A Sustainable Reverse Logistics System: A Retrofit Case

Abstract

This chapter presents a real case study of a recyclable waste collection system aiming at redesigning service areas and associated vehicle collection routes to support a sustainable operation. Not only economic objectives are to be considered, but also one should account for environmental and social aspects. The economic dimension is modeled through traveling distance that directly influences the global cost. The environmental one is modeled throughout the calculations of the CO₂ emissions. Finally, the social aspect is considered by aiming to define a balanced solution regarding working hours among drivers. A multi-objective solution approach based on mixed-integer linear programming models is developed and applied to real data.

Appendices

Annex A: Multi-objective Formulation for the MDPVRPI

The multi-objective MDPVRPI is formulated as a set partitioning problem (Balas and Padberg 1976), where K represents the set of all feasible routes (closed and inter-depot routes) and τ_ktg is a binary variable that equals 1 if route k is performed on day t by vehicle g; and 0 otherwise. The mathematical formulation considers the following indices and sets.

Indices

k	Route indices
t	Time period (days) indices
g	Vehicle indices
i, j	Node indices
m	Recyclable material indices

Sets

K	Route set $K=\sum \limits_{m\in M}{K}_{\mathrm{m}}$, K = Kin ∪ Kcl
Km	Route subset to collect material m
Kin	Inter-depot route subset
Kcl	Closed route subset
T	Time period set
G	Vehicle set
V	Node set V = Vc ∪ Vd ∪ Vs
Vc	Collection site subset
Vd	Depot subset
Vs	Sorting station subset
M	Recyclable material set

Each route k ∈ K  is characterized by (1) a distance dis_k; (2) a duration dur_k including travel, service, and unloading times; (3) a load Lo_k; and (4) CO₂ emissions Co_k. The collection sites belonging to route k are given by a binary parameter μ_ik that equals 1 if collection site i belongs to route k and 0 otherwise. The starting and ending depots for route k are also given by binary parameters St_ki and En_ki, respectively; St_ki equals 1 if route k starts at depot i, and En_ki equals 1 if route k ends at depot I and 0 otherwise.

The vehicles are fixed at the depots. If vehicle g belongs to depot i, the binary parameter α_gi equals 1 and 0 otherwise.

The collection frequency of each collection site i with recyclable material m is given by fr_im representing the number of times a collection site needs to be visited within the planning horizon. The minimum and maximum interval between two consecutive collections for recyclable material m are given by Imin_m and Imax_m, respectively.

Three objective functions are addressed in this work to tackle the three sustainability dimensions: the economic objective (z¹(S) ), the environmental objective (z²(S) ), and the social objective (z³(S) ). Let S  be the vector of decision variables; z¹(S) , z²(S) , and z³(S)  the three objective functions; and Ω  the feasible region; the multi-objective problem can be written in the following form:

$$ {\displaystyle \begin{array}{l}\min \kern1.5em \left\{\;{z}^1(S),\kern0.5em {z}^2(S),\kern0.5em {z}^3(S)\right\}\\ {}\mathrm{st}\kern1em S\in \Omega \end{array}} $$

(14.1)

The total distance traveled (z¹(S) ) is given by Eq. (14.2).

$$ {z}^1(S)=\sum \limits_{k\in K}\sum \limits_{t\in T}\sum \limits_{g\in G}{\mathrm{dis}}_k{\tau}_{ktg}+ $$

(14.2a)

$$ \kern0.6em \sum \limits_{j\in {V}_{\mathrm{s}}}\sum \limits_{i\in {V}_{\mathrm{d}}}\sum \limits_{m\in M}\sum \limits_{k\in {K}_{\mathrm{m}}}\sum \limits_{t\in T}\sum \limits_{g\in G}{\mathrm{En}}_{ki}{\tau}_{ktg}{\mathrm{Lo}}_k/{QT}_m2{d}_{ij}- $$

