Abstract-This paper deals with a lateral transshipment models involving two-echelon supply chain network, with a single supplier at the higher echelon and two retail locations at the lower. Lateral transshipment is considered as an option at each reorder decision under the periodic review standard (R, s, S) replenishment policy. The purpose of this paper is twofold. Firstly, a metamodel-based simulation optimization approach is applied to find the optimal values of s and S, for each retailer. Secondly, a series of simulation experiments are performed to find the best transshipment policy, in terms of smallest total cost and disservice rate. The tested policies are no pooling, complete pooling and various partial pooling policies according to what the threshold levels of physical stock are selected. An important finding is that each tested transshipment policy is considerably superior to a policy of no such transshipments, although at the expense of increased transportation activity. The best transshipment policy is such partial pooling with exactly s as the well-chosen value of the threshold level. Partial pooling is very interesting transshipment policy and should be further addressed in future research.
Keywords-emergency transshipment; pooling; periodic (R, s, S) replenishment policy; discrete event simulation; metamodel-based simulation; Desirability function approach
Introduction
A Supply Chain (SC) is a network of facilities and distribution entities such as materials vendors, manufacturers, distributors, wholesalers and retailers that performs the functions of procurement of raw materials, transformation of raw materials into intermediate and finished products and distribution of finished products to customers. A SC is typically characterized by a forward flow of materials and a backward flow of information. End user demand information suffers from delay and distortion as it moves upstream in a SC. This phenomenon has become well-known as Bullwhip effect.
Effective supply chain management (SCM) is currently recognized as a key determinant of competitiveness and success for most manufacturing and retailing organizations, because the implementation of SCM has significant impact on cost, service level, and quality. For instance, the coordination between organizations in the SC, through sharing of demand information, is a possible solution to counter the SC distortion. As a result, enterprises have shown a growing interest for an integrated SC management. An important issue in integrated logistic network management is to control the inventory at different entities while meeting end-customer service level requirements, therefore quantifying the trade-off between inventory investment and end-customer service levels.
Numerous strategies for archiving best SC performances have been proposed and investigated in both practice and academic over the past decades. One such strategy, commonly practiced in multi-location SC systems facing stochastic demand, allows movement of stock between locations at the same echelon level or even across different levels. These stock movements are termed lateral transshipments. As a demand occurs under the implementation of transshipment strategy, there will be three possible activities: the demand is met from the stock on-hand or it is met via transshipment from another location in the system or it is backordered.
In others words, the purpose of a transshipment is to realign inventory balances to ensure the right quantities are available in the right location to satisfy either expected demand or backorders. One location may have customer demand for an item but no inventory while another location may have one or more items on hand and no demand at neither present nor expected in the near future. Transshipment could then be used to transfer items from the location with inventory to the location that is out of stock in order to meet demand and effectively use inventory.
Transshipment research is motivated by observations from various industries. It has gained increasingly attention in medicine, apparel, and fashion goods, particularly by those retailers with brick and click outlets, or critical repairable spare parts of equipment-intensive industries such as airlines and complex machines [1].
Abundant literature is available on the topic of lateral transshipment. For an overviews examples, see [2, 3], where lateral transshipment literature is categorized in terms of the number of echelons, the number of items, the number of retailers (Stock points or locations), periodic or continuous review, replenishment policy (known as, inventory control policy), and the type of analysis done: exact or approximate evaluation, optimization or approximation. Although a significant amount of research has been done studying various aspects of lateral transshipments in inventory systems, most of it deals with continuous review (R, Q) and (S-1, S) replenishment policies or with periodic review (R, S) and (S-1, S) replenishment policies. The inventory control policy (R, s, S) has not received the same degree of attention. This work can be described as a two-echelon, two retailers, single-item, periodic review model with (R, s, S) replenishment policy with R = 1. In addition, as a good analysis tool to complex and dynamic systems, discrete event simulation (DES) is applied to multi-echelon inventory system to find the best transshipment policy.
The remainder of the paper is organized as follows: In the following section, a background and a literature overview of lateral transshipment policies, simulation-based approaches, and desirability function approach, are presented. Section 3 is devoted the two-steps solution methodology used in this study. Afterwards, section 4 presents the obtained simulation and optimization results. Lastly, conclusion is made in Section 5.
