Due to heavy transportation for singlesetup multidelivery (SSMD) policy in supply chain management, this model assumes carbon emission cost to obtain a realistic behavior for world environment. The transportation for buyer and vendor is considered along with setup cost reduction by using an investment function. It is assumed that the shipment lot size of each delivery is unequal and variable. The buyer inspects all received products and returns defective items to vendor for reworking process. Because of this policy, end customers will only obtain nondefective items. The analytical optimization is considered to obtain the optimum solution of the model. The main goal of this paper is to reduce the total cost by considering carbon emission during the transportation. A numerical example, graphical representation, and sensitivity analysis are given to illustrate the model.
Carbon emission cost affects the capital investment of any manufacture/production industry. There are several research papers in the literature where carbon emission cost is taken as fixed. But this assumption is unrealistic as transporting of lot size may be variable as per demand of buyers. Hence, carbon emission cost may also be treated as variable. Nag and Parikh [
Setup cost is the cost for setting a production system and configure all production batchrelated works. It is important for many manufacturing industries to reduce the setup cost as this cost is directly related to the total cost. Most of the existing literatures stated that setup cost is a fixed cost. This setup cost can be reduced by a small capital investment. Hong et al. [
It is assumed that whenever the buyer places an order to the vendor, the vendor shipped those products in equal delivery lot sizes. The produced lot may be shifted in partial batches to balance holding cost and setup cost. Goyal and Szendrovits [
Many earlier research works considered an unrealistic assumption that all the produced items are absolutely nondefective. That means after the production process, all the manufactured good is nondefective. But this assumption is not applicable always in reality. During longrun production, imperfect products may occur. With the help of inspection procedure, buyer can obtain nondefective and defective products. After inspection, buyer keeps the nondefective quality items and returned the defective items to vendor for reworking process. By using inspection policy, manufacturing industries are able to provide good quality items into market. Wang and Sheu [
Supply chain defines a management linking the organizations in order to fulfill demand across the whole chain as efficiently as possible. It generally minimizes transportation costs of inventories and manages inventories needed across the supply chain. The aim of supply chain is to satisfy all customers with more facilities, less cost, and time, as well as good quality. Asghari [
Contribution of the different authors.
Author(s)  Setup cost 
Unequal delivery 
Carbon emission cost  Inspection policy  Supply chain 

