Research Article  Open Access
Amir Mohajeri, Mohammad Fallah, "ClosedLoop Supply Chain Models with Considering the Environmental Impact", The Scientific World Journal, vol. 2014, Article ID 852529, 23 pages, 2014. https://doi.org/10.1155/2014/852529
ClosedLoop Supply Chain Models with Considering the Environmental Impact
Abstract
Global warming and climate changes created by large scale emissions of greenhouse gases are a worldwide concern. Due to this, the issue of green supply chain management has received more attention in the last decade. In this study, a closedloop logistic concept which serves the purposes of recycling, reuse, and recovery required in a green supply chain is applied to integrate the environmental issues into a traditional logistic system. Here, we formulate a comprehensive closedloop model for the logistics planning considering profitability and ecological goals. In this way, we can achieve the ecological goal reducing the overall amount of CO_{2} emitted from journeys. Moreover, the profitability criterion can be supported in the cyclic network with the minimum costs and maximum service level. We apply three scenarios and develop problem formulations for each scenario corresponding to the specified regulations and investigate the effect of the regulation on the preferred transport mode and the emissions. To validate the models, some numerical experiments are worked out and a comparative analysis is investigated.
1. Introduction
The issue of supply chain management has received increasing attention among the researchers over the last few decades or so. Nowadays, due to the existence of global and competitive market, it is necessary that enterprises work together to enhance their adaptive ability and viability in the market. Hereby, these enterprises achieve common goals such as minimizing the total costs and the delay of deliveries in the whole chain [1–3]. Three main flows exist in the chain: the material flow, the information flow, and the fund flow. Coordination and integration of these flows across enterprises are called a supply chain management (SCM) [4]. The global economic growth from the 20th to the 21st century has led to a rise in consumption of goods. Consequently, large streams of goods all over the world have been founded. In this way, the production and all aspects of logistics such as transportation, warehousing, and inventories have created large environmental problems such as global warming and climate changes [5]. In 1955, an assessment was published by the Intergovernmental Panel on Climate Change (IPCC). This assessment claimed that Earth’s surface warming is a result of increase in greenhouse gas concentrations [6, 7]. Greenhouse gases are a collection of gases among which are CO_{2} (carbon dioxide), CH_{4} (methane), N_{2}O (nitrous oxide), HFCs (hydrofluorocarbons), PFCs (perfluorocarbons), and SF_{6} (sulphur hexafluoride) [8]. Department of the Environment, Transport and the Regions (DETR) estimated that, among these greenhouse gases, CO_{2} is present in the atmosphere in significant quantities and accounts for twothirds of global warming [9]. CO_{2} is released from several sources such as transportation, industrial processes, and other commercial sectors. As a result, a greenhouse effect is increased [10]. Integration of SCM concept with the issue of environment protection confirms sharp decline in pollution problem. Research on this approach has received considerable attention recently and led to the creation of new research agenda: green supply chain management (GSCM). So, GSCM is a new paradigm where the supply chain will have a direct relation to the environment. Due to the quality and supply chain revolution in the late 1980s and 1990s, respectively, most enterprises have been motivated to become environmentally conscious and have been faced with pressure to protect the environment in their supply chains [11, 12]. Nowadays, most research on GSCM has had a tendency to the reverse logistics and closedloop supply chains such as researches done by Blumberg [13] and Pochampally et al. [14]. In the reverse logistics/closedloop supply chain systems, a product returns to the manufacturer after use and can be repaired or remanufactured to be delivered again to the end consumers. A top environmental issue for an enterprise is how to reduce the utilization of the materials by reusing and remanufacturing the used products. This brings about the GSCM concept and has led to a problem of the closedloop supply chain management. The closedloop logistics are divided into two parts. These two parts are given as follows.(i)Forward logistics: after manufactory, the distributors are responsible to deliver the final products to the end consumers satisfying their demands.(ii)Reverse logistics: the flow of used products is processed from the customers back to the dismantlers to do the sorting or disassembling for recovery, reuse, or disposal [13, 14].
