Statement of the problem

This paper aims at solving the MDHFPCVRP-TW problem presented in [2] using a novel formulation. The problem reflects a real-life scenario in the vending industry in Medellín, Colombia in which the objective is to plan routes a fleet of heterogeneous vehicles in a week (six days in this specific case) to minimize the total elapsed time fulfilling the clients’ demands. The final solution must indicate to the vending company how to route every vehicle each day in order to fulfill the clients’ demands.

Some fundamental considerations to take into account in this problem are:

• Although clients have hard time windows which are not day-dependent, vehicles are allowed to arrive before according to a maximum stand-by time allowed for each client. The service must end before the time window closes.
• Clients have some allowed delivery patterns indicating the specific days of the week in which demand can be delivered.
• Clients have a fixed demand in terms of vending machines. This demand is the same in every visit.
• Depots have a time window which comprehend the maximum duration of a route.
• Depots have a determined capacity in terms of number of vending machines that they can attend per day.
• Each vehicle operates under capacity constraints, limiting the number of vending machines it can service within a daily route. All routes must originate from and terminate at the same depot, forming a closed circuit.

New mathematical formulation

In this new formulation, variables take the value one if arc is used on day h and zero otherwise (note that a vehicle index is not required). What differentiates it from the model 3IF presented in [2] is that instead of relying in binary variables like and to assign a node and a depