1. Introduction

This paper presents a real-life problem for the design of an opening hazardous material production and distribution network by optimizing capital expenditures (CAPEX) and operating expenses (OPEX). The problem presents a high degree of complexity, mostly because both CAPEX and OPEX play a major role in the feasibility of the venture. CAPEX includes buying machinery, acquiring permits, and investment in transport vehicles (fleet size), whereas OPEX involves a hazardous material to the customers (truck drivers and truck fuel) and the raw material costs.

The problem considers that a company has evaluated several locations on which to install the processing of a product and distribute from there to different customer’s locations. The transportation is considered as in-house, reason why the fleet size and vehicles capacities are a decision of the owners or investors of the company. The company business model requires the customers to sign a take-or-pay off contract in which it is obligated to pay for a specific amount of product independent on whether it is consumed or not. This business model allows for the planning of the whole contract period by the selection of supply stations, machinery and transport routes.

In this paper, a new version of the Multi Depot Multi Period Vehicle Routing Problem with heterogeneous fleet (MDMPVRPHF) is proposed. In this version, CAPEX are included to the MDMPVRPHF model as a set of restrictions that must be considered to design a better freight distribution network. We called this model the Multi Depot Multi Period Vehicle Routing Problem with heterogeneous fleet and management restrictions (MDMPVRPHFMR).

The MDMPVRPHF and the MDMPVRPHFMR are variants of the vehicle routing problem (VRP). The VRP is introduced by ^{Dantzig and Ramser (1959)}. It is the generalization of the Traveling Salesman Problem (TSP) presented by ^{Flood (1956)}. The classic VRP aims to design a network by minimizing distances, travel times or transportation costs. The network is defined on a graph

G = (V, Ɛ, C), where V = {V ₀ , …, V _n } is a set of vertices, Ɛ = {(V _i , V _j )│(V _i , V _j ) ϵ V ² , i ≠ j } is the arc set; and

C = (C _ij ) _{(Vi, Vj)ϵƐ} is a cost matrix defined over Ɛ. The depot is vertex V ₀ and the customers to be served are represented by the remaining vertices V (^{Pillac, Gendreau, Guéret, & Medaglia, 2013}). The classic VRP consists in designing a network by finding a set of routes for a fleet of vehicles with the same capacity, customer´s demands are known and supplied by only one vehicle (^{Archetti & Speranza, 2008}).

Different VRP variants have been developed. The most studied are the Capacitated VRP (CVRP), the VRP with time Windows (VRPTW), the VRP with Pick-up and Delivery (PDP), the Split Delivery Vehicle Routing Problem (SDVRP), and the Heterogeneous fleet VRP (HVRP).

In the CVRP, a set of customers have a different demand for a good and the fleet of vehicles have finite capacity (^{Pillac et al., 2013}). The VRPTW designs routes from one depot to a set of dispersed customers who can be supplied once by only one vehicle in a time interval, every route starts and ends at the depot, the total demand transported per route (sum of the demands of all points in a route) cannot exceed the capacity of the vehicle (^{Braysy & Gendreau, 2005}). In the PDP, a specific amount of goods must be picked-up and delivered to customers (^{Pillac et al., 2013}). Contrary to the classical VRP, the SDVRP does not consider the restriction that costumers that costumers are supplied by only one different capacities and costs is available for the distribution of goods (^{Baldacci, Battarra, & Vigo, 2008}).

In the literature, there are many variations of the HVRP problem with vehicles capacities constraints and with time window constraints to consider multiple depots, multiple trips to be operated by the vehicles, multiple vehicles with different capacities and other operational constraints. These HVRP variations are the Heterogeneous VRP with Fixed Costs and Vehicle Dependent Routing Costs (HVRPFD), the Heterogeneous VRP with Vehicle Dependent Routing Costs (HVRPD), the Fleet Size and Mix VRP with Fixed Costs and Vehicle Depending Routing Costs (FSMFD), the Fleet Size and Mix VRP with Vehicle Dependent Routing Costs (FSMD), the Fleet Size and Mix VRP with Fixed Costs (FSMF), and the Site-Dependent VRP (SDVRP) (^{Baldacci et al., 2008}).

^{Goel and Gruhn (2008)} study the VRP in real-life applications, and they find different difficulties to be considered. In real-life, the VRP must consider a heterogeneous fleet of vehicles, time window restrictions, differing travel times and costs, vehicles capacity, facility capacity, vehicle compatibility with specific orders, multiple pick-ups per order, delivery locations, service locations, orders where a vehicle can start and finish a journey at different locations, and vehicle route restrictions such as maximum sizes and weights. They include these difficulties as restrictions into the classic VRP and therefore, they formulate the General Vehicle Routing Problem (GVRP). Based on the GVRP, ^{Mancini (2016)} develops the MDMPVRPHF. In her study, Mancini explains that real-life cargo distribution problems have a high degree of complexity because of multi-dimensional vehicle capacity constraints, characteristics of the vehicles, route lengths and travel times, time windows, the compatibility between products, the compatibility between products and vehicles, the compatibility between customers and vehicles, and objective functions which consider different costs such as transportation costs, inventory costs, opportunity costs, etc.

In this article, the MDMPVRPHFMR recognizes the MDMPVRPHF restrictions and adds and proves that CAPEX transport vehicles) and OPEX (raw material cost and transportation of product to the customers such as truck drivers and truck fuel).

This article presents the application of the MDMPVRPHFMR to a real-life problem for the design of an opening liquefied natural gas (LNG) distribution network for a planning time horizon. In this study case, the CAPEX and OPEX information of a real company is used to design its distribution network prior to establishing a contract with the client. The model simultaneously optimizes location allocation, production capacity and vehicle routing decisions. To solve the problem, we present optimal solutions for different random variables small instances using the optimization software CPLEX from IBM. The customers’ demands and the distances between nodes (suppliers and customers) have been generated using Mersenne Twister which is a random number generator. These values are different for each instance and they are presented in Appendix A.

The paper is organized as follows. In Section 2, the MDMPVRPHF and the MDMPVRPHFMR problems are described and the mixed integer linear programming models are presented. In Section 3, the study case is presented together with the computing results of the application of the MDMPVRPHF and the MDMPVRPHFMR models. Section 4 presents the aleatory instances used to test the MDMPVRPHF and the MDMPVRPHFMR models and their computing results. Finally, conclusions and references are included.

2. MDMPVRPHFMR problem description and mathematical

3. Study case

In this Section, a real-life study case is presented to evaluate the performance of the proposed MDMPVRPHFMR model against the performance of the MDMPVRPHFMRWP model and the MDMPVRPHF model. For the real-life study case, the names of the locations are confidential and therefore not shown.

In this study case, the currency is USD and the input data is as follows: a contract period of 5 years, the maximum route travelling time α is equal to 10 [h], the vehicles average speed s is equal to 50 [km/h], the planning time horizon Θ is equal to 24 [h], the transport capacity per vehicle Cv is equal to 23,128 m3 of LNG, the cost of usage per vehicle μv is equal to 0.526 [$/km], the time to discharge/charge the hazardous material from the vehicle to each customer δj is equal to 0.5 [h], the penalty cost in time γ is 20 [$], the cost of renting/buying the vehicle βv is equal to 6.36 [$/h] and the machines production capacity of LNG cim is equal to 863.33 [m3/h].

The cost of opening the supply stations are shown in Table 1. This costs correspond to the legal paperwork and a physical installations needed to connect the station to a natural gas supply, which is the raw material. In the case of Supply Station 2 (SS_2), there is no cost because the client already has a connection to the natural gas pipe line. The supply station opening cost ρi for Supply Station 1 (SS_1) and Supply Station 3 (SS_3) for a 5-year contract period is ρ1=ρ3= 500,000 / (5*365) = 273.97 [$/day].

