The horizontal interaction between retailers, coupled with replenishment rules and time delays, makes the dynamics in supply chain systems highly complicated. This paper aims to explore the impacts of lateral transshipments on the stability, bullwhip effect, and other performance measurements in the context of a two-tiered supply chain system composed of one supplier and two retailers. In particular, we developed a unified discrete-time state space model to address two different scenarios of placing orders. Analytical stability results are derived, through which we found that inappropriate lateral transshipment policies readily destabilize the supply chain system. Moreover, the lead time of lateral transshipments further complicates the stability problem. Theoretical results are validated through simulation experiments and the influences of system parameters on performance measures are investigated numerically. Numerical simulations show that lateral transshipments help improve the customer service level for both retailers. It is also interesting to observe that the demand of the two retailers can be satisfied even if only one retailer places orders from the upstream supplier.
The dynamics of a supply chain system has grown increasingly complicated due to numerous factors, such as global economic downturn, e-commerce development, and disruptive events. These factors bring great challenges to supply chain management in uncertain environments. In uncertain conditions, it is usually very hard to derive analytically optimal policies in maximizing the benefits for the whole supply chain system over a long duration. However, understanding how different factors affect the dynamic complexities of supply chain systems will be very useful for guiding the selection of parameters. To deal with demand variability and reduce the risk of stock-out, one of the options is to implement collaborative programs between supply chain members by exploiting advanced information technologies, such as Vendor Managed Inventory (VMI) and Collaborative Planning Forecasting and Replenishment (CPFR) [
The benefits of lateral transshipments have been well discussed, such as reducing inventory cost and improving customer service level [
The majority of the existing literatures on lateral transshipments have hitherto focused on deriving optimal or suboptimal transshipment policies as well as replenishment decisions under specific assumptions [
Actually, a supply chain system is a dynamical system by nature, because its inventory and order fluctuate over time. The dynamics of supply chain systems, such as the bullwhip effect [
To bridge the aforementioned gap, this paper focuses on exploring the dynamic complexities in a supply chain system with lateral transshipments between two retailers. Specifically, we focus on how the vertical replenishment policy and horizontal lateral transshipment policy affect the fluctuations of inventory and order. We derived the stability conditions for the supply chain system in two different scenarios. In the first scenario, each of the two retailers places orders from the upstream supplier. In the second scenario, only one retailer assumes the task of placing orders. This scenario is motivated by our real-life observations in both electrical and PC retail industries in China. To reduce the ordering cost, stock-out cost, and inventory holding cost, the retailers in such industries may satisfy customer demand just by lateral transshipments from nearby retailers without placing orders with upstream suppliers. This scenario can be very effective in reducing inventories and occurs frequently in two situations: either a retailer attempts to sell new products or customers want to buy new products but their retailer does not have temporarily. The most critical contribution of this research lies in our provision of both delay-dependent and delay independent stability conditions through analytical study, from which we show that lateral transshipments make the dynamics of supply chain systems more complicated. These theoretical results are significant for the selection of both replenishment parameters and later transshipment parameters in achieving better performance. Through simulation experiments, we demonstrate the advantages of lateral transshipments in improving demand satisfaction and mitigating the bullwhip effect. Furthermore, an interesting observation is that both retailers can better fulfill customer demand even when one of them places orders from the upstream supplier. The derived results are helpful in providing general guidelines on system planning and operation.
The structure of this paper is organized as follows. In Section
Consider a two-echelon supply chain system comprised of one external supplier and two retailers with lateral transshipments. The structure of the supply chain system is depicted by Figure
The structure of a supply chain system with lateral transshipments.
During each period
We consider two different scenarios in our model: scenario
For simplicity, we assume that the replenishment lead time for the two retailers to place orders with the upstream supplier is only one period. We assume that lateral transshipments incur a fixed lead time of
We shall firstly develop a state space model based on difference equations corresponding to the event sequence introduced before. We use
Due to its excellence in short-term predictions [
The replenishment rule is one of the core factors underlying system performance. In general, inventory decisions should be made accounting for demand trend, on-hand inventory, stock-out, and pipeline stock. In our model, however, there is no pipeline stock because we assume that replenishment lead time is only one period. Thus, the inventory rule is viewed as a function of demand forecast and inventory level. The inventory polices corresponding to the aforementioned two scenarios are quite different.
