A multiobjective model for the green capacitated location-routing problem considering drivers’ satisfaction and time window with uncertain demand

Abstract

Location-routing problem is a combination of facility location problem and vehicle routing problem. Numerous logistics problems have been extended to investigate greenhouse issues and costs related to the environmental impact of transportation activities. The green capacitated locating-routing problem (LRP) seeks to find the best places to establish facilities and simultaneously design routes to satisfy customers’ stochastic demand with minimum total operating costs and total emitted carbon dioxide. In this paper, features that make the problem more practical are: considering time windows for customers and drivers, assuming city traffic congestion to calculate travel time along the edges, and dealing with capacitated warehouses and vehicles. The main novelty of this study is to combine the mentioned features and consider the problem closer to the real-world case uses. A mixed-integer programming model has been developed and scenario production method is used to solve this stochastic model. Since the problem belongs to the class of NP-hard problems, a combination of the progressive hedging algorithm (PHA) and genetic algorithm (GA) is considered to solve large-scale problems. It is the first time, as per our knowledge, that this combination is implemented on a green capacitated location routing problem (G-CLPR) and resulted in satisfactory solutions. Nondominating sorting genetic algorithm II (NSGA-II) and epsilon constraints methods are used to face with the bi-objective problem. Finally, sensitivity analysis is performed on the problem’s input parameters and the efficiency of the proposed method is measured. Comparing the results of the proposed solution approach with those of the exact method indicates that the solution approach is computationally efficient in finding promising solutions.

Appendix

Section a) CO₂ emission formula

In Toro et al. (2017), all the forces toward the truck are discussed. Below the schematic of such forces on the truck is depicted. Using mathematical relations and static mechanics, the total amount of energy consumed, and the total amount of fuel consumed, followed by the total amount of carbon dioxide emissions, are calculated (Fig. 11).

Fig. 11.

Forces act on the truck

$\overrightarrow{F_R}$ represents the forces that are opposed to the movement of the vehicle, $\overrightarrow{F_M}$ represents the forces created by the engine and transmitted to the tire of the vehicle, mg is the weight of the vehicle, and $\overrightarrow{N}$ is the surface force on the vehicle.

$$ \sum {F}_x=m{a}_x\kern2.25em {a}_x=0\kern0.5em {F}_M-{F}_R- mgsin{\beta}_{ij}=0 $$

(63)

$$ \sum {F}_y=m{a}_y\kern2.25em {a}_y=0\kern0.5em N- mgcos{\beta}_{ij}=0 $$

(64)

$\overrightarrow{F_{R, tires}}$ represents the force created between the wheels without traction and the ground, which is opposite to the movement of the vehicle. $\overrightarrow{F_{R, wind}}$ is the force exerted by the wind against the motion of the vehicle. $\overrightarrow{F_{R, internal}}$ represents the equivalent force of internal forces opposing the motion of the vehicle, and $\frac{m\ {v}^2}{2{d}_{ij}}$is the force required by the vehicle to achieve steady state dynamic energy. The mass of the loaded vehicle is the sum of the unloaded vehicle's mass (the mass of the vehicle itself) m₀ and the load carried between nodes i and j.

$$ \overrightarrow{F_R}=\overrightarrow{F_{R, tires}}+\overrightarrow{F_{R, wind}}+\overrightarrow{F_{R, internal}}+\frac{m\ {v}^2}{2{d}_{ij}} $$

(65)

$$ m={m}_0+{t}_{ij} $$

(66)

$$ {F}_{R, tires}= Nb $$

(67)

$$ \overrightarrow{F_M}=\left( mgcos{\beta}_{ij}\right)b+\overrightarrow{F_{R, wind}}+\overrightarrow{F_{R, internal}}+\frac{m\ {v}^2}{2{d}_{ij}}+ mgsin{\beta}_{ij} $$

(68)

In this step, we calculate the work function (U_ij) done by the truck, which is the same force in the displacement amount. Next, the total amount of gas emissions will be obtained.

$$ {U}_{ij}=\left[\left({m}_0+{t}_{ij}\right) gbcos{\beta}_{ij}+\overrightarrow{F_{R, wind}}+\overrightarrow{F_{R, internal}}+\frac{\left({m}_0+{t}_{ij}\right){v}^2}{2{d}_{ij}}+\left({m}_0+{t}_{ij}\right)g\ \mathit{\sin}\ {\beta}_{ij}\Big]\right]{d}_{ij} $$

(69)

$$ {U}_{ij}=\left[{m}_0g\ \left( bcos{\beta}_{ij}+\mathit{\sin}{\beta}_{ij}\frac{v_{ij}^2}{2g{d}_{ij}}\right)+{F}_{R, wind}+{F}_{R, internal}\right]{d}_{ij}+\left[g\left( bcos{\beta}_{ij}+\mathit{\sin}{\beta}_{ij}\frac{v_{ij}^2}{2g{d}_{ij}}\right)\right]{t}_{ij}{d}_{ij} $$

(70)

The amount of fuel (diesel) required to do the whole work function, i.e., ∑_{i, j ∈ V}U_ij, is obtained by the conversion factor E₁ (gallon diesel/joule). Another conversion factor obtains each fuel unit’s emission rate, E₂ (grams of CO₂/gallon diesel). Therefore, the amount of carbon dioxide emissions is calculated as follows:

$$ {E}_1\times {E}_2\times {\sum}_{i,j\in V}{U}_{ij}=E\times {\sum}_{i,j\in V}{U}_{ij} $$

(71)

Another part of the second objective function that needs to be addressed is the amount of carbon dioxide emitted in terms of the amount of time the vehicle is in traffic congestion. We can calculate this by the following equation of carbon dioxide emissions.

$$ {E}_2\times {E}_3\times \sum \limits_{i,j\in V,l\in L}{\sigma}_l\left({LT}_i\right)\ast {time}_{ij}{x}_{ij}= EPM\sum \limits_{i,j\in V,l\in L}{\sigma}_l\left({LT}_i\right)\ast {time}_{ij}{x}_{ij} $$

(72)

In this regard, the EPM, which is the conversion factor of the idle hours of truck operation to the amount of carbon dioxide produced, is obtained using two conversion factors. The first conversion factor, E₃, is for gallons of diesel and the second conversion factor is equal to the same factor E₂.

Section b) GA parent selection and operators

• 1. Parent selection

In the proposed GA, two types of parent selection procedures, including random selection and roulette wheel selection, are involved in the model. However, based on the results of the genetic algorithm’s implementation in several identical examples, the roulette wheel selection method has performed better than another method (Yu et al. 2019). According to (Amal and Chabchoub 2018), half of the chromosomes with the highest fitness function will be selected for the next generation.

• 2. Crossover

Two different types of crossover are used randomly with the same probability to produce offspring from two selected parents: single-point crossover and two-point crossover (See (Asefi et al. 2014) for more details). The two types of crossover are depicted in Figs. 12 and 13.

Fig. 12

An example of single-point crossover on location variable

Fig. 13

An example of two-point crossover on location variable

• 3. Mutation

Three mutation operators will be used for location variables, and two mutation operators will be used for routing variables in the proposed GA. Mutation operators for location variables are change, swap, and insertion, and routing variables are swap and insertion.

In the change mutation operator, two random points are selected from each chromosome. Two points are chosen randomly along the chromosome. Then, as depicted in Fig. 14, if the chosen point is 0, it will be converted to 1, and vice versa. Other mutation operators are illustrated in (Asefi et al. 2014). Figs. 15 and 16 clearly show how such operators perform.

Fig. 14.

An example of change mutation for location variables

Fig. 15.

An example of swap mutation for location variable

Fig. 16.

An example of insertion mutation

Section c) NSGA-II

Ranking means that we want to rank the solutions according to the concept of quality. To rank the initial population solutions, we put the solutions that never dominated in the first place. Then, we remove non-dominated solutions from the solution set and again compare the rest of the solutions. Once again, we put non-dominated solutions on the second frontier. In this way, all the solutions will be categorized into different boundaries.

If the solutions are ranked only by the rank criterion, there is no need for the crowding distance defined by the neighbors of a solution and the first (best) and last (worst) chromosome of the population (Van Veldhuizen and Lamont 1999).