Nag and Parikh [ 


Butler et al. [ 


Ma et al. [ 


Wygonik and Goodchild [ 


Hua et al. [ 


Bachmann and Van Der Kamp [ 


Zhang et al. [ 


Hong et al. [ 



Ouyang et al. [ 


Chuang et al. [ 


Hou [ 


Sarkar and Majumder [ 

 
Diaby et al. [ 


Sarkar and Moon [ 


Sarkar et al. [ 


Goyal and Szendrovits [ 


Hoque and Kingsman [ 


Bogaschewsky et al. [ 


Siajadi et al. [ 


Zhou and Wang [ 


Hoque [ 


Hariga et al. [ 


Wang and Sheu [ 


Wang and Sheu [ 


Wang [ 


BenDaya and Noman [ 


Konstantaras et al. [ 


Yoo et al. [ 


Sarkar and Saren [ 


Asghari [ 
 
Lin et al. [ 
 
Watanabe and Kusukawa [ 
 
Chen [ 
 
Kusukawa [ 
 
Watanabe and Kusukawa [ 
 
This paper 





This study is considered to discuss the effect of carbon emission cost reduction during transportation in industry sector. During transporting items, fixed and variable carbon emission costs for both vendor and buyer are used in this paper. It is assumed that the vendor’s setup cost is variable and delivery lot sizes are unequal and variable. After receiving the lot, the buyer conducts an inspection procedure and defective items are returned to vendor for reworking operation. The main purpose for developing this model is to reduce the carbon emission cost for vendorbuyer system. This paper continues with mathematical model in Section
The following notations are used to develop the model.
A singlevendor singlebuyer model is considered for a single type of product.
Whenever the buyer places an order, the vendor shipped the lot size in an unequal sized delivery. Delivery rate for lot size is assumed as
Vendor’s setup cost is taken as variable instead of taking as constant. This setup cost is reduced by using an investment to decrease total cost function of vendor.
After receiving each lot, the buyer commences an inspection process to detect the defective items. While the next lot has been received from the vendor, the buyer sent back all defective items of previous lot to vendor for reworking procedure.
Carbon emission costs for both vendor and buyer are included during transporting lots. Two types of carbon emission costs, that is, fixed and variable, are considered.
Demand is assumed to be deterministic.
Shortages are not considered in this model as production rate is greater than demand rate; that is,
Lead time is assumed to be negligible.
Using SSMD policy, vendor sends ordered lot size in
Therefore, the production batch which transported from vendor to buyer is obtained by summing the total of shipment lots as follows:
Hence, buyer’s total inspection cost during product inspection is
The total number of nondefective items for each production cycle is obtained by the area of the triangle given in Figure
Buyer’s inventory model.
Buyer’s fixed carbon emission cost can be calculated for the entire production cycle as
Hence, total carbon emission cost for the buyer by assuming fixed and variable carbon emission costs is
The number of production cycles is
Hence, vendor’s total setup cost is
Vendor’s rework cost for retread defective items is given by
The total stock in the production system is
The total carbon emission cost for vendor is determined by adding fixed and variable carbon emission costs. That is,
The total inventory cost for the vendor can be calculated by adding setup cost, rework cost, fixed and variable carbon emission cost, holding cost, and investment for the setup cost reduction
Therefore, the joint total cost for vendorbuyer system is formulated by
The necessary conditions to minimize the joint total cost for vendorbuyer system
The first order partial derivative of joint total cost for vendorbuyer system
Now, the first order partial derivative of joint total cost for vendorbuyer system
(See the Appendix for the values of
By equating
Now, the first order partial derivative of joint total cost for vendorbuyer system
Now, the first order partial derivative of joint total cost for vendorbuyer system
From the equation
The joint total cost always contain the global minimum solution as the Hessian matrix for
The first order partial derivatives of joint total cost for vendorbuyer system
The second order partial derivatives at the optimal values are given by
Now,
Therefore,
Similarly, it can be obtained that
From the two conditions
Now,
Now,
Therefore,
As
Therefore,
Now,
Finally, it is seen that all principal minors are positive. Hence, the Hessian matrix
The parametric values for this model are chosen as
Then, the joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
Joint total cost for vendorbuyer system
The sensitivity analysis is given for the key parameters of the model in Table
Sensitivity analysis for key parameters.
Parameters  Changes (in %)  JTC 








 

 









 

 









 

 









 

 









 

 









 

 









 

 









 

 









 

 




— 




 


“—” refers to infeasible solution.
In this section, sensitivity analysis is performed to obtain the effect of several parameters such as
The joint total cost for vendorbuyer system
If rework cost
For the increasing value of the parameter
The percentage change in buyer’s fixed carbon emission cost
It is seen that if the buyer’s variable carbon emission cost
From Table
The paper developed an integrated vendorbuyer model with the setup cost reduction for vendor and effect of carbon emission during transporting items from vendor to buyer. The model considered a logarithmic investment function to reduce the setup cost of vendor. Instead of lotforlot (LFL) policy, SSMD policy was used to reduce the holding cost of buyer. In case of LFL policy, holding cost of the buyer was more than the SSMD policy. Increasing holding cost indicated higher joint total cost. Thus, for less joint total cost, SSMD was very useful to reduce the holding cost for buyer. Based on this situation, industry sector will be more beneficial if it use SSMD policy rather than LFL policy. Another major contribution in this model was the unequal delivery lot sizes, while, in the literature, generally SSMD policy was considered with equal lot sizes. Inspection policy is incorporated to this model such that there is a probability to obtain less imperfect items. Beside that, the proposed model reduced the setup cost of vendor by using an investment function. This model minimized the joint total cost for vendorbuyer system considering the setup cost reduction and carbon emission cost. A lemma was constructed to show the global optimum solution of the model. This model extended several models which considered SSMD policy without carbon emission cost or SSMD policy with equal lot sizes, or SSMD policy without setup cost reduction of vendor. This paper can be extended by adding other realistic features such as inspection errors, shortages, and inflation.
The expressions are given for
Dr. Biswajit Sarkar is in leave on lien from Vidyasagar University.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the research fund of Hanyang University (HY2013P).