Recall that the two retailers assume their tasks of placing orders independently in the first scenario, while only retailer 1 places order with the upstream supplier in the second scenario. In the first scenario, the inventory policy for the two retailers is represented as
In the second scenario, only retailer 1 makes replenishment decisions based on the systematic inventory, which is the sum of the two retailers’ inventory levels:
Based on the above difference equations, we develop a unified state model for the two scenarios by substituting the inventory policies (
Lateral transshipments critically affect the system performance by taking role in redistribution of inventory to the retailers. In this research, we will consider the average total inventory cost (
The average total inventory cost (
The average ordering cost for the supply chain system is computed by
The average lateral transshipment cost for the two retailers is obtained by
By using the sign function introduced previously, the service level for the two retailers is defined as
The bullwhip effect refers to the amplification of demand fluctuations as one moves up a supply chain from downstream to upstream [
In this section, we analyze the dynamic complexities of the state space model (
Analyze the steady states of the inventories of the two retailers, which essentially determine the equilibrium points due to the interaction between supply chain members [
Derive the stability conditions for the two scenarios. As investigated in the literature, stability is a fundamental problem for any dynamic systems including inventory systems [
Without loss of generality, we assume that the steady states of the demands of the two retailers are expressed as
Firstly, consider the scenario
In the following, we will consider the second scenario
According to the scenarios aforementioned, the supply chain system can be analyzed under 5 different cases.
There exist no lateral transshipments and the two retailers make replenishment decisions independently. This case is selected for comparison and highlights the dynamic complexities brought by lateral transshipments.
The retailers also make independent inventory decisions, and the lateral transshipments are incorporated. Note that the lead time of lateral transshipments is neglected as in [
The retailers also make independent inventory decisions and the lateral transshipments are incorporated. However, we consider the impact of the lead time in lateral transshipments on the stability problem (
In this case, we focus on scenario
In this case, we focus on scenario
It is noteworthy that we consider 3 cases for
In the following, we will perform detailed stability analysis for the above 5 cases.
Assume that
We firstly consider
From Theorem
Assume that
Using
Using Rouches’ Theorem [
The stability conditions in Theorems
Stable boundaries in scenario
Exact stable boundaries when
Stable boundaries independent of
Figure
When
The proof for Theorem
Under the scenario
For the scenario
Theorem
This section will validate the stability results. The customer demand of the two retailers is exogenous and thus exerts no effect on system stability. The step signal is selected to test the dynamics because of its wide use in studying the dynamics and system performance of supply chain systems. For example, the step signal has been used to model sudden changes in customer demand after price promotion activities [
We mainly validate Theorems
Simulation designs for stability validation.
Simulation design | Scenario |
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Theorem |
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Design 1 |
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1.5 | 1.5 | - | 0.2 | 0 | Theorem |
Design 2 |
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1.5 | 1.5 | - | 0.3 | 0 | Theorem |
Design 3 |
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0.8 | 1.2 | - | 0.2 | 2 | Theorem |
Design 4 |
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0.8 | 1.2 | - | 0.2 | 12 | Theorem |
Design 5 |
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0.8 | 0.6 | - | 0.4 | 2 | Theorem |
Design 6 |
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0.8 | 0.6 | - | 0.4 | 12 | Theorem |
Design 7 |
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- | - | 0.6 | 0.12 | 5 | Theorem |
Design 8 |
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- | - | 0.6 | 0.12 | 6 | Theorem |
Design 9 |
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- | - | 0.6 | 0.12 | 7 | Theorem |
Stability validation for Theorem
Theorem
Stability validation for Theorem
Stability validation for Theorem
The question of stability is of fundamental significance for the selection of parameters. To further improve system performance, we will numerically disclose the impacts of parameters on the measurements introduced in Section
The computational results for our performance measurements are shown in Table
The computational results of performance measurements in scenario
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0 | 0 | 0 | 13.41 | 66.67 | 0 | 0.99 | 0.99 | 1.05 |
0 | 0 | 0.05 | 13.32 | 66.69 | 0.04 | 0.99 | 0.99 | 1.00 |
0 | 0 | 0.15 | 13.32 | 66.69 | 0.11 | 0.99 | 0.99 | 0.94 |
0 | 0 | 0.25 | 13.32 | 66.69 | 0.18 | 0.99 | 0.99 | 0.91 |