Background And Literature Overview
Lateral Transhipment policies
Two main strands of literature on lateral transshipments can be identified that differ in the timing of transshipments. Lateral transshipments can either be restricted to take place at predetermined times before all demand is realized, or they can take place at any time to respond to stock outs or potential stock outs. We will refer to these two types as proactive transshipment and reactive transshipment [2]. In reactive transshipment (known as emergency lateral transshipment) models, lateral transshipments are realized after the arrival of demand but before it is satisfied. If there is inventory at some of the stocking locations while some have backorder, lateral transshipments between stocking locations can work well. This kind of lateral transshipment is suitable in an environment where the transshipment costs are relatively low compared to the costs associated with holding large amounts of stock and with failing to meet demands immediately. In proactive transshipment (known as preventive lateral transshipment) models, lateral transshipments are used to redistribute stock amongst all stocking points in an echelon at predetermined moments in time. This can be arranged in advance and organized such that the handling costs are as low as possible. Since handling costs are often dominant in the retail sector, this type of lateral transshipment is most useful in that environment. Some authors combine reactive transshipment and proactive transshipment policies together (known as service level adjustments) to reduce the risk of stock outs in advance and efficiently respond to actual stock outs. In fact, emergency lateral transshipment responds to actual stock outs while preventive lateral transshipment reduces the risk of possible future stock outs.
A significant amount of literature in transshipment assumed that complete pooling policy is to be applied. This is part of the agreement between the cooperating companies. When the demand at a location cannot be met from on-hand inventory, it is met via transshipment(s) from other outlet(s) in a way that minimizes the transshipping cost. A unit demand is backordered if it cannot be satisfied via transshipment, in other words when there are no units in the system. In case companies do not want to share their last parts, one may introduce threshold parameters, known as partial pooling, and agree that a company does not supply a part by a lateral transshipment if the physical stock of the requested item is at or below the threshold level. A rule has to be added for how the values of the threshold parameters are chosen, or one may consider them as additional decision parameters. Reference [4] classified the transshipment policy as complete pooling and partial pooling for lateral transshipment.
As mentioned above, this paper can be described as a two-echelon, two retailers, single-item, periodic review model with (1, s, S) replenishment policy. It deals with simulation based-approach to study reactive lateral transshipments under periodic review and particularly in the case of two echelon centralized systems. Therefore, in the next sub-section, the literature review will be limited to previous researches which are similar to this paper.
Reference [5] evaluated the impact of four different emergency transshipment policies using the (s, S) inventory system, where an order is placed to bring the inventory level up to the desired maximum stock level (S) when the inventory on hand is equal to or less than reorder point (s). Reference [6] focuses on the sensitivity of the policy based on the variability within the demand distribution. Considering a two location system, with a redistribution point that is optimized using simulation for each specific case, they find that preventive transshipments are generally beneficial. However, major benefits are only obtained when demand is highly variable.
Moreover, References [7], [8] and [9] compare the performance of a proactive redistribution policy to a simple reactive transshipment method. In these studies it is assumed that replenishment orders are placed according to a periodic base-stock policy. Rather than focusing on what policy is best, [10] look at the cost structure of a system and when using transshipments would be beneficial. Reference [11] presents a specific model for repairable spare parts. In this model items can be repaired at a central base-depot which supplies the individual locations with the repaired items. These locations use a one-for-one ordering system to replenish their stocks. The demand processes is assumed Poisson and the replenishment policy is (s, S). A simulation based method is proposed to find optimal values for s and S.
Simulation and simulation-based metalodel
In SC modeling, the simplistic assumptions are necessary, nevertheless, for keeping the computations tractable in the process of finding optimal solutions, albeit at the expense of loss of realism. In contrast, in view of the complexities involved in the analytical modeling and solution of SC problems, some researchers in this area have attempted simulation approaches and/or heuristic approximations, in efforts to preserve at least some degree of realism in their analyses. Indeed, it is very difficult to develop mathematical models for multi-echelon inventory system, especially the system with complex interactions.