(14.2b)

$$ \sum \limits_{j\in {V}_{\mathrm{s}}}\sum \limits_{i\in {V}_{\mathrm{d}}}\sum \limits_{m\in M}\sum \limits_{k\in {K}_{\mathrm{m}}}\sum \limits_{t\in T}\sum \limits_{\begin{array}{l}g\in G\\ {}{\alpha}_{gj}=1\end{array}}{\mathrm{St}}_{ki}{\mathrm{En}}_{ki}{\tau}_{ktg}{\mathrm{Lo}}_k/{QT}_m2{d}_{ij}+ $$

(14.2c)

$$ \kern0.6em \sum \limits_{\begin{array}{l}g\in G\\ {}{\alpha}_{gi}=1\end{array}}\sum \limits_{\begin{array}{l}k\in K\\ {}{\mathrm{En}}_{ki}=0\\ {}{\mathrm{St}}_{kj}=1\end{array}}\sum \limits_{t\in T}\sum \limits_{\begin{array}{l}i,j\in {V}_{\mathrm{d}}\\ {}\end{array}}2{\tau}_{ktg}{d}_{ij} $$

(14.2d)

The total distance traveled involves, as mentioned, the inbound distance (14.2a), the outbound distance (14.2b and 14.2c), and also a possible extra distance since it is allowed to vehicles based at depot i to perform closed routes from and to depot j (14.2d). The distance (d_ij) of moving a vehicle between depots is then penalized. The outbound distance considers the ending depot of each route and the load collected, to compute the number of needed round-trips to the sorting station. Note that the number of round-trips is not round upward since it is being accounting for the number of round-trips that occur within a finite time period. These are to be repeated in the next period. When, for instance, 10.4 round trips are considered within the period, it means that 10 round trips occur within the period and the 11th occurs in the next period, but some of the load is related to the previous period. It is also considered that if a vehicle, belonging to the sorting station performs closed routes from depot i, the load collected will be unloaded at the sorting station and not at depot i. Therefore, no outbound distance will be accounted for. Term (14.2c) decreases the objective function of such value.

The environmental objective is related to the CO₂ emissions associated with the collection routes and the outbound transportation between depots and the sorting station. Its total value (z²(S) ) given by Eq. (14.3).

$$ {z}^2(S)=\sum \limits_{k\in K}\sum \limits_{t\in T}\sum \limits_{g\in G}{\mathrm{Co}}_k{\tau}_{ktg}+ $$

(14.3a)

$$ \kern0.6em \sum \limits_{j\in {V}_{\mathrm{s}}}\sum \limits_{i\in {V}_{\mathrm{d}}}\sum \limits_{m\in M}\sum \limits_{k\in {K}_{\mathrm{m}}}\sum \limits_{t\in T}\sum \limits_{g\in G}{\mathrm{En}}_{ki}{\tau}_{ktg}{\mathrm{Lo}}_k/{QT}_m\left(\mathrm{Co}{F}_{ijm}+\mathrm{Co}{E}_{ji}\right)- $$

(14.3b)

$$ \sum \limits_{j\in {V}_{\mathrm{s}}}\sum \limits_{i\in {V}_{\mathrm{d}}}\sum \limits_{m\in M}\sum \limits_{k\in {K}_{\mathrm{m}}}\sum \limits_{t\in T}\sum \limits_{\begin{array}{l}g\in G\\ {}{\alpha}_{gj}=1\end{array}}{\mathrm{St}}_{ki}{\mathrm{En}}_{ki}{\tau}_{ktg}{\mathrm{Lo}}_k/{QT}_m\left(\mathrm{Co}{F}_{ijm}+\mathrm{Co}{E}_{ji}\right)+ $$