With wellmanaged reverse logistics, the environment protection can be achieved with minimizing of total costs in the whole closedloop supply chain. Most of the previous studies focused on reverse logistics and only formulated models corresponding to this field. Some researchers presented the closedloop models, but they did not consider the relation between forward and reverse flows in their proposed models [15–17]. These models often assumed the unlimited capacities for the reverse logistics, which is not a valid assumption for representing the real situations. In real life situations, the DC also plays such role as a collector in a recovery system. So, the capacity of DC is restricted to both distribution and collection. Now, there is an interaction between amounts of the distribution and the collection so that, when the amounts of the collection are larger, the amounts of distribution must decrease under the same capacity. The closedloop supply chain is characterized by these interactions. With the lack of such kind of relations, the model can be separated into two parts independently and become a supply chain including forward and reverse chains but not a loop. There exist a few studies in which closedloop models were proposed with realistic assumption. In these studies, researchers shared the same capacity for the reverse logistics and stated the relation between forward and reverse flows [18]. These authors proposed a generalized closedloop model for the logistics planning. They formulated an integer linear programming model in which the integration between forward and reverse logistics and the decisions for selecting the places such as DCs was considered. Due to NPhard nature of their model, a genetic algorithm based on spanning tree structure was developed.
Reviewing the literature on closedloop supply chain, it is concluded that a few studies consider the relations between forward and reverse logistics. In this study, we extend Wang and Hsu’s model [18] doing more to protect the environment. First of all, in addition to managing properly reverse logistics to reduce negative impact of greenhouse gases emissions, we suggest another strategy for achieving an expected goal, simultaneously. Here, we focus on a different and important aspect of green supply chains: we focus on transport mode selection as a way to reduce emissions. For this, in addition to minimizing the total cost in the whole closedloop chain, we consider two types of regulations to reduce carbon emissions coming from freight transport. The first mechanism specifies a cost for carbon emissions and the second one is a constraint on emissions. In this study, we pursue three scenarios and develop problem formulations for each scenario corresponding to these regulations and investigate the effect of the regulation on the preferred transport mode and the emissions. These scenarios are given as follows:(i)model without emissions (basic),(ii)model with emissions (emissionconstraint problem),(iii)model with emissions (emission costminimization problem).
Here, we use empirical data to estimate the carbon emissions for various transport modes accurately. The transport modes differ with respect to unit transportation cost, lead time, and unit emissions. For the first scenario in which carbon emissions resulting from freight transport are not considered, our model trades off between long lead times and lower transport costs and short lead times and higher transport costs for transport modes. The carbon emissions are taken into account for the rest of the scenarios where a tradeoff exists between lead time, unit transportation cost, and unit emissions for transport mode. For example, air transport has a shorter lead time, higher unit transportation costs, and carbon emissions than water transport. Defining the three scenarios, we analyze the effect of the regulation on the preferred transport mode and the emissions. Second, we focus on structure of closedloop supply chain. Many procedures are available in this field. One of these procedures is related to traveling salesman problem (TSP) concept in which, having N cities, a salesman should start from home city, visit all customers once, and come back to the home city finding a minimal route. Multiple traveling salesman problem (mTSP) is a wellknown problem in which several salesmen should start and return to a single home city somehow all customers are visited exactly once. Now, we suppose that there are multiDCs in the proposed supply network. Any of them has a number of salesmen. Multiple DCs, multiple traveling salesmen problem (MDMTSP) finds tours for all salesmen such that all customers are visited exactly once and the total cost of the tours is minimized, while salesmen departure from DCs and arrival to the single destination is called the multiple departures single destination multiple TSP [19]. Third, we consider time windows in our proposed closedloop supply chain. There are four layers of supply network (manufacturers, DCs, customers, and dismantlers). Customers send order lists and wait to deliver them. The purposes are determination of proper locations of manufactories, DCs, and dismantlers among candidates set and a suitable distribution of goods throughout the network minimizing cost of all tours. Selection of proper manufactories, DCs, and dismantlers to supply customers depends on satisfying time windows on customer’s viewpoints. So, embedding the transport mode selection, MDMTSP, and time window concepts in a closedloop system with respect to the overall amount of CO_{2} emitted from journeys, it is noted that our closedloop network design is more precisely planned with the aim of protecting environment. To our knowledge, this study is the first paper which considers these concepts simultaneously in the closedloop supply chain. The remainder of our work is organized as follows. The proposed problem is fully explained and justified in Section 2. The methodology based on empirical data to estimate the carbon emissions for different modes of transport is also discussed. Next, the mathematical formulation for three scenarios is developed. In Section 4, the numerical experiments to illustrate the effectiveness of the proposed methodology are given. A comparative analysis is presented in Section 5. Finally, conclusions are drawn.
2. Problem Description
There are essentially four stages along a green logistic network: manufacturers, DCs, customers, and dismantlers. Here, we consider multiple manufactories, DCs, dismantlers, and customers being serviced with one supplier, various transport modes, and one commodity with deterministic demands. The initial problem is making decisions for choosing the proper places of manufactories, DCs, and dismantlers among candidates set while pursuing minimal operations cost, carbon emission, and maximal profits, considering inventory constraints, and satisfying customer demands. Distribution of products from DCs to customers plays critical role. MDMTSP approach can be appropriate for this problem. Any salesman located at DC must depart and visit customers and then go back to the similar or dissimilar DC. In this problem, we suppose that any customer is supplied by only one DC. Meanwhile, the total demands are satisfied. We use the basic conditions for our closedloop chain and consider them as our assumptions in modeling. These basic conditions are given as follows.(i)The customers’ demands must be satisfied.(ii)The flow transferred between two inconsecutive stages must be prevented.(iii)The number of opened facilities and their capacities are limited.Recycling rate issue is only discussed in the closedloop logistics literature. This contains the recovery and landfilling rates. In our model, the recovery amount is assumed to be a percentage of the customer demand corresponding to the Van Der Laan et al. [20] assumption based on the dependence of the amounts of returned products on the demand of the products. So, the following assumption is considered by our model.(iv)The recovery and landfilling rates are given.The framework of proposed closedloop chain is illustrated in Figure 1. One of the main advantages of our proposed model is integrating the transport mode selection and closedloop logistics in the supply network. In this study, we design a closedloop supply chain with the aim of both minimizing the total cost and reducing the environmental impact in the whole chain by choosing the optimal locations of the facilities, the flows of operation units, and the transportation modes along each capacityconstrained stage when the demand of customers and the recycling rates are given. In relation to the transportation issue, it has a significant impact on air pollution so that the overall amount of CO_{2} emitted from it is about 14% of total emissions at the global level [21, 22]. Transportation mode is one of the main choices in transport. There is a variety of transportation modes in our closedloop chain such as transport by plane, ship, truck, or rail. Costs, transit time, and environmental performance are factors by which each mode is distinguished from other modes. Here, the transport mode is chosen using financial and environmental considerations. Besides, the time window constraints play a key role in selecting the transport mode. Due to the air pollution impacts resulting from freight transportation, this paper pays a special attention to this issue from CO_{2} emission’s viewpoint. With respect to the emission calculation issue, there are several methodologies to measure carbon emissions accurately: Greenhouse Gas (GHG) Protocol [23], Artemis [24], EcoTransIT [25], NTM [26], and STREAM [27]. Here, we use the NTM method which specifies emissions for four types of transport: air, rail, road, and water. The NTM method has a high level of detail and focuses on Europe. In this section, we describe the calculation method for the total emissions for each type of transport. This method calculates the total emissions for an averageloaded vehicle and allocates part of the emissions to one unit of product. Below, emissions calculated for four types of transport based on NTM method are given.
Air Transport. The emission factor and the distance are the two main elements determining the total emissions coming from the air transport. The emission factor is in two parts: a constant emission factor (CEF) and a variable emission factor (VEF). Estimation of the emission factors from aircraft is based on aircraft type, engine type, and maximum load. With respect to this type of transport, the flight distance is considered to calculate the distance between the origin and destination location. The bend of the earth is taken into account when we need to calculate the flight distance. The total emissions for an aircraft are calculated by the following equation: Defining (1), the total emissions for an averageloaded vehicle have been calculated. If we want to allocate part of the emissions to one unit of product , we also have to define the dimensional weight which is determined by the density multiplied by volume of one unit of product. Corresponding to [28], if a product has a higher density than 167 (kg/m^{3}), the actual weight is considered to calculate the dimensional weight. In contrast to this, the volume multiplied by 167 (kg/m^{3}) is substituted for the actual weight when a product has a low density. Then, Since the amount of goods carried by a vehicle depends on the weight and the volume of the load, the emissions allocated to one unit of the product ( in kg) are calculated as follows: where and LF are the maximum load of an aircraft (in kg) and the average load factor of the aircraft, respectively.