The computer simulation is just a model or a function that transforms the inputs into the outputs. The operational parameters and their variables are described as the inputs and the performances, which are derived from simulation, are described as the outputs. The operational conditions are then tested on this model to achieve the objectives. One objective of the application of simulation is to search for a set of operational parameters so that system performance is improved. Simulation is essentially a trial-and-error approach. It is merely a tool for problem solving; by itself, it cannot provide an answer. In addition to a good model, one also needs a sound technique to utilize the information from a simulation to make a decision. One such technique is named Simulation optimization. Several excellent surveys have been written on this topic and a different classification has proposed by various researchers such as [12] and [13]. Statistical selection methods, metamodel-based methods, stochastic gradient estimation based methods, and global search methods represents the most widely used methods for simulation optimization.
The most used methods are metamodel-based methods [13]; Indeed, a metamodel-based optimization strategy consists of choosing a metamodel form, designing an experiment to fit the metamodel, fitting the metamodel and validating the quality of its fit, optimizing the metamodel (or using it to provide a search direction), and checking the performance of the simulation at the metamodel-predicted optimum (or in the metamodel-determined search direction). A metamodel, or model of the simulation model, simplifies the simulation optimization in two ways: the metamodel response is deterministic rather than stochastic, and the run times are generally much shorter than the original simulation [14]. Various modeling forms have been introduced for metamodeling [15], such as Response surface metamodel, Regression spline metamodels, spatial correlation (kriging) metamodels, radial basis function metamodels and neural network metamodels.
Desirability function approach
The desirability function approach is one of the most widely used methods in industry for dealing with the optimization of multiple-response problems [16]. It is based on the idea that the quality of a product that has multiple quality characteristics is completely unacceptable if one of the characteristics lies outside the desired limits. This method assigns a score to a set of responses and chooses factor settings that maximize that score.
The first step in defining a desirability function is to assign values to the responses that reflect their desirability. The Multi-objective desirability optimisation method involves transformation of each predicted response, Å·, to a dimensionless partial desirability function, di, which includes the researcher's priorities and desires when building the optimization procedure. One or two-sided functions are used, depending on whether each of the n responses has to be maximized or minimized, or has an allotted target value [18]. If the response i is to be maximized the quantity di is defined as presented in (1). Likewise, di can be defined when the response is to be minimized or if there is a target value for the response. In (1), A and B are, respectively, the lowest and the highest values obtained for the response i, and wi is the weight. di ranges between 0, for a completely undesired response, and 1, for a fully desired response. In both cases, di will vary non-linearly while approaching the desired value. But with a weight of 1, di varies linearly.
(1)
(2)
In this work, we chose weights equal to 1 for all responses. The partial desirability functions are then combined into a single composite response, the so-called global desirability function D, defined as the geometric mean of the different di values as indicated in (2).
(2)
A value of D different from zero implies that all responses are in a desirable range simultaneously and, consequently, for a value of D close to 1, the combination of the different criteria is globally optimum, so the response values are near the target values. Note that the use of desirability requires the designation of performance targets. In addition, maximizing desirability is a multi-objective optimization problem.
In this work, response optimizer tool of the Minitab 14 software package is used for finding optimal values s and S of the periodic (1, s, S) replenishment policy. The search of the maximum desirability function is iterative which is based on reduced gradient search algorithm with multiple starting points. Detailed description of this local search algorithm can be found in [19].
Solution Methodology
The supply network studied in this paper can be described as a two-echelon, two retailers, single-item, periodic review model with (s, S) replenishment policy. The problem is about how choose the values of s and S of each retailer, and what is the best transshipment policy would be implemented.
The proposed solution methodology is described through Fig. 1. It can be divided successively in two steps. The objective of the first step is to apply a metamodel-based simulation optimization approach for finding the optimal values of s and S, which will be apply by each retailer. For this first purpose, a DES model was firstly implemented as a tool in estimating total cost and disservice performances. Secondly, Factorial Design of Experiment (FDoE) is applied to conduct simulation experiments. Finally, multi-objective optimisation is achieved by applying the desirability function approach. The objective of the second step is to apply a simulation-based methodology to assess the impact of various lateral transshipment policies between two competing retailers, which are relatively close to each other compared to the supplier. Each retailer use the periodic review standard (s*, S*) replenishment policy.