(14.3c)

$$ \kern0.6em \sum \limits_{\begin{array}{l}g\in G\\ {}{\alpha}_{gi}=1\end{array}}\sum \limits_{\begin{array}{l}k\in K\\ {}{\mathrm{En}}_{ki}=0\\ {}{\mathrm{St}}_{kj}=1\end{array}}\sum \limits_{t\in T}\sum \limits_{\begin{array}{l}i,j\in {V}_{\mathrm{d}}\\ {}\end{array}}2{\tau}_{ktg}\mathrm{Co}{E}_{ij} $$

(14.3d)

The CO₂ emissions for the inbound transportation (routes to collect all collection sites) are given by the first term (14.3a), where the emission value of each route k  is given by parameter Co_k . The CO₂ emissions from the outbound transportation are also considered (terms 14.3b and 14.3c) where larger vehicles are used. Notice that round trips between the sorting station and the depots are performed, with vehicles traveling empty from the sorting station to the depot and in full truckload (FTL) back to the sorting station. The amount of CO₂ emissions for outbound transportation is given by parameter CoF_ijm when the vehicle travels in FTL from depot i to sorting station j with material m and CoE_ij when the vehicle travels empty in the opposite direction. The last term (14.3d) accounts for the CO₂ emissions of a vehicle, based at depot i, traveling empty to depot j to perform closed routes from and to depot j.

As mentioned above, the social objective minimizes the maximum working hours among drivers. The maximum value of vehicle’s total working hours in the planning horizon is given by a positive decision variable DMax when assuming a fixed driver-vehicle assignment (constraint 14.4).

$$ \kern0.5em D\operatorname{Max}\ge \sum \limits_{k\in K}\sum \limits_{t\in T}{\tau}_{ktg}{\mathrm{d}\mathrm{ur}}_k+\sum \limits_{\begin{array}{l}k\in K\\ {}{\mathrm{St}}_{kj}=1\\ {}{\mathrm{En}}_{ki}=0\end{array}}\sum \limits_{\begin{array}{l}i,j\in {V}_{\mathrm{d}}\\ {}i\ne j\end{array}}{\tau}_{ktg}2{b}_{ij},\kern1em \forall g $$

(14.4)

Then, the function for the social objective is given by Eq. (14.5).

$$ {z}^3(S)=D\operatorname{Max} $$

(14.5)

With the objective functions defined, the constraints for the multi-objective model of the MDPVRPI are expressed in constraints (14.6) to (14.13).

$$ \sum \limits_{k\in {K}_{\mathrm{m}}}\sum \limits_{t\in T}\sum \limits_{g\in G}{\tau}_{ktg}{\mu}_{ik}={\mathrm{fr}}_{im}\kern1em \forall i\in {V}_{\mathrm{c}},\forall m $$

(14.6)

$$ \sum \limits_{k\in K}{\tau}_{ktg}{\mathrm{d}\mathrm{ur}}_k+\sum \limits_{\begin{array}{l}k\in K\\ {}{\mathrm{St}}_{kj}=1\\ {}{\mathrm{En}}_{ki}=0\end{array}}\sum \limits_{\begin{array}{l}j\in {V}_{\mathrm{d}}\\ {}j\ne i\end{array}}{\tau}_{ktg}2{b}_{ij}\le H\kern1em \forall t,\forall g,\forall i\in {V}_{\mathrm{d}}:{\alpha}_{gi}=1 $$

(14.7)

$$ \sum \limits_{\begin{array}{l}k\in {K}_{\mathrm{in}}\\ {}{\mathrm{St}}_{ki}=1\end{array}}{\tau}_{ktg}=\sum \limits_{\begin{array}{l}{k}^{\prime}\in {K}_{\mathrm{in}}\\ {}{\mathrm{En}}_{k^{\prime }i}=1\end{array}}{\tau}_{k^{\prime } tg}\kern1em \forall g,\forall t,\forall i\in {V}_{\mathrm{d}} $$

