  1 < s < 2  Zero at α = α^{0}  Positive if α ≤ α^{ b } Negative if α > α^{ b }  Convex if α ≤ α^{ b } Concave if α > α^{ b } α^{*} = min (max (α_{min}, α^{0}), α_{max})  
1.3  s ≥ 2  Zero at α = α^{0}  Always positive  Convex, α^{*} = α^{0} α^{*} = min (max (α_{min}, α^{0}), α_{max})  
2.1  $T<\frac{{\sum}_{i=1}^{n}{d}_{i}{l}_{i}}{{d}_{\mathrm{max}}}$  s < 1  Zero at α = α^{0}  Positive if α ≥ α^{ b } Negative if α < α^{ b }  Concave if α ≤ α^{ b } Convex if α > α^{ b } ${\mathit{\alpha}}^{\ast}=arg{min}_{\{{\mathit{\alpha}}_{\mathrm{min}},{\mathit{\alpha}}_{\mathrm{max}}\}}TRC{\left(\mathit{\alpha},TZ\right)}_{{M}_{1}}$ 
2.2  1 ≤ s ≤ 2  Always positive  Always negative  α^{*} = α_{min}  
2.3  s > 2  Always positive  Positive if α ≥ α^{ b } Negative if α < α^{ b }  Concave if α ≤ α^{ b } Convex if α > α^{ b } α^{*} = α_{min}  
3.1  $T=\frac{{\sum}_{i=1}^{n}{d}_{i}{l}_{i}}{{d}_{\mathrm{max}}}$  s < 1  Always negative  Always positive  α^{*} = α_{max} 
3.2  s = 1  Always zero  Always zero  Any α ∈ [α_{min}, α_{max}]  
3.3  1 < s < 2  Always positive  Always negative  α^{*} = α_{min}  
3.4  s = 2  Always positive  Always zero  α^{*} = α_{min}  
3.5  s > 2  Always positive  Always positive  α^{*} = α_{min} 
For both Cases 1.2 and 2.1, we note that the inequality α^{0} < α^{ b } is always satisfied, as is shown in Appendix 3. In addition, a more detailed analysis of the patterns of $TRC{\left(\mathit{\alpha},TZ\right)}_{{M}_{1}}$ for Cases 1.2 and 2.1 is provided in Appendix 4.
We establish the following solution procedure to solve the proposed model:
Solution procedure
 Step 1.
(Shipment schedule)
First, the optimal delivery sequence within a single cycle, Z^{*}, is established by sorting the retailers according to the sequencing rule $\frac{{d}_{\left[i\right]}}{{l}_{\left[i\right]}}\ge \frac{{d}_{\left[i+1\right]}}{{l}_{\left[i+1\right]}},i=1,2,\dots ,n1.$
 Step 2.
(Container capacity and cycle length)
Secondly, iteratively determine both container capacity, α, and cycle length, T. In the initialization step, select α_{min} as a starting value for α. Step 2.1.
(Initialization step) k = 1, ${\mathit{\alpha}}_{(1)}^{\ast}={\mathit{\alpha}}_{\mathrm{min}}$
 Step 2.2.
If $\left{T}_{(k)}^{\ast}{T}_{(k1)}^{\ast}\right\le \mathit{\epsilon}$, then go to Step 3. Otherwise, go to Step 2.2.
 Step 2.1.
 Step 3.
(Shipment lot size)
Using both cycle length and container capacity, calculate the shipment lot size for finished goods and RTIs: ${q}_{\text{i}}^{\ast}={d}_{\text{i}}{T}^{\ast},{r}_{\text{i}}^{\ast}=\u2308\frac{{q}_{i}^{\ast}}{{\mathit{\alpha}}^{\ast}}\u2309,\forall i.$
5 Model 2: Early shipments
5.1 The objective function
5.2 Solution of the model
By comparing Eqs. (1) and (8), it can be seen that both cost functions differ only in a single cost term. This difference is caused by the different inventory carrying costs of the supplier in both models. As a result, the solution of both models is similar.
The solution procedure developed in Sect. 4.2 can now easily be adopted to find an optimal solution to Model 2.
Solution procedure
 Step 1.
(Shipment schedule)
Generate all possible shipment sequences using full enumeration. Run Step 2 and Step 3 consecutively and then select the best shipment schedule among the feasible sequences.
 Step 2.(Container capacity and cycle length)
 Step 2.1.
Select a shipment sequence and check its feasibility with the help of condition (10). If the shipment sequence violates this condition, repeat Step 2.1 for evaluating the next sequence.
 Step 2.2.
(Initialization step) k = 1, ${\mathit{\alpha}}_{(1)}^{\ast}={\mathit{\alpha}}_{\mathrm{min}}$.
 Step 2.3.
(Iteration step) k = k + 1.
First, calculate the optimal cycle length using the current value of ${\mathit{\alpha}}_{(k1)}^{\ast}$ with the help of Eq. (11). Secondly, obtain the optimal value for ${\mathit{\alpha}}_{(k)}^{\ast}$ from ${T}_{(k)}^{\ast}$ according to Table 1. If $\left{T}_{(k)}^{\ast}{T}_{(k1)}^{\ast}\right\le \mathit{\epsilon}$, then go to Step 3. Otherwise, go to Step 2.2.
 Step 2.1.
 Step 3.
(Shipment lot size)
Using both cycle length and container capacity, calculate the shipment lot size for finished goods and RTIs: ${q}_{\text{i}}^{\ast}={d}_{\text{i}}{T}^{\ast},{r}_{\text{i}}^{\ast}=\u2308\frac{{q}_{i}^{\ast}}{{\mathit{\alpha}}^{\ast}}\u2309,\forall i.$
6 Numerical studies
6.1 Illustrative example
Sample data set
Vendor  Retailer  

p  10,000  1  2  3  4  
S  60  d _{ i }  1,200  720  820  600 
h _{ R }  5.0  h _{ i }  8.0  7.4  8.2  8.1 
h _{ F }  5.2  A _{ i }  63  51  39  63 
c _{ α }  0.2  l _{ i }  0.009  0.008  0.007  0.008 
${\mathit{\alpha}}_{\mathrm{min}}=2,$${\mathit{\alpha}}_{\mathrm{max}}=30$, s = 2 
To analyze the benefit that results for the supply chain from coordination, we compare the costs of the integrated models with the case of noncoordination. In the noncoordinated models, we assume that the supplier determines the production–delivery policy without considering the retailers’ cost information (cf. Appendix 6 for the noncoordinated models).
 Case 1.
Late shipments
First, the case where the production lot has to be finished before a shipment can be made from the lot is considered. Using the proposed solution procedure for Model 1, the other decision variables of the optimal shipment policy were obtained as follows:
 Case 1.1.
Integrated model with late shipments (M1)
Z^{ * } = {1, 3, 2, 4}, α^{*} = 4.5132, T^{*} = 0.1219, q^{*} = (146, 88, 100, 73), r^{ * } = (33, 20, 23, 17) and $TR{C}_{{M}_{1}}=4,670.9.$
The results indicate that it is optimal to design containers with a capacity of approximately 4.5 units and to operate a total of 33 containers in the system to accommodate the retailers’ demand. Shipment quantities for each delivery to the retailers are set as above.
 Case 1.2.
Noncoordinated model with late shipments (N1)
Z^{ * } = {1, 3, 2, 4}, α^{*} = 4.4368, T^{*} = 0.1062, q^{*} = (127, 76, 87, 63), r^{ * } = (29, 18, 20, 15) and $TR{C}_{{N}_{1}}=4,713.9.$
As can be seen, in the case on noncoordination, smaller batch sizes are transported from the supplier to the buyers, which results in fewer containers that are operated in the system. The total cost of the system is higher as compared to the coordinated case.
 Case 2.
Early shipments
In a second step, we consider the case where shipments from a lot can be made while the production process is still in progress. Using the proposed solution procedure for Model 2, the other decision variables of the optimal shipment policy are obtained as follows:
 Case 2.1.
Integrated model with early shipments (M2)
Z^{ * } = {1, 2, 4, 3}, α^{*} = 4.4683, T^{*} = 0.1168, q^{*} = (140, 84, 96, 70), r^{ * } = (32, 19, 22, 16) and $TR{C}_{{M}_{2}}=4,261.0.$
The results indicate that it is optimal to design containers with a capacity of approximately 4.5 units and to operate a total of 32 containers in the system to accommodate the retailers’ demand. Thus, it becomes clear that in this example, the size of the containers is not changed; permitting early batch shipments, however, reduces total cost, which can be attributed to lower inventory carrying costs that result from earlier inventory depletions.
 Case 2.2.