Railway Transport. Here, the emission calculation method for only diesel engine in railway transportation is described [29]. The unit emissions are calculated based on the emission factor, the distance, and the weight of the product. The amount of CO_{2} emitted when transporting 1 net tonne over 1 km in way is known as the emission factor (EF in kg CO_{2}/net tonne km). It depends on several factors outlined as follows.(i)The gross weight of the train ( in tonne) includes the weight of the locomotive and the carriages.(ii)An emission constant determines the fuel consumption for a way.(iii)A correcting factor for the terrain is different based on the topography of the way. For example, the factor for hilly and mountainous terrain is greater than for flat. Hence, and , where .(iv)The load factor (LF) equals the ratio of net and gross weight of the train.(v)The fuel emissions (FE) denote the emissions per liter of fuel burnt.
The emission factor for the diesel rail transport ( in (kg CO_{2}/net tonne km)) is defined by the following equation: The emissions allocated to one unit of the product ( in kg) are a function of the distance ( in km), the weight of the product ( in tonne), and the emission factor. The formula for the unit emissions for the diesel engine in railway transportation is then
Road Transport. In this section, the fuel consumption, the fuel emissions, and the distance are three main factors to calculate the total emissions of the vehicle. Below, each factor is given in more detail.(i)The fuel consumption (FC in L/km) is based on two factors, load factor (LF) and the type of vehicle, and is calculated as follows: where and are the fuel consumption for a full loaded vehicle and the fuel consumption for an unladen vehicle, respectively.(ii)The fuel emissions (FE) are defined as gram of CO_{2} emitted per liter of fuel.(iii)The distance ( in km) is the distance between the locations.
Combining these factors yields the following equation for the total emissions of the vehicle for road transport (EM_{total} in g): Defining (7), the emissions of the entire vehicle have been calculated. If we want to allocate part of the emissions to one unit of product , we also have to define the dimensional weight of one unit of product, which is defined as where 250 is a default density used by transport companies [30]. So, if a product has a density higher than 250 (kg/m^{3}), the actual weight is considered to calculate the dimensional weight. In contrast to this, the volume multiplied by 250 (kg/m^{3}) is substituted for the actual weight when a product has a low density. The emissions allocated to one unit of the product ( in g) are calculated as follows: where and are the maximum load of a vehicle (in kg) and the average load factor of the vehicle, respectively.
Water Transport. Shortsea transport with diesel oilpowered vessels is known as water transport [31]. Here, the total emissions (EM_{total} in kg) depend on three factors: the fuel consumption (FC), the fuel emissions (FE), and the distance . The fuel consumption (FC) (in L per km) is given in [31] for both a given vessel type and an average load factor. The distance (in km) is the distance between two locations over waterways which is larger than the distance over road. The fuel emissions (FE) factor (in kg) is also the amount of CO_{2} emitted when 1 L of diesel is burnt. The total emissions (EM_{total} in kg) of the vessel are calculated by the following equation: The unit emissions for the vessel in waterway transportation are obtained defining the allocation fraction as follows: where the type of ship plays a critical role in determining the unit of capacity; here, it can be weight for bulk vessels. The formula for the unit emissions ( in kg) of the vessel is then
3. Supply Chain Models with Considering the Environmental Impact
Here, we pursue three scenarios and develop problem formulations for each scenario. A mixed integer linear programming (MILP) optimization model to minimize the total construction cost of this network is presented.
3.1. Basic Model
In order to simplify this problem, we suppose that there is only one product in the concerned closedloop chain, and the carbon emission resulting from freight transport is not considered. In order to formulate this simplified problem mathematically, the following notations are necessary.
