This model extends a two-echelon supply chain model by considering the trade-credit policy, transportations discount to make a coordination mechanism between transportation discounts, trade-credit financing, number of shipments, quality improvement of products, and reduced setup cost in such a way that the total cost of the whole system can be reduced, where the supplier offers trade-credit-period to the buyer. For buyer, the backorder rate is considered as variable. There are two investments to reduce setup cost and to improve quality of products. The model assumes lead time-dependent backorder rate, where the lead time is stochastic in nature. By using the trade-credit policy, the model gives how the credit-period would be determined to achieve the win-win outcome. An iterative algorithm is designed to obtain the global optimum results. Numerical example and sensitivity analysis are given to illustrate the model.
Supply chain indicates that the relation among the supply chain players is forever to obtain maximum profit together and individual profit always. The aim of this model is to reduce the total supply chain cost.
Supply chain management (SCM) is the collaboration among suppliers, manufacturers, retailers, and customers. Practically, the aim of the SCM model is to minimize the total cost or to maximize the total profit throughout the channel. In this direction, the idea of integrated vendor-buyer inventory management has been successfully considered since last few decades. In some practical situations, lead time and setup cost can be controlled and reduced in various ways. It is a trend by shortening the lead time and reducing setup cost; the safety stock can be minimized. Thus, the target is always to decrease the stockout loss and improve the service level for the customer as to increase the competitive edge in business within the SCM environment. Thus, the controllable lead time and setup cost reduction are the key concepts to obtain successful business and have attracted extensive research attention [
Ouyang et al. [
In reality, transportation cost is not always constant. But, many papers used the concept of constant transportation costs. Thus, it is too much important to consider the cost as variable. By using the single-setup-multidelivery (SSMD), the number of types of transportation increases always. The primary aim of using SSMD policy is to reduce the holding cost of buyer, but as a result, the transportation cost increases. Therefore, there will be a trade-off between them to control the cost of the whole system. Ganeshan [
Nowadays transportation mode selection with positive manufacturing lead time is more effective in SCM system [
In real life situation, everybody prefers the best quality of products with the cheapest price. As a result, all industries have to make good quality products with at least cheaper price. That is why, in many cases, some investment can be done to reduce the setup cost and improve the quality of products. In this direction, Ouyang et al. [
In this highly competitive business environment, companies always desire for trade-credit policy for the entire customers. Thus, trade-credit plays an important role in modern business system. Vendors offer trade-credit-period to buyers to encourage sales, promote market shares, and reduce on-hand stock. Goyal [
Jaber and Osman [
In some inventory system, such as fashionable items, the length of the waiting time for the next replenishment would determine whether the backorder is accepted or not. Therefore, backorder rate is variable and dependent on the waiting time for the next replenishment [
This paper illustrates a channel coordination mechanism between transportation discounts, trade-credit financing, number of shipments, quality improvement of products, and reduced setup cost in a two-echelon supply chain model. Table
Distinction between previous and this model.
Author(s) | Supply chain model | Variable lead time | Variable backorder | Setup cost reduction | Quality improvement | Transportation discounts | Trade-credit financing |
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Porteus, 1986 | √ | √ | |||||
Ouyang et al., 1996 | √ | ||||||
Moon and Choi, 1998 | √ | ||||||
Hariga and Ben-Daya, 1999 | √ | ||||||
Ouyang and Chuang, 2001 | √ | √ | |||||
Ouyang et al., 2002 | √ | √ | √ | ||||
Lee, 2005 | √ | √ | |||||
Lin, 2008 | √ | √ | |||||
Sarkar and Majumder, 2013 | √ | √ | √ | ||||
Sarkar and Moon, 2014 | √ | √ | √ | √ | |||
Sarkar et al., 2014 | √ | √ | √ | ||||
Sarkar et al., 2015 | √ | √ | |||||
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The following notation are used to develop the model.