Noncoordinated model with early shipments (N2)
Z^{ * } = {1, 3, 2, 4}, α^{*} = 4.4683, T^{*} = 0.1121, q^{*} = (135, 81, 92, 67), r^{ * } = (31, 19, 21, 16) and $TR{C}_{{N}_{2}}=4,269.8$.
Again, batch sizes are smaller in the noncoordinated case as compared to the coordinated case, and fewer containers are operated in the system. Again, noncooperation leads to an increase in total relevant cost.
6.2 Analysis of $TR{C}_{{M}_{2}}/TR{C}_{{M}_{1}}$
Parameter ranges used in the generation of the random data sets
S  h _{ R }  h _{ F }  c _{ α }  s  p 

[50, 60]  [2, 6]  [2, 6]  [0.1, 4.0]  [0.01, 5.0]  [1.5 × d, 3.0 × d] 
h _{ i }  d _{ i }  l _{ i }  A _{ i }  α _{min}  α _{max} 

[h_{ F } + 2, h_{ F } + 3]  [500, 1,500]  [0.001, 0.04]  [30, 70]  [1, 9]  [α_{min} + 20, α_{min} + 30] 
Results of a regression analysis between the model parameters and the ratio $TR{C}_{{M}_{2}}/TR{C}_{{M}_{1}}$
Parameters  Standardized beta  t  Significance 

S  0.607  20.663  0.000 
h _{ R }  0.037  4.968  0.000 
h _{ F }  −0.984  −20.108  0.000 
c _{ α }  −0.071  −16.652  0.000 
p  0.676  61.710  0.000 
s  −0.276  −66.031  0.000 
A _{1}  0.080  8.734  0.000 
A _{2}  0.060  6.637  0.000 
A _{3}  0.065  7.198  0.000 
A _{4}  0.060  6.564  0.000 
h _{1}  0.365  8.058  0.000 
h _{2}  0.321  7.056  0.000 
h _{3}  0.416  9.128  0.000 
h _{4}  0.344  7.600  0.000 
d _{1}  −0.161  −20.422  0.000 
d _{2}  −0.170  −21.454  0.000 
d _{3}  −0.164  −20.935  0.000 
d _{4}  −0.178  −22.582  0.000 
l _{1}  −0.042  −9.877  0.000 
l _{2}  −0.038  −8.872  0.000 
l _{3}  −0.029  −6.735  0.000 
l _{4}  −0.035  −8.127  0.000 
Adjusted R^{2} = 0.957 
For a practical application, this result implies the following: Typically, the question whether Model 1 or 2 can be implemented in practice is dictated by the technical requirements of the production process. However, these requirements can often be changed if an investment is made. Thus, based on our results, we recommend that companies check whether a shift to a production process that permits early batch shipments is profitable in case the demand rates of the retailers are high or the RTI return lead times are long. Such a scenario could arise, for example, if products are distributed internationally.
6.3 Analysis of the relative advantage of coordination
Results of a regression analysis between the model parameters and the ratio $TR{C}_{{N}_{1}}/TR{C}_{{M}_{1}}$
Parameters  Standardized beta  t  Significance 

S  0.447  85.871  0.000 
h _{ R }  0.016  12.428  0.000 
h _{ F }  −0.588  −67.846  0.000 
c _{ α }  0.004  5.041  0.000 
p  0.004  1.914  0.056 
s  0.004  5.165  0.000 
A _{1}  0.025  15.571  0.000 
A _{2}  0.025  15.454  0.000 
A _{3}  0.026  16.534  0.000 
A _{4}  0.025  15.740  0.000 
h _{1}  0.249  31.069  0.000 
h _{2}  0.229  28.408  0.000 
h _{3}  0.240  29.793  0.000 
h _{4}  0.234  29.231  0.000 
d _{1}  0.014  10.135  0.000 
d _{2}  0.014  9.921  0.000 
d _{3}  0.013  9.424  0.000 
d _{4}  0.013  9.079  0.000 
l _{1}  0.000  0.165  0.869 
l _{2}  0.002  2.152  0.031 
l _{3}  0.001  1.966  0.049 
l _{4}  0.001  1.318  0.188 
Adjusted R^{2} = 0.999 
Results of a regression analysis between the model parameters and the ratio $TR{C}_{{N}_{2}}/TR{C}_{{M}_{2}}$
Parameters  Standardized beta  t  Significance 

S  0.435  36.364  0.000 
h _{ R }  0.017  5.745  0.000 
h _{ F }  −0.582  −29.193  0.000 
c _{ α }  0.000  −0.204  0.838 
p  0.022  4.954  0.000 
s  −0.014  −8.521  0.000 
A _{1}  0.029  7.670  0.000 
A _{2}  0.027  7.343  0.000 
A _{3}  0.027  7.270  0.000 
A _{4}  0.035  9.447  0.000 
h _{1}  0.243  13.143  0.000 
h _{2}  0.223  12.019  0.000 
h _{3}  0.242  13.059  0.000 
h _{4}  0.245  13.322  0.000 
d _{1}  0.015  4.686  0.000 
d _{2}  0.012  3.717  0.000 
d _{3}  0.012  3.729  0.000 
d _{4}  0.011  3.361  0.001 
l _{1}  −0.004  −2.544  0.011 
l _{2}  0.002  1.057  0.290 
l _{3}  0.000  0.038  0.970 
l _{4}  −0.003  −1.514  0.130 
Adjusted R^{2} = 0.993 
As can be seen in Fig. 4, the relative efficiency of the early shipment policy is, on average, much higher than the relative efficiency of the late shipment policy.