Notations : Set of candidate manufactories : Set of candidate DCs : Set of customers : Set of candidate dismantlers : Set of transport mode types : Set of transport mode types at manufactory; : Set of transport mode types at DC; : Set of transport mode types at dismantler; .
Parameters : Capacity of manufactory : Total capacity of (forward and reverse) : Capacity of dismantler : The percentage of total capacity for reverse logistics in : The percentage of recovery of customer : The percentage of landfilling of dismantler : Demand of customer : Unit cost of production in manufactory : Unit cost of transportation from manufactory to DC by vehicle per km : Unit cost of transportation from DC to customer by vehicle per km : Unit cost of transportation from dismantler to manufactory by vehicle per km : Fixed cost for operating manufactory : Fixed cost for operating : Fixed cost for operating dismantler : Fixed cost for landfilling per unit : Distance between manufactory and : Distance between and customer : Distance between customer and customer : Distance between and dismantler : Distance between dismantler and manufactory : The time of transportation from to customer using vehicle : The time of transportation from customer to customer using vehicle : The lower bound of expected time for delivering product at customer : The upper bound of expected time for delivering product at customer : The recovery cost in from customer : Number of vehicles at manufactory : Number of vehicles at : Number of vehicles at dismantler : Capacity of vehicle : Capacity of vehicle : Capacity of vehicle : Maximum load for vehicle : Maximum load for vehicle : Maximum load for vehicle : Average load factor for vehicle : Average load factor for vehicle : Average load factor for vehicle vol: Volume of product : Density of product for vehicle wp: Weight of product capw: Total capacity of cargo vessel : The maximum number of nodes a salesman may visit : The minimum number of nodes a salesman must visit : A large number.
Decision Variables : 1, if a product can be shipped by vehicle from manufactory to ; 0, otherwise : 1, if a product can be shipped by vehicle from to customer ; 0, otherwise : 1, if a recovered product can be shipped by vehicle from to dismantler ; 0, otherwise : 1, if a vehicle returned from customer to ; 0, otherwise : 1, if a reused product can be shipped by vehicle from dismantler to manufactory ; 0, otherwise : 1, if production takes place on manufactory ; 0, otherwise : 1, if is opened; 0, otherwise : 1, if dismantler is opened; 0, otherwise : 1, if a product can be shipped by vehicle from customer to customer ; 0, otherwise : Amount shipped by vehicle from manufactory to : Amount shipped by vehicle from to customer : Amount of recovered product shipped by vehicle from to dismantler : Recovered amount shipped by vehicle from customer to : Reused amount shipped by vehicle from dismantler to manufactory : Recovered amount shipped by vehicle from customer to customer : Quantity produced at manufactory : The number of nodes visited by travelers from DC to node : Amount of congested product at customer : Amount of congested recovered product at customer : The arrival time of product at customer .
Using these definitions, the basic model for the proposed closedloop chain can be described as follows.
Objective Function
Consider the following:
Constraints
Consider the following: Equation (13) is the objective function which minimizes cost of opening manufactory, distribution center, and dismantler, minimizes the total cost of both forward and backward distances, and minimizes the total cost of operations. Constraints (14) and (15) show that there exist at least one activated manufactory and one DC in the chain, respectively. Constraint (16) ensures that each manufactory can produce an amount of product just after it is selected. Each activated manufactory covers at least one DC, and Constraint (17) represents this goal. On the contrary, each DC receives at least one link from manufactories just after it is selected (Constraint (18)). Constraint (19) represents the amount of flow between manufactory and DC. Constraint (20) represents the limit of the capacity for manufactories in forward logistics. Constraint (21) imposes that the number of traveling vehicles from manufactory would not exceed the existing vehicles. Constraint (22) prevents the route between manufactory and DC from accepting its vehicle more than once. The capacity constraint of each vehicle that traveled from manufactory to DC is shown by Constraint (23). Constraint (24) guarantees that each activated DC covers at least one customer. Each activated DC receives at least one link from customers, and Constraint (25) represents this goal. Constraint (26) represents that a salesman from DC must visit at least two customers. Constraint (27) requires that any customer be supplied by either DC or other customers. In addition, it either comes back to DC or supplies other customers. This concept is represented by Constraint (28). Each customer is supplied and supplies by the same vehicle. This is represented by Constraint (29). Constraints (30), (31), and (32) prevent any subtour in network. Constraint (33) indicates the amount of congested product for supplying other customers by each customer. Constraint (34) represents the amount of flow between DC and customer. Constraint (35) is to satisfy the customer demand. The capacity constraint of each vehicle traveling from DC to customer is shown by Constraint (36). Constraint (37) imposes that the number of traveling vehicles from DC would not exceed the existing vehicles. Constraint (38) satisfies the law of the flow conservation by inflow equal to outflow. The amount of congested product for recovering from other customers by each customer is indicated by Constraint (39). Constraints (40)(41) represent the amount of flow between customer and DC. The amount of flow among customers is represented by Constraint (42). Constraint (43) guarantees that each activated dismantler receives at least one link from DCs. Constraint (44) prevents the route between DC and dismantler from accepting its vehicle more than once. Constraint (45) guarantees that each activated DC covers at least one dismantler. The amount of flow between DC and dismantler is shown by Constraint (46). Constraint (47) satisfies the law of the flow conservation by inflow equal to outflow. Constraint (48) indicates that the total forward and backward flows cannot exceed the total capacity of DC. Constraint (49) imposes that the number of traveling vehicles from DC to dismantler would not exceed the existing vehicles. The capacity constraint of each vehicle traveling from DC to dismantler is shown by Constraint (50). Constraint (51) means the reverse limit of the capacity for DCs. Constraint (52) ensures that each activated dismantler covers at least one manufactory. Constraint (53) prevents the route between dismantler and manufactory from accepting its vehicle more than once. Constraint (54) guarantees that each activated manufactory receives at least one link from dismantlers. The amount of flow between dismantler and manufactory is shown by Constraint (55). Constraints (56) and (57) satisfy the law of the flow conservation by inflow equal to outflow. Constraint (58) means the reverse limit of the capacity for dismantlers. Constraint (59) imposes that the number of traveling vehicles from dismantler to manufactory would not exceed the existing vehicles. The capacity constraint of each vehicle traveling from dismantler to manufactory is shown by Constraint (60). Constraints (61)–(66) satisfy time windows. Constraint (67) denotes the binary variables, and Constraint (68) restricts all other variables from taking nonnegative values.
3.1.1. Linearization
To improve the performance of the proposed mathematical model we act out the following linearization for the nonlinear equations. As Constraint (33) is nonlinear, we turn it into the following equations: As Constraint (39) is nonlinear, we turn it into the following equations:
3.2. Emissions
Here, we describe how the carbon emissions are incorporated into our model and the methodology to calculate the emissions. In Section 3.2.1 we define the emission costminimization problem in which a unit cost for emission is charged. In Section 3.2.2 we define the emissionconstraint problem in which we have a hard constraint on the carbon emissions.
3.2.1. Emission CostMinimization Model
The objective of the proposed basic model is to minimize the total construction and operations costs while considering structural, product flow, capacity, customers’ demands, and time windows constraints. It has ignored the carbon emission as an important factor for green supply chain. Below, we will extend the basic model by adding a cost for carbon emissions. In the Emission Trading Scheme the carbon cost is expressed in €/(metric) tonne emissions. We therefore specify a carbon emission cost per tonne of CO_{2} emitted. For any transportation mode, let and , and , and , and , and , and and denote its total carbon emissions and the emissions allocated to one unit of the product for transportation from the th manufactory to the th DC, from the th DC to the th customer, from the th customer to the th customer, from the th customer to the th DC, from the th DC to the th dismantler, and from the th dismantler to the th manufactory, respectively. In this model, the following objectives and constraints are added to the proposed basic closedloop model to consider the carbon emission issue. In order to formulate this emission costminimization model mathematically, the following notations are necessary.