Table 2 shows the distance between nodes or between supply stations and customers in km. Where Customer 1 (C_1), Customer 2 (C_2) and Customer 3 (C_3) are three demand nodes for the same customer and SS_1, SS_2 and SS_3 are the three possible supplier stations.

The total amount of fuel consumed by the three demand nodes (C_1, C_2 and C_3) for a 5-year contract period is 412,836,900 [m3].

The customer demand nodes are shown in Table 3.

Fig. 1 shows the results for the application of the MDMPVRFHF model. In Fig. 1, Fig. 2 and Fig. 3, the dark circles indicate supply stations that are not part of the solution, the big dark dots indicate the supply stations that are part of the solution, the little light dots indicate the customer locations and the medium size dots indicate the customer locations where LNG is delivered. Each row corresponds to a vehicle v∈V whereas the columns correspond to the route k∈K. Each route has a title, e.g. “V1-R2 T=5.5h” with the following notation: “V” corresponds to the vehicle, “R” corresponds to the route, “T” corresponds to the time of the route. Vehicle routes are consecutive, it means “R1” happens before “R2”, and so on. The quantity delivered of LNG is indicated by the number with an arrow pointing to its location in [m3]. The subscript of the quantity delivered corresponds to the supply station number where that quantity is produced.

Fig. 1. Transport routes of the solution obtained with the MDMPVRFHF model. 

Fig. 2. Transport routes of the solution obtained with the MDMPVRFHFMRWP model. 

Fig. 3. Transport routes of the solution obtained with the MDMPVRFHFMR model. 

The production needed in SS_1, SS_2 and SS_3 to satisfy the customer demands at C_1, C_2 and C_3 are shown in Table 4.

The transport routes using the MDMPVRFHF model are shown in Fig. 1. Vehicle 1 (V1) operates five routes per day and vehicle 2 (V2) operates six routes per day.

Although, in the MDMPVRFHF model only TRA and VEH are minimized, all CAPEX and OPEX costs are considered for the calculation of the customers fuel priceas shown in Table 5.

The transport routes using the MDMPVRFHFMRWP model are shown in Fig. 2. Vehicle 1 (V1) operates five routes per day, vehicle 2 (V2) operates one route per day, and vehicle 3 (V3) operates five routes per day. The production needed in SS_2 and SS_3 to satisfy the customer demands at C_1, C_2 and C_3 are shown in Table 6. The results indicate that SS_1 is not required to operate, therefore there are no opening costs for this station.

Table 7 shows all the costs considered for the customer’s fare when using the MDMPVRFHFMRWP model. By comparing the total cost per m3 of LNG from Table 5 and Table 7, it is possible to conclude that the total cost is reduced from $0.24786 to $0.23893 USD. The results demonstrate that the MDMPVRFHFMRWP model achieves lower costs than the MDMPVRFHF model.

Finally, the transport routes using the MDMPVRFHFMR model are shown in Fig. 3. Vehicle 1 (V1) operates four routes per day, vehicle 2 (V2) operates two routes per day, and vehicle 3 (V3) operates five routes per day. The production needed in SS_1 and SS_2 to satisfy the customer demands at C_1, C_2 and C_3 are shown in Table 8. The results indicate that SS_3 is not required to operate, therefore there are no opening costs for this station.

Table 9 shows all the costs considered for the customer’s fare when using the MDMPVRFHFMR model. By comparing the total cost per m3 of LNG from Table 5 ($0.24786), Table 7 ($0.23893), and Table 9 ($0.23), it is possible to conclude that the minimum total cost, and hence the minimum fuel price (Sj), is reached when using the MDMPVRFHFMR model. Therefore, the results obtained with the proposed MDMPVRFHFMR model indicates that TRA, VEH, PEN, RAX, INV, and MCH costs must be considered. It also demonstrates that the model proposed by ^{Mancini (2016)} (the MDMPVRFHFMR model) does not achieve the lowest possible cost.