(14.8)

$$ \sum \limits_{g\in G}{\tau}_{kt g}{\mu}_{ik}+\sum \limits_{g\in G}{\tau}_{kt\hbox{'}g}{\mu}_{ik}\le 1\kern1em \forall i\in {V}_{\mathrm{c}},\forall k\in {K}_{\mathrm{m}},\forall m,\forall t,{t}^{\prime}\in T,t>{t}^{\prime },\left(t-{t}^{\prime}\right)\le I{\min}_m $$

(14.9)

$$ \sum \limits_{g\in G}{\tau}_{ktg}{\mu}_{ik}+\sum \limits_{g\in G}{\tau}_{k^{\prime }{t}^{\prime }g}{\mu}_{i{k}^{\prime }}\le 1\kern1em \forall i\in {V}_{\mathrm{c}},\forall k,{k}^{\prime}\in {K}_{\mathrm{m}},\forall m,\forall t,{t}^{\prime}\in T,t>{t}^{\prime },\left(t-{t}^{\prime}\right)\le I{\min}_m $$

(14.10)

$$ \sum \limits_{g\in G}{\tau}_{ktg}{\mu}_{ik}+\sum \limits_{g\in G}{\tau}_{k{t}^{\prime }g}{\mu}_{ik}\le 1\kern1em \forall i\in {V}_{\mathrm{c}},\forall k\in {K}_{\mathrm{m}},\forall m,\forall t,{t}^{\prime}\in T,t>{t}^{\prime },\left(t-{t}^{\prime}\right)>I\mathrm{m}{\mathrm{ax}}_m,\left(t-{t}^{\prime}\right)\le I{\max}_m+I{\min}_m $$

(14.11)

$$ \sum \limits_{g\in G}{\tau}_{ktg}{\mu}_{ik}+\sum \limits_{g\in G}{\tau}_{k^{\prime }{t}^{\prime }g}{\mu}_{i{k}^{\prime }}\le 1\kern1em \forall i\in {V}_{\mathrm{c}},\forall k,{k}^{\prime}\in {K}_{\mathrm{m}},\forall m,\forall t,{t}^{\prime}\in T,t>{t}^{\prime },\left(t-{t}^{\prime}\right)>I\mathrm{m}{\mathrm{ax}}_m,\left(t-{t}^{\prime}\right)\le I{\max}_m+I{\min}_m $$

(14.12)

$$ {\tau}_{ktg}\in \left\{0,1\right\}\kern1em \forall k\in K,\forall t\in T,\forall g\in G\kern1em $$

(14.13)

Constraint (14.6) ensures that a collection site i with material m has to be collected fr_im times over the time horizon. Constraint (14.7) states that the total route duration performed by vehicle g on day t will not exceed the maximum time allowed for a working day (H). If a vehicle g, belonging to depot i, performs a route starting at depot j, the travel time between i and j is considered.

Since all vehicles have to return to their origin depot, constraint (14.8) guarantees that an inter-depot route k, starting at depot i, is part of the solution only if another inter-depot route k′ ends at depot i. Considering all depots i ∈ V_d, constraint (14.8) ensures continuity among inter-depot routes enabling a vehicle rotation.

Constraints (14.9) to (14.12) model the minimum and maximum intervals between consecutive collections which can be performed by the same route or by two different routes. Therefore, constraint (14.9) states that the same route for material m has to be performed with a minimum time interval of Imin_m, while constraint (14.10) considers the case of two different routes collecting the same site i at consecutive collections. Analogously, constraints (14.11) and (14.12) ensure the maximum interval Imax_m between consecutive collections. Variable’s domain is given in constraint (14.13).

Annex B: Solution Procedure

1.1 B.1 Step 1: Routes Generation Procedure

The set of recyclable materials M is involved, and given that each material has to be collected in separated routes, each procedure of step 1 is run independently for each material.

