Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 469486, 13 pages

http://dx.doi.org/10.1155/2015/469486

## Effect of Unequal Lot Sizes, Variable Setup Cost, and Carbon Emission Cost in a Supply Chain Model

^{1}Department of Industrial & Management Engineering, Hanyang University, Ansan, Gyeonggi-do 426 791, Republic of Korea^{2}Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

Received 21 August 2015; Accepted 5 October 2015

Academic Editor: Paulina Golinska

Copyright © 2015 Biswajit Sarkar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to heavy transportation for single-setup 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.

#### 1. Introduction

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 [1] considered several important matters as time series estimates of indirect carbon emissions per unit of power consumption and baseline emissions for the power sector till 2015. Butler et al. [2] obtained differences between the contributions of various sectors to the total emissions from each city and connect these differences to different methodologies. Ma et al. [3] examined if the global climate change effects were taken into consideration during rapid economic development of China. They described a study of energy consumption and carbon emissions in Tianj during the period from 1995 to 2007. Their research model determined the primary causes of carbon emissions and put forward suggestions for carbon emission reduction in energy consumption. Wygonik and Goodchild [4] designed an emission minimization vehicle routing problem (VRP) with time windows. In their model, they provided a stable relationship between monetary cost and kilograms of . Their results suggest the most effective way to reduce cost and emissions. Hua et al. [5] observed the way of carbon footprints in inventory management under the carbon emission trading mechanism. Bachmann and Van Der Kamp [6] addressed an approach to quantify monetised environmental benefits related to reduction in air pollutant emissions. Zhang et al. [7] studied an evolutionary game model to obtain the promotional effect of the rising oil price on companies behaviour of carbon emission reduction. Their research observed the theoretical reference for promulgating rational low carbon policies.

Setup cost is the cost for setting a production system and configure all production batch-related 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. [8] investigated three production policies under nonconstant, deterministic demand, and dynamic setup cost reduction. They developed a lot sizing and an investment solution procedure to decrease the dynamic setup cost. Ouyang et al. [9] considered that lead time demand follows a normal distribution and optimized lot size, reorder point, process quality, setup cost, and lead time. They used min-max distribution-free approach to solve the problem. They explained the way of setup cost reduction by investing some initial investment. Chuang et al. [10] derived periodic review inventory models with a mixture of backorders and lost sales by controlling lead time and setup cost simultaneously to minimize inventory operating cost. On the other hand, it is considered that probability distribution of the protection interval, that is, review period plus lead time, demand is unknown, but its first two moments are given. Hou [11] investigated an economic production quantity (EPQ) model with imperfect production processes to reduce setup cost. Later, Sarkar and Majumder [12] developed vendor-buyer supply chain model with vendors setup cost reduction strategy. Diaby et al. [13] discussed the issue of investing in reduced setup times and defect rate reductions along with the corresponding optimal levels of investments and optimal production cycle time for each product. They assumed several cases of product-specific quality improvements and joint-product quality improvements. Sarkar and Moon [14] provided an imperfect production process in which they discussed the relationship between quality improvement, reorder point, and lead time with backorder rate. They assumed that lead time demand follows a normal distribution and applied the distribution free approach for the lead time demand to minimize total system cost. In their model, they optimized setup cost, lot size, lead time, reorder point, and process quality parameter. Sarkar et al. [15] presented a continuous-review inventory model with quality improvement, service level constraint, and setup cost reduction. They considered some distribution-free approach and minimized the total system cost against the worst possible distribution scenario.

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 [16] considered a constant lot size model to obtain economic lot size and batch sizes for each stage. They assumed in their research model that equal or unequal sized batches can be shipped from one stage to the next and the total number of batches may differ across stages. Transportation of partial lots is provided between stages in their proposed model. Hoque and Kingsman [17] obtained a new heuristic solution procedure for the constant lot size model for the production of a single product requiring processing through a fixed sequence of manufacturing stage. Bogaschewsky et al. [18] extended previous research works in this field by considering multistage production model in which unequal sized batches are produced. An optimization method is constructed to measure the economic lot size and optimal batch sizes for each stage by assuming setup costs, inventory holding costs, and transportation costs in their model. Siajadi et al. [19] proposed a shipment policy to minimize the joint total relevant cost (JTRC) for both vendor and buyer. Considering two-buyer and more than two-buyer cases, they obtained exact and approximate optimum solutions. Though several models considered SSMD policy, many models assumed only equal shipments. Zhou and Wang [20] generated a production-inventory model with deteriorating item for a single-vendor single-buyer integrated system. Their model considered the structure of shipment policy. In addition, their model extended to the situation with shortages permitted, based on shortages being allowed to occur only for the buyer. Hoque [21] developed a manufacturer-buyer integrated inventory model by assuming equal/unequal-sized batches delivery. Hariga et al. [22] developed a mixed integer nonlinear program that minimizes total supply chain costs and allows unequal shipment frequencies to the retailers.

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 long-run 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 [23] used an inspection policy for the batch not for a single item. Wang and Sheu [24] discussed a deteriorating production system with product inspection policy. In addition, production-maintenance policy is also discussed in their model. Wang [25] optimized the production run length and product inspection policy by obtaining an efficient solution procedure. Ben-Daya and Noman [26] formulated an integrated inventory model based on the assumption that a lot is received, buyer uses some special type of inspection policies. The fraction nonconforming is considered to be a random variable which follows a beta distribution. Konstantaras et al. [27] deduced a classical economic order quantity model with the assumption that all received items may be damaged due to transportation or production condition while screening is usually a manual task performed by inspectors which may improve with learning. Yoo et al. [28] addressed an imperfect production and inspection system with customer return and defective disposal. They considered production and inspection quality investment with all quality costs. They assumed Type I and Type II inspection error proportions which minimize the total quality cost and maximize the total profit. Recently, Sarkar and Saren [29] extended an economic production quantity model with warranty, inspections, and inspection errors.

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 [30] examined the applicability of numerous measures and metrics in a multiobjective optimization problem of supply chain network design to allocate customers’ orders. He determined important aspects of strategic planning of manufacturing in a supply chain model. In Lin et al.’s [31] model, a hybrid approach, including applied interpretive structural modeling to build a hierarchical structure, and application of analytic network process to examine dependence relations are discussed. In addition, their model used fuzzy set theory to analyze linguistic preferences. They also provided that the financial aspect and life cycle assessment are the most essential performance and weighted criteria. Watanabe and Kusukawa [32] generated an optimal operational policy for both decentralized GSC (DGSC) and an integrated GSC (IGSC). They described that a retailer pays an incentive for collection of used items from customers and formulates optimal order quantity of a single product under uncertainty in product demand. In their model, some mathematical models are observed to obtain collection incentive of used products, lower limit of quality level for recycling affectability. Chen [33] produced how green operations affect firms environmental performance with green innovation. His model determined the positive relationships existing among green operations, green innovation, and environmental performance. Kusukawa [34] deduced decision-making approaches for two situations which made a decentralized supply chain (DSC). Decentralized supply chain (DSC) maximizes the retailers profit and an integrated supply chain (ISC) is used to increase the whole systems profit. On the other hand, supply chain coordination is established to set the unit wholesale price at each order time with Nash bargaining solutions. Watanabe and Kusukawa [35] studied a dual-sourcing supply chain (DSSC) in which two scenarios of product’s demand are considered as known demand distribution and known mean and variance of demand. They analyzed an optimal ordering policy under DSSC to increase the total expected profit and an optimal ordering policy under the integrated DSSC to maximize the whole systems total expected profit. See Table 1 for the contribution of several authors.