0 | 0 | 0.35 | 13.32 | 66.69 | 0.25 | 0.99 | 0.99 | 0.90 |
0 | 0 | 0.45 | 13.32 | 66.69 | 0.35 | 0.99 | 0.99 | 0.94 |
1 | 1 | 0 | 14.13 | 66.67 | 0 | 0.93 | 0.94 | 1.05 |
1 | 1 | 0.05 | 14.13 | 66.65 | 0.04 | 0.93 | 0.94 | 1.04 |
1 | 1 | 0.15 | 14.12 | 66.65 | 0.11 | 0.93 | 0.94 | 1.02 |
1 | 1 | 0.25 | 14.11 | 66.65 | 0.18 | 0.93 | 0.94 | 1.01 |
1 | 1 | 0.35 | 14.10 | 66.65 | 0.26 | 0.93 | 0.94 | 1.01 |
1 | 1 | 0.45 | 14.11 | 66.65 | 0.36 | 0.93 | 0.94 | 1.02 |
1 | −1 | 0 | 14.09 | 66.68 | 0 | 0.93 | 0.94 | 1.05 |
1 | −1 | 0.05 | 14.034 | 66.67 | 0.10 | 0.93 | 0.94 | 0.97 |
1 | −1 | 0.15 | 13.94 | 66.67 | 0.28 | 0.94 | 0.95 | 0.87 |
1 | −1 | 0.25 | 13.89 | 66.67 | 0.45 | 0.94 | 0.95 | 0.81 |
1 | −1 | 0.35 | 13.88 | 66.67 | 0.64 | 0.95 | 0.95 | 0.80 |
1 | −1 | 0.45 | 13.93 | 66.67 | 0.88 | 0.94 | 0.94 | 0.86 |
The impact of
Table
The results on system performance in scenario
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0 | 0 | 0.05 | 150.01 | 40 | 3.33 | 1 | 0 | 0.94 |
0 | 0 | 0.15 | 20.08 | 40 | 3.33 | 1 | 0.92 | 0.94 |
0 | 0 | 0.25 | 19.99 | 40 | 3.33 | 1 | 1 | 0.94 |
0 | 0 | 0.35 | 20 | 40 | 3.33 | 0.99 | 1 | 0.94 |
0 | 0 | 0.45 | 20.07 | 40 | 3.33 | 0.97 | 1 | 0.94 |
1 | 1 | 0.05 | 150.08 | 40.01 | 3.33 | 1 | 0 | 1.73 |
1 | 1 | 0.15 | 25.75 | 40.01 | 3.33 | 0.99 | 0.58 | 1.73 |
1 | 1 | 0.25 | 20.39 | 40.01 | 3.33 | 0.98 | 0.96 | 1.73 |
1 | 1 | 0.35 | 20.60 | 40.01 | 3.33 | 0.91 | 0.99 | 1.73 |
1 | 1 | 0.45 | 21.25 | 40.01 | 3.33 | 0.82 | 0.99 | 1.73 |
1 | −1 | 0.05 | 150.31 | 39.99 | 3.34 | 1 | 0 | 0.15 |
1 | −1 | 0.15 | 24.81 | 39.99 | 3.34 | 0.99 | 0.58 | 0.15 |
1 | −1 | 0.25 | 20.17 | 39.99 | 3.34 | 0.98 | 0.99 | 0.15 |
1 | −1 | 0.35 | 20.55 | 39.99 | 3.34 | 0.92 | 0.99 | 0.15 |
1 | −1 | 0.45 | 21.19 | 39.99 | 3.34 | 0.83 | 1 | 0.15 |
This paper aims to explore the dynamic complexities of a supply chain system with lateral transshipments. The horizontal transshipments between two retailers, coupled with the replenishment policies and lead time, render the structure of the whole system. These inventory rules culminate in multiple feedback loops and lead to dynamic complexities in terms of stability and system performance. To understand the complexities in such a system, we developed a unified state space model to incorporate two different scenarios. Based on the state space model, we analyzed the steady state of the supply chain system and analytical stability conditions are derived. The stability results demonstrate that lateral transshipments complicate the system dynamics more than do the traditional supply chain systems. Exactly, increasing the magnitude of lateral transshipments easily destabilizes a supply chain system, especially in the case with a long lead time of lateral transshipments. In the second scenario, we discerned the interesting observation that the demand of the two retailers can be satisfied even though only one retailer places orders with the upstream supplier.
Through simulation experiments, we have validated our theoretical results using a step signal demand. Based on the stability results, we selected different settings of parameter to disclose the advantages of lateral transshipments. The simulation results reveal that, in a decentralized supply chain, lateral transshipments improve customer service level in the scenario where each of the two retailers places orders with the upstream supplier. Improving customer service level is one of the main motivations for lateral transshipments, which has been validated widely in the existing literature [
Our periodic model is very general by incorporating 5 different cases in two scenarios without imposing any specific assumptions in demand. The results obtained are applicable to different industries, such as electrical and PC industries. In particular, we observe that the second scenario
This research can be extended from several aspects. One interesting problem is that replenishment lead times can be included in our model, which will further complicate the dynamics. Another interesting subject is the dynamic complexities of a supply chain network with multiple retailers and multiple suppliers and sees how demand uncertainty propagates along a network system.
Index of periods
Index of retailers
The set of scenarios:
Lead time in lateral transshipments
The parameter to determine the magnitude of lateral transshipments
The smoothing coefficient in the exponential smoothing algorithm
The replenishment parameter to adjust the inventory discrepancy of retailer
The replenishment parameter to adjust systematic inventory discrepancy in scenario
Inventory holding cost per unit
Stock-out cost per unit
Ordering cost per unit
Lateral transshipment cost per unit
The amount of the customer demand of retailer
Inventory level of retailer
The amount of the order placed by retailer
The amount of the demand forecast by retailer
The amount of lateral transshipment received by retailer
Bullwhip effect of the supply chain system
Average total inventory cost
Average total ordering cost
Average lateral transshipment cost
Customer service level of retailer
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the National Natural Science Foundation of China (nos. 71401181, 71701213, and 71501151) and the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (nos. 14YJC630136 and 15YJC630008).