7 Conclusions and suggestions for future research
This paper studied a supply chain consisting of a single vendor and multiple retailers that uses returnable transport items to facilitate shipping products from the vendor to the retailers. The paper developed models for two different shipment strategies and derived optimal solutions for the cycle time, the container size, the individual order quantities of the retailers and the shipment sequence with the intention to minimize the average total costs of the system. The behavior of the models was analyzed with the help of numerical examples.
The results of the paper indicate that the individual demand rates of the retailers and their respective RTI return lead times are critical for the performance of the models. It was shown that shipping batches to the retailers while the production process of the supplier is still in progress is especially beneficial in case the individual demand rates of the retailers are low (as compared to the supplier’s production rate) and the return lead times for RTIs are high. Further, the analysis indicated that the supplier should use few containers with a relatively large capacity if the individual demand rates are low and the RTI return lead times are high, and many containers with a relatively low capacity in the opposite case.
The model developed in this paper has limitations. First, it was assumed that the cost of keeping RTIs in inventory are independent of the capacity (and therewith size) of the containers. In practice, however, we may assume that different container sizes require different amounts of storage spaces (e.g., a 20′ container vs. a 40′ container), which would result in inventory carrying costs that depend on the capacity of the RTI. Secondly, this paper assumed that the capacity of an RTI can be chosen freely within given limits. In a practical application, however, the supplier may face a discrete number of different types of RTIs out of which he or she has to choose (e.g., in the case of maritime containers), or he or she may face restrictions imposed by a transport service provider. In such a case, it would be necessary to include these constraints in the optimization problem developed in this paper, or to test a given set of container sizes for optimality. We note, however, that our model could act as a heuristic in such a case, as it would give the decision maker an idea as to which container types could in principle be suitable candidates for the supply chain. Thirdly, we assumed in developing the proposed models that the supplier tries to minimize the overall amount of RTIs in the system and that only as many RTIs are kept in stock as are needed to ship the largest batch quantity to the retailers. A different strategy would be to increase the maximum RTI level in the system, which would enable the supplier to ship a second batch before the RTI return shipment of the first batch has been received. Furthermore, our analysis showed that the return lead time of empty RTIs is critical for the performance of both models. In a next step, it would be interesting to study different practices that help the buyers to return empty RTIs quicker to the supplier. In addition, it would be interesting to introduce a stochastic component into the model to analyze how long RTI return lead times influence the performance of the system if demand is uncertain or how production and distribution should be coordinated if the return lead times are stochastic themselves. Finally, the centralized models developed in this paper presuppose that a central planner exists that has access to cost parameters both at the supplier and the buyers. It is obvious that in a practical scenario, getting access to the required data may be very difficult. We note, however, that the cost reductions that can be obtained with the help of cooperative planning may act as an incentive for the parties involved to disclose the required information, as all parties that participate in the cooperation can improve their individual positions if the cooperation gain is adequately distributed. If the parties are not willing to cooperate, then incentive mechanisms, such as trade credits or quantity discounts, might have to be implemented. These, and other related issues, should be addressed in an extension of this paper.
Notes
Acknowledgments
The authors are grateful to the anonymous reviewers for their valuable comments on an earlier version of this paper.
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