The following assumptions are considered to formulate this model. These assumptions are mainly adopted from Sarkar and Moon [ The study considers a supply chain model for a single type of products with the single-setup-multidelivery (SSMD) policy and controllable lead time. The lead time Let This model considers the variable backorder rate Logarithmic expressions are assumed for both quality improvement and setup cost reduction (Porteus, [ The trade-credit financing is considered to make it a cost-reduced supply chain. The supplier provides a transportation cost discount, when the buyer places the order of
The model considers the single-setup-multidelivery (SSMD) policy in a single-supplier single-buyer supply chain model. If the buyer orders quantity
If the inventory level reaches the reorder point
The model assumes that the lead time demand
The concept of Ouyang et al. [
In reality, the fixed or constant backorder rate is very rare and it is found only in case of life saving drugs, costly products, or others. But for any low-cost products, it is generally variable. Thus, based on lead time of this model, we use the concept of Sarkar and Moon [
Thus, total expected cost per unit time for the buyer, considering the partial backorder, can be expressed as
Using the above, the expected shortage at the end of the cycle can be expressed as
In this model, under the SSMD policy, the cycle length for supplier is
Inventory pattern under the SSMD policy (see for reference Ouyang et al. [
In this model, there are two investments to reduce the total supply chain cost to make the supply chain more profitable. An investment is used to improve the quality of products and another investment is used to reduce setup cost. We consider the concept of Porteus [
Using the concept of defective items, the expected annual total cost is
Therefore, the total expected cost per unit time for supplier can be expressed as follows:
To make the profitable supply chain, an attempt of trade-credit policy is used. By using the trade-credit policy, buyer saves his/her total interest during the credit-period and the supplier lost opportunity cost. We define the trade-credit cost for buyer offered by the supplier as follows:
Nowadays, for highly competitive business market, transportation cost is a major issue of the total operational cost in SCM. For appropriate incorporation of transportation cost into the total annual cost function, it should identify the exact transportation cost which relates the reality. In many SCM models, the transportation cost is only considered implicitly as a part of fixed setup or ordering cost and thus, it is assumed to be the independent of the size of the shipment. In this section, we address the case, where the transportation cost is explicitly considered in the model. The structure of all-unit-discount transportation cost is adopted, which is similar to Ertogral et al. [
Structure of all-unit-discount transportation cost.
Range | Unit transportation cost |
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where |
Another attempt of transportation cost discount is considered to make a SCM forever. For selling large quantities, the supplier offers a transportation cost discount to the buyer. In this model, the transportation cost is dependent on
Hence, the expected annual total cost per unit time includes the receiving of uncertain quantity and the transportation cost for the SCM model with partial backorder, setup cost, quality improvement, and trade-credit. Therefore, this problem reduces to
Now the optimum cost of the whole supply chain model is calculated. To do that optimization, we initially ignore all constraints and calculate all the partial derivatives which are necessary for the optimization; then all restrictions are applied on it. The values of all the partial derivatives are as follows:
To obtain the global minimum solution of the supply chain model, the following second-order partial derivatives are used to calculate all minors:
It is found that
Thus, by taking the values of
For a given
See Appendix.
It is a nonlinear program. Thus, the following algorithm is employed to obtain the optimum results.
The input parameters are taken from Sarkar and Moon [
Lead time data.
Lead time component |
Normal duration |
Minimum duration |
Unit crashing cost |
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1 | 20 | 6 | 0.4 |
2 | 20 | 6 | 1.2 |
3 | 20 | 9 | 5.0 |
Transportation cost structure.
Range | Unit transportation cost |
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0.4 |
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0.25 |
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0.17 |
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0.01 |
The optimal cost
Sensitivity analysis for the total cost of supply chain is executed with changing parameters by −10%, −5%, +5%, and +10% in (Table The holding cost for supplier is the most sensitive cost in the supply chain. Negative changes are more than positive changes; that is, when supplier’s holding cost increases total cost increases and vice versa. Its effects are more in supply chain than any other parameters. The holing cost of buyer is 2nd most sensitive comparing other costs of the supply chain. Negative changes are more than positive changes. Decreasing value of buyer’s holding cost affects more than the increasing value of buyer’s holding cost in the total supply chain cost. From the sensitivity analysis, it is found that if initial setup cost increases, total cost also increases. It follows that negative and positive changes are almost similar for two changes. Negative changes are slightly more than positive change. Thus, this model considered the reduction of this setup cost by some investment function and by the numerical study, the obtained reduced setup cost with reduced total supply chain cost. The increasing value of the buyer’s ordering cost indicates the increasing value of the total cost. By comparing the changes within positive and negative direction, two changes are similar. Positive and negative percentage changes are almost same. If rework cost increases or decreases, then the total cost increases or decreases and negative percentage change and positive percentage change are almost the same (see Figure The percentage changes for annual fractional cost are less sensitive than rework cost. Total supply cost change increases for the increase of this parameter. This is the least sensitive parameter among all parameters (see Figure
Sensitivity analysis.
Parameters | Changes of parameters (in %) |
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The paper developed a supply chain model with a stochastic lead time demand, trade-credit policy, quality improvement of products, setup cost reduction of supplier, and variable backorder rate. The backorder rate was lead time-dependent. The aim was to minimize the total supply chain cost with simultaneous optimization of six decision variables as number of shipments, lot size, lead time, setup cost of supplier, quality improvement parameters, and safety stock. Sarkar and Moon [
For given concave function
For the 1st minor, one can obtain easily as
For 3rd minor, the value is obtained as
Finally, for 4th minor, the optimum value is obtained as
Hence,
From the above calculations, all principal minors of the Hessian matrix are positive. Therefore, the Hessian matrix
The authors declare that there are no conflicts of interest regarding the publication of this paper.