Parameters CEF: Constant emission factor VEF: Variable emission factor : The fuel consumption for vehicle : The fuel emissions for diesel fuel for vehicle : The fuel consumption for semitrailer stated in manufactory : The fuel emissions for diesel fuel for semitrailer stated in manufactory : The fuel consumption for semitrailer stated in dismantler : The fuel emissions for diesel fuel for semitrailer stated in dismantler : The fuel consumption factor for diesel train : The fuel emissions for diesel train : The gross weight of the train : The fuel emissions for cargo vessel : The fuel consumption for cargo vessel : The price of carbon emission (expressed in €/(metric) tonne emissions).
Decision Variables : Total emissions of the vehicle from manufactory to : Total emissions of the vehicle from to customer : Total emissions of the vehicle from customer to : Total emissions of the vehicle from customer to customer : Total emissions of the vehicle from to dismantler : Total emissions of the vehicle from dismantler to manufactory : Unit emissions of the vehicle from manufactory to : Unit emissions of the vehicle from to customer : Unit emissions of the vehicle from customer to : Unit emissions of the vehicle from customer to customer : Unit emissions of the vehicle from to dismantler : Unit emissions of the vehicle from dismantler to manufactory .
Objective Function
Consider the following:
Constraints
Consider the following: Nonlinear equation (77) is the objective function which minimizes total cost of the carbon emissions allocated to whole units of the product for transportation from manufactory to DC, DC to customer, customer to customer, customer to DC, DC to dismantler, and dismantler to manufactory, respectively. Constraints (78)–(85) show the emissions allocated to one unit of the product for transportation from the th manufactory to the th DC, where are the binary variables to link carbon emissions constraints to the related types of transport. Constraints (78)(79), (80)(81), (82)(83), and (84)(85) measure carbon emissions of the aircraft, vehicle, diesel train, and vessel based on NTM method for air transport, road transport, rail transport, and water transport.
Constraints (86) and (87) show the emissions allocated to one unit of the product for transportation from the th DC to the th customer. Constraints (88) and (89) show the emissions allocated to one unit of the product for transportation from the th customer to the th customer. Constraints (90) and (91) show the emissions allocated to one unit of the product for transportation from the th customer to the th DC. Constraints (92) and (93) show the emissions allocated to one unit of the product for transportation from the th DC to the th dismantler. Constraints (86)–(93) measure carbon emissions of the vehicle based on NTM method for road transport. Constraints (94)–(97) show the emissions allocated to one unit of the product for transportation from the th dismantler to the th manufactory, where and are the binary variables to link carbon emissions constraints to the related types of transport. Constraints (94)(95) and (96)(97) measure carbon emissions of the vehicle and diesel train based on NTM method for road transport and rail transport.
(1) Linearization. To improve the performance of the proposed mathematical model we act out the following linearization for the nonlinear equations. As (77) is nonlinear, we turn it into the following equation and turn the constraints related to the definition of into the following equations:
3.2.2. EmissionConstraint Model
This problem extends the basic model by constraining the carbon emissions, which is denoted by (in Kg). We note that the constraint for carbon emissions is equal to the average of total carbon emissions for the basic model and emission costminimization problem. In this model, the objectives are the same as the basic model ones. Constraints (79), (81), (83), (85), (87), (89), (91), (93), (95), (97), (99)–(108) and the new one are added to the proposed basic closedloop model to define and limit the carbon emissions issue. In order to formulate this emissionconstraint model mathematically, the following notation is necessary.
Parameters
_Average: The average carbon emissions of the entire system.
The new constraint added to the emission costminimization model is as follows.
The New Constraint
Consider the following: The carbon emissions constraint is shown by Constraint (109).
4. Numerical Experiments
Here, we propose a numerical example to indicate the effectiveness of the proposed mathematical models. Our models are tested in small scale of data. Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are the given data. The numbers of potential locations for the manufactory, DC, and dismantler are three, four, and two, respectively. Manufactories, DCs, and dismantlers are selected to secure 57 customers having definite demands. While the applied optimization software is not able to provide solutions for 57 customers in a reasonable time, we categorized the customers into 7 more comprehensive zones with aggregated demands. There are four types of transportation mode (air, rail, road, and water) used to transfer product from manufactories to DCs; one type of transportation modes (road) is used to transfer product from DCs to customers and dismantlers, and two types of transportation modes (rail and road) are used to transfer product from dismantlers to manufactories.