The best fuel price for the customer (S _j ) is obtained when using the proposed MDMPVRFHFMR model. It is important to notice that the machine utilization increases when we consider the MCH. The unitary costs TRA, VEH and INV for the three models are compared in Fig. 4. The unitary costs RAW, MCH and the sum of all costs are compared in Fig. 5 for the three models under study.

Fig. 4 The RAW, MCH and the total costs for the MDMPVRPHF model. 

Fig. 5 The TRA, VEH and VEH costs for the MDMPVRPHF model. 

Although OPEX increases when all costs are minimized, the MCH costs decreases and therefore the total cost is minimized and the best fuel price for the customer (S _j ) is obtained.

4. Computational Results

In this section, we test the performance of the MDMPVRPHF model, the MDMPVRPHFMRWP model, and the MDMPVRPHFMR model. These tests study how suitable the models are to solve small and medium instances. A description of the instances used in the computational study is given in Appendix A. Table 10 shows the computation results for each instance tested.

Fig. 6 shows the relative gap between the upper and lower bounds. Here, it is possible to conclude that the MDMPVRPHFMR model achieves the lowest relative gap in the same amount of computing time. For solving the MDMPVRPHFMRWP, the relative gap increases probably because the MDMPVRPHF does not narrow the possible best solutions. In the case of the MDMPVRPHFMR model, OPEX and CAPEX narrows the feasible solutions region.

Fig. 6. Relative gap for instances with 15 and 20 demand points. 

Fig. 7 shows the LNG fuel prices with the MDMPVRPHF model, the MDMPVRPHFMRWP model and the MDMPVRPHFMR model. The LNG fuel price are minimized for all instances when using the MDMPVRPHFMR model. Therefore, we can conclude that companies must consider CAPEX and OPEX for designing their supply chain networks when the contract period is fixed between the supplier and the customer.

Fig. 7. Fuel cost for instances with 10, 15 and 20 demand points. 

5. Conclusions and Future Work

The Multi Depot Multi Period Vehicle Routing Problem with heterogeneous fleet and management restrictions (MDMPVRPHFMR) has been introduced and formulated in this paper. This is a modification of ^{Mancini (2016)} Multi Depot Multi Period Vehicle Routing Problem with heterogeneous fleet (MDMPVRPHF) to consider capital expenditures and operating expenses (MDMPVRPHFMR). In the MDMPVRPHFMR, the goal is to carry out delivery operations at the minimum costs by considering transport costs, vehicle rent costs, time services, raw material, investments, and machine costs. In this paper, we test the proposed MDMPVRPHFMR model and the MDMPVRPHF model in a real case scenario and by solving different instances with random parameters to test the effectiveness and efficacy of these models. The results allows to compare the performance of the proposed MDMPVRPHFMR model with the results obtained using the model proposed by ^{Mancini (2016)} or MDMPVRPHF model. By comparing results, it is possible to conclude that the minimum total cost, and hence the minimum fuel price (Sj), is reached when using the MDMPVRFHFMR model. The results indicates that CAPEX and OPEX must be considered. It also demonstrates that the model proposed by ^{Mancini (2016)} (the MDMPVRFHFMR model) does not achieve the lowest possible cost in a real company scenario.

The major contribution of this paper is the proposition of a model capable of minimizing CAPEX and OPEX at the same time with the aim of designing a LNG supply chain network considering must of the variables presented in a real company scenario. By considering more variables and having more real restrictions the feasible solutions region is narrowed and therefore the relative gap between the upper and lower bound is reduced. Finally, it is possible to conclude that companies must consider CAPEX and OPEX for designing real supply chain networks when the contract period is fixed between the supplier and the customer.

As future work, a Periodic Multi Period Vehicle Routing Problem with heterogeneous fleet and management restrictions can be developed for companies that require periodic deliveries. Such a model can be an extension of the periodic vehicle routing problem (PVRP) and the MDMPVRPHFMR.