The models involved in each procedure are formulated through MILP formulations based on the two-commodity flow formulation (Baldacci et al. 2004). In such formulations, the network is defined by a direct graph GR = (V, E) with V = V_c ∪ V_d ∪ V_f ∪ V_s, being V_c = {1, …, n} a set of n customers, V_d = {n + 1, …, n + w} a set of w depots, V_f = {n + w + 1, …, n + 2w} a replica of the depots set, V_s = {n + 2w + 1, …, n + 2w + s} a set of s sorting stations, and E = {(i, j) : i, j ∈ V_c ∪ V_d ∪ V_f ∪ V_s, i ≠ j} the edge set.

Each site i ∈ V_c is characterized by a demand p_i and a service duration t_i. The service duration depends on the average time to collect a container (U), on the average distance between containers within a locality (B), on the average speed within localities (vw) and on the number of containers at each locality (c_i), being ${t}_i={c}_i\left(U+\frac{B}{vw}\right)$. The inbound vehicles have a weight capacity of Q and the outbound vehicles QT. The maximum duration for a working day is given by H. Every edge (i, j) has an associated distance d_ij and a travel time b_ij, where ${b}_{ij}=\frac{d_{ij}}{vb}$ and vb is the average speed between localities. An unloading time L is also considered to account for the time to unload a vehicle at the end of each route.

The depot replica set (V_f) is needed since, in the two-commodity flow formulation, routes are defined by paths starting at the real depots and ending at the replica ones. To establish the routes, this formulation requires two flow variables defining two flow paths for any route. One path from the real depot to the replica one modeled by the flow variable representing the vehicle load (variable y_ij). In a collection problem, this load increases along the route. The other path from the replica depot to the real one is given by the second flow variable (y_ji) that models the vehicle empty space which decreases along the route.

These sets, parameters, and variables are the baseline to all route generating procedures which are briefly described in the next sections.

1.1.1 B.1.1 Procedure 1: MDVRP

In the MDVRP only closed routes are defined. A set of routes K is considered and partitioned by depot, K = K₁ ∪ … ∪ K_i, where K_i is the subset of routes belonging to depot i. Decision variables are the binary variables x_ijk that equal 1 if site j is visited immediately after site i on route k (x_ijk = 0, otherwise) and the corresponding reverse variable x_jik when the reverse path is being defined and the flow variables y_ijk and y_jik; and a binary variable δ_ik is defined to assign site i to route k. The objective function also considers the distance to be traveled within each collection site (second term of Eq. (14.14)) and the outbound distance (third term of Eq. (14.14)).

$$ \operatorname{Min}\kern0.5em \frac{1}{2}\sum \limits_{i\in V}\sum \limits_{j\in V}\sum \limits_{k\in K}{x}_{ij k}{d}_{ij}+\sum \limits_{i\in {V}_{\mathrm{c}}}{c}_i\kern0.1em S+2\sum \limits_{i\in {V}_{\mathrm{c}}}\sum \limits_{j\in {V}_{\mathrm{f}}}\sum \limits_{h\in {V}_{\mathrm{s}}}\sum \limits_{k\in K}\frac{y_{ij k}}{QT}{d}_{hj} $$

(14.14)

subject to

$$ \sum \limits_{\begin{array}{l}j\in V\\ {}j\ne i\end{array}}\left({y}_{ijk}-{y}_{jik}\right)=2{p}_i{\delta}_{ik},\kern1em \forall i\in {V}_{\mathrm{c}},\forall k $$

(14.15)

$$ \sum \limits_{i\in {V}_{\mathrm{c}}}\sum \limits_{j\in {V}_{\mathrm{f}}}\sum \limits_{k\in K}{y}_{ijk}=\sum \limits_{i\in {V}_{\mathrm{c}}}{p}_i $$

(14.16)

$$ \sum \limits_{i\in {V}_{\mathrm{c}}}\sum \limits_{j\in {V}_{\mathrm{f}}}\sum \limits_{k\in K}{y}_{jik}\le \mid K\mid Q-\sum \limits_{i\in {V}_{\mathrm{c}}}{p}_i\kern1em $$

(14.17)

$$ \sum \limits_{i\in {V}_{\mathrm{c}}}{y}_{ijk}\le Q\kern1em {\forall}_j\in {V}_{\mathrm{f}},\forall k\in {K}_j\kern1em $$

(14.18)

$$ \sum \limits_{\begin{array}{l}i\in V\\ {}i\ne j\end{array}}{x}_{ijk}=2{\delta}_{jk},\kern1em \forall j\in {V}_{\mathrm{c}},\forall k $$

(14.19)

$$ {y}_{ijk}+{y}_{jik}={Qx}_{ijk}\kern1em \forall i,j\in V,i\ne j,\forall k $$

(14.20)

$$ \sum \limits_{k\in K}{\delta}_{ik}=1\kern1em \forall i\in {V}_{\mathrm{c}}:{p}_i>0\kern1em $$

(14.21)

$$ {\delta}_{ik}={\delta}_{\left(i+w\right)k}\kern1em \forall i\in {V}_{\mathrm{d}},\forall k\in {K}_i $$

(14.22)

$$ \sum \limits_{i\in {V}_{\mathrm{c}}}\sum \limits_{j\in V}{t}_i{x}_{ij k}+\sum \limits_{i\in V}\sum \limits_{j\in V}{b}_{ij}{x}_{ij k}\le 2\left(H-L\right)\kern1em \forall k\in K\kern1em $$

(14.23)

$$ \sum \limits_{j\in {V}_{\mathrm{c}}}{x}_{ijk}\le 1\kern1em \forall i\in {V}_{\mathrm{d}},\forall k\in {K}_i\kern1em $$

(14.24)

$$ \sum \limits_{i\in {V}_{\mathrm{c}}}{x}_{ijk}=0\kern1em \forall j\in {V}_{\mathrm{f}},\forall k\notin {K}_j\kern1em $$

(14.25)

$$ \sum \limits_{j\in {V}_{\mathrm{c}}}{x}_{ijk}=0\kern1em \forall i\in {V}_{\mathrm{d}},\forall k\notin {K}_i\kern1em $$

(14.26)

$$ {y}_{ijk}\ge 0\kern1em \forall i,j\in V,k\in K\kern1em $$

(14.27)

$$ {x}_{ijk}\in \left\{0,1\right\}\kern1em \forall i,j\in V,k\in K\kern1em $$

(14.28)

$$ {\delta}_{ik}\in \left\{0,1\right\}\kern1em \forall i\in {V}_{\mathrm{c}},k\in K\kern1em $$

(14.29)

The above formulation is an extension for the MDVRP of the formulation proposed by Baldacci et al. (2004) for the CVRP. Constraints (14.15) to (14.20) are rewritten since it is considered index k and the binary variable δ_ik. Constraints (14.21) to (14.26) are new constraints that deal with multiple depots and duration constraints. Equation (14.21) guarantees that each locality with positive demand has to be visited by a single route. Constraint (14.22) matches the real depots with their replica, ensuring that a route will start at the real depot and will end at the corresponding replica. Constraint (14.23) guarantees that the duration of each route does not exceed the maximum allowed routing time. Constraint (14.24) ensures that each route will leave its home depot at most once. Finally, constraints (14.25) and (14.26) jointly ensure that a vehicle route cannot leave and return to a depot other than its home depot (real and replica depot). The new variable definition is given in Eq. (14.29).

The proposed formulation, when applied to large instances, is computationally difficult to solve. Therefore, a solution method is proposed to solve the MDVRP (see Fig. 14.12). First, a problem where both closed and open routes are allowed, is solves, the MDVRP with mixed closed and open routes (MDVRP-MCO). The MDVRP-MCO formulation is proposed in the work of Ramos et al. (2013) and is capable of dealing with large instances. Moreover, the majority of the routes in the solution for the MDVRP-MCO are feasible for the MDVRP – the closed routes. For “(the open routes)”, the MDVRP formulation is applied having, as input data, only the sites belonging to each open route.