(1) SO based metamodel applied to (s, S) replenishment policy
(2) Simulation of various lateral transshipment policies
The best lateral transshipment policy
(for the SC network)
Optimal values for s* and S*
(for each retailer)
Architecture of the SC simulation model
The (s, S) replenishment policy settings
Stochastic inventory model have received considerable attention in inventory literature. We consider one of the most common practical stochastic inventory control problems, known as the (s, S) model. This model (with s < S) is a model of an inventory management (or control) system in which the inventory I is replenished whenever it decreases to a value smaller than or equal to the reorder level s; the order quantity Q is such that the inventory is raised to the order-up-to level S:
(3)
The model and some of its variants have been analyzed by several studies. For example, review of the inventory (I in (3)) may be either continuous (in real time) or periodic. The lead time of the order may be either a nonnegative constant or a nonnegative random variable. Random demand (say) D that exceeds the inventory at hand (so D > I) may be either lost or backlogged. Costs may consist of inventory, ordering, and out-of-stock costs. These cost components are specific mathematical functions; for example, inventory carrying (or holding) cost may be a constant per item unit, per time unit. In practice, however, out-of-stock costs are hard to quantify so a service (or fill rate) constraint may be specified instead. For instance, the expected fraction of total demand satisfied from stock on hand should be at least 90% (a disservice level 10 %).
The following assumptions are used in the set of experiments conducted by several research teams. These assumptions were selected also by [20] and [21].
Demands are exponentially distributed with mean 100.
Lead times are Poisson distributed with mean 6 (so the probability of order crossing is relatively high).
The maximum disservice level c is 0.10.
The holding cost is 1, the variable ordering cost is 1, and the fixed ordering cost is 36.
The tested transshipment policies
It should be noted that, complete pooling policy is when the demand at a retailer cannot be met from on-hand inventory; it is met via transshipment(s) from the other retailer in a way that minimizes disservice levels. Partial pooling policy is when retailer does not supply a part by a lateral transshipment if the physical stock of the requested item is at or below the threshold level. In this study four different scenarios are evaluated: (S1) without transshipment; (S2) complete pooling policy, (S3) partial pooling policy with threshold level is equal to s*, (S4) partial pooling policy with threshold level is equal to s*/2, and (S5) partial pooling policy with threshold level is equal to s*/4.
The demand experienced by a retailer is fulfilled from its existing stock. When the on hand inventory quantity at a retailer reaches its reorder point, a replenishment order is placed. When the on-hand inventory quantity at a retailer reaches its reorder point, the stock level at other retailer is checked. If the stock level of other retailer is more than a predetermined threshold level (the stock level above which the stock can be transferred from one retailer to another retailer), the order is placed on to the other retailer.
The main simulation model
The Arena simulation software was used to develop the simulation model of the supply network. Arena, developed by Rockwell Automation, is a simulation and automation software based on SIMAN processor and simulation language. Fig. 2 shows the simulation model using the software Arena 10. The main portion of the model's operation will consist of logic sub-models to represent the (s, S) replenishment policy for each retailer and the transshipment policy parameters.
Retailer 1
Transshipment policy (S1, or S2, or S3, or S4, or S5) parameters:
Supplier
(s*, S*) replenishment policy
Retailer 2
(s*, S*) replenishment policy
Architecture of the SC simulation model
Simulation Results
Step 1: the optimal (s, S) replenishment policy
Befeore simulation results collection and analysis, it is essential that the steady state is reached. As shown, in Fig. 3., the steady state is rapidly established before less than 200 days. Therefore, the warm-up period is insignificant compared to the simulation length 300, 000 days.
Steady state
Warm-up period
Average stock level behavor during 1000 days simulation
The goal of FDoE investigation is to obtain information as efficiently as possible. An experiment is a series of planned trials in which factors (s and S) that are thought to affect the outcome are varied systematically and the outputs (Total cost and disservice) are measured and recorded. In this study, we have chosen, for each variables s and S, two levels. For s, level 1 consists in 500 units, and level 2 consists in 1000 units. For S, level 1 consists in 1000 units, and level 2 consists in 2000 units. Several simulation runs were made for each SC configuration, each run length is fixed at 300 000 days. The result of these runs is shown in Table I. In this paper, the "total inventory cost" means the sum of inventory cost and ordering cost.
The 22 Factorial Design Configurations Of (s, S) Replenishment policy
Exp.