Fig. 14.12

Solution method for the MDVRP

1.1.2 B.1.2 Procedure 2: MDVRPI

The MDVRPI allows inter-depot routes, where vehicles have to return to the home depot on the same working day. Therefore, a vehicle rotation is limited by the maximum duration of a working day (H). To solve the MDVRPI, the solution methodology proposed by Ramos (2012) was used, considering an unlimited vehicle fleet. A MDVRPI Relaxation is solved where inter-depot and closed routes are obtained (see Fig. 14.13). This formulation corresponds to the MDVRP-MCO formulation to which adds constraint (14.30).

$$ \sum \limits_{j\in V}{x}_{ij}+\sum \limits_{j\in V}{x}_{ji}=\sum \limits_{j\in V}{x}_{\left(i+w\right)j}+\sum \limits_{j\in V}{x}_{j\left(i+w\right)}\kern1em \forall i\in {V}_{\mathrm{d}} $$

(14.30)

Fig. 14.13

Solution method for the MDVRPI

Constraint (14.30) guarantees that the number of routes departing from one depot is equal to the number of routes arriving at that depot. This ensures connectivity between the inter-depot routes and the rotation concept, i.e., a vehicle returns to its home depot. However, it is not guaranteed that the vehicle returns within a working day since no duration constraints for rotation are considered in the MDVRPI Relaxation. Notice that in the two-commodity formulation, to any real depot i ∈ V_d, a corresponding copy depot assumed i + w ∈ V_f (w is the number of depots), and the x_ij and x_ji modeled the opposite paths.

For the inter-depot routes obtained from the solution of the MDVRPI Relaxation, rotations are defined by linking the inter-depot routes until one reaches the starting depot. The duration of each rotation is then assessed. For rotations that do not respect the working-day time limit, the MDVRPI formulation is solved and rotations redefined to comply with the imposed limit. As a solution, one can have inter-depot routes belonging to rotations that satisfy the maximum duration for a working day and/or closed routes. More details can be founded in Ramos (2012).

1.1.3 B.1.3 Procedure 3: MDVRPI Extension

The MDVRPI Extension solves the problem by visiting all sites only by inter-depot routes. For that, the MDVRPI Relaxation is used, but instead of considering all depots and all collection sites at the same time, only two depots are considering in each run, and only the closest sites to those depots are made available to be collected. Moreover, a constraint is added to enforce routes to start and end at different depots. As a result, only inter-depot routes are defined.

A pair of depots [dp, dp^′] ∈ V_d is considered at a time, and constraints (14.31) and (14.32) are added to the MDVRPI Relaxation formulation, imposing that all routes have to start at depot dp and end at depot dp^′ to obtain a solution with only inter-depot routes between each pair of depots.

$$ \kern0.5em {x}_{ij}=0,\kern0.5em \forall i\in {V}_{\mathrm{c}},j= dp+w $$

(14.31)

$$ \kern0.5em {x}_{ij}=0,\kern0.5em \forall j\in {V}_{\mathrm{c}},i=d{p}^{\prime } $$

(14.32)

Regarding the maximum duration for each inter-depot route in this procedure, it is considered the value $\left(H-L-{b}_{dp,d{p}^{\prime }}\right)$ to guarantee that the vehicle can return to the origin depot within a working day.