Factors levels
Total inventory cost
Disservice
s
S
1
500
1000
426.9
18.8%
2
500
2000
932.8
8.9%
3
1000
1000
564.9
4.4%
4
1000
2000
1060.6
2.8%
After planning the experiments and identifying the most important factors of the model, these factors are used as input data for multi-objective desirability optimization. This optimization tool is integrated in Minitab software. Applying (1) for each response measure, we obtain in optimal configuration that the individual desirability for each performance measure (Inventory cost and disservice) is equal to 1.
The response optimization consists in determining how the solution has satisfied the combined goals for all the responses. Composite desirability has a range of zero to one. One represents the ideal case; zero indicates that one or more responses are outside their acceptable limits. Composite desirability is the weighted geometric mean of the individual desirability for the responses as presented in (2). The composite desirability for all these two variables is 1. To obtain this desirability, we would set the factor levels at the values shown under global solution in the Fig. 4. That is, each retailer would set its s* level at 905 units, and its S* level at 1033 units.
The response optimization consists in determining how the solution has satisfied the combined goals for all the responses. Composite desirability has a range of zero to one. One represents the ideal case; zero indicates that one or more responses are outside their acceptable limits. Composite desirability is the weighted geometric mean of the individual desirability for the responses as presented in (2). The composite desirability for all these two variables is 1. To obtain this desirability, we would set the factor levels at the values shown under global solution in the Fig. 3. That is, each retailers would set s at 905 units, and S at 1033 units.
Multi-objective optimization based on desirability functions
Step 2: The best transshipment policies selection
Each configuration (S1, S2, S3, S4, and S5) is modeled and simulated a period of 300,000 days. In addition of the previous assumptions, the following ones are used in this second step:
Each retailer use (s*, S*) replenishment policy, which is optimized in the first step.
The transshipment times are negligible.
The variable transshipment cost is 1.
As presented in Table II, all tested emergency transshipment policies are beneficial for the supply chain. In fact, the average rate disservice of the two retailers is smaller when transshipment policy where used and not exceeds 2.5%. The extra cost due to each transshipment policy ranges from 9% to 20%. It should be noted that the best transshipment policy is the partial pooling with a threshold level s*. In this perfect case, the disservice level is null. However, it is trivial that the transshipment cost is the largest. This numerical case study shows the importance of transshipment in supply chains. An emergency transshipment strategy represent one way in which logistics managers can maintain inventories cost while simultaneously reducing customer disservice rates. It seems that, the findings of this study are obviously valid only under the specific operating assumptions of the model studied in this paper. Nevertheless, it is clear that systems with partial pooling are more difficult to control and optimize than systems with complete pooling, as there is the additional managerial decision of how much inventory to reserve as the threshold level.
Simulation Results of The Five Tested Transshipment Policies
(S1)
Without transshipment
With transshipment
(S2)
Complete pooling
Partial pooling
(S3) TL is s*
(S4) TL is s*/2
(S5)
TL is s*/4
Retailers inventory cost
1107.9
1039.3
1107.9
1064.1
1042.8
Transshipment cost
0
178.1
230.7
217.5
191.7
Total cost
1107.9
1217.4
1338.6
1281.6
1234.5
Disservice
5.55%
2.45%
0 %
0.11%
0.64%
TL: threshold level
Finally, transshipments increase transportation costs; therefore, the impact of this increase needs to be considered in determining the cost-effectiveness of variance reduction through transshipments. The chosen of each unit cost have a significant effect in results. Extending the proposed approach in this direction is our interesting research perspective.
Conclusion
The purpose of this paper is twofold. Firstly, a metamodel-based simulation optimization approach is applied to find the optimal values of s and S, for each retailer. Secondly, a series of simulation experiments are performed to find the best transshipment policy, in terms of smallest total cost and disservice rate. The studied network is a two-echelon supply chain network, with a single supplier and two retailers. Each retailer use the periodic review standard (s, S) replenishment policy. An important finding is that each lateral transshipment policy is considerably superior to a policy of no such transshipments, albeit at the expense of increased transportation activity. The best transshipment policy is such partial pooling with a well-chosen value of the threshold level.
We considered the usage of a periodic review system for this study, but additional studies of lateral transshipments are needed to include the characteristics that incorporate inventory policy systems according to the characteristics of products in the supply chain. There are many variations of that model which present practical interests and constitute potential topics of future research. Some of the most extensions are: non-negligible transshipment times; different costs at each base; dependent demand; more than two retailers; etc.