After running the three procedures, the set K is built. Each route k ∈ K is characterized by mileage (dis_k), duration (dur_k), load (Lo_k), and CO₂ emissions (Co_k). The first three parameters are provided by the solutions of the problems solved. The last one, the CO₂ emissions, has to be assessed a posteriori. For that, the emission model proposed by Barth et al. (2004) was used. When a vehicle travels over an arc (i,j), it is assumed that it emits a certain amount of CO₂, which depends on the fuel consumption that, in turn, is a function of many factors (such as, distance traveled, vehicle load – curb weight plus load – speed, road angle, engine features, vehicle frontal surface area, coefficients of rolling resistance and drag, and air density, among others (see Barth et al. 2004)). The conversion factor of 1 l of diesel fuel containing 2.6676 kg of CO₂ was assumed (as proposed in Defra). Note that CO₂ emissions were considered on arcs and nodes since nodes represent collection sites aggregating one or more containers and a certain mileage is traveled within each node.

The computation of the CO₂ emissions for all routes k ∈ K concludes step 1.

1.2 B.2 Step 2: Solution Method for the Multi-objective Problem

In step 2 the multi-objective problem defined in Sect. 14.3 is solved (Fig. 14.4). In such problems it is rarely the case a single point optimizes simultaneously all objective functions (Coello and Romero 2003); therefore trade-offs between the objectives have to be analyzed in line with the notion of Pareto optimality. A solution is Pareto optimal if there exists no feasible solution, which improves one objective without causing a deterioration in at least one other objective. This concept generally does not apply to a single solution, but rather a set of solutions called the Pareto optimal set. The image of the Pareto optimal set under the objective functions is called Pareto front.

The improved version of the traditional ε-constraint method is applied to the problem so that the Pareto front is generated. Mavrotas (2009) proposes that the objective function constraints are transformed into equations (instead of inequalities as in the conventional method) by incorporating slack or surplus nonnegative variables, which are then used as penalization factors in the objective function. This augmented ε-constraint method produces only efficient solutions. In this work three objective functions exists; therefore a total of (q₂ + 1) × (q₃ + 1) runs are performed to obtain the Pareto front, when q₂ and q₃ are the equal amplitude intervals partitioning the range of each objective function. When the problem becomes infeasible, it means that there is no need to further constrain the corresponding objective function as it will from then on lead to infeasibility (more details in Mavrotas 2009).

When solving the problem under analysis in this work, where three objectives are being tackled, an approximation to the Pareto front is designed by using the augmented ε-constraint method, where the economic objective is optimized and the social and environmental constrained (see Table 14.4).

Table 14.4 Pseudo-code of the augment ε-constraint method

Finally, to propose a sustainable solution, that is, a compromise solution between the three objectives, a compromise solution method (Yu 1985) is applied, where the Pareto optimal solution closest to the ideal point is obtained. The ideal point (z_I) is defined according to the individual minima of each objective ${z}_{\mathrm{I}}=\left({z}_{\mathrm{min}}^1,{z}_{\mathrm{min}}^2,{z}_{\mathrm{min}}^3\right)$, while the nadir point (z_N) is defined according to the worst values obtained for each objective (${z}_{\mathrm{N}}=\left({z}_{\mathrm{max}}^1,{z}_{\mathrm{max}}^2,{z}_{\mathrm{max}}^3\right)$. To apply this method, the objective functions are normalized by the differences between the nadir and ideal points, measuring the variability of the objective function within the Pareto set. Afterward, the compromise solution is obtained by minimizing the distance from the Pareto front to the ideal point, where the Tchebycheff metric is used as distance measure:

$$ \min \left\{\underset{j=1}{\overset{\phi }{\max }}\left\{{\lambda}_j\left|{z}^j(S)-{z}_{\mathrm{I}}^j\right|\right\}:S\in \Omega \right\} $$

(14.33)

where ϕ is the number of objective functions in study and λ_j the normalized factor for each objective function:

$$ {\lambda}_j=\frac{1}{r_j}{\left[\sum \limits_{i=1}^{\phi}\frac{1}{r_i}\right]}^{-1} $$

(14.34)

$$ {r}_j={z}_{\mathrm{max}}^j-{z}_{\mathrm{min}}^j $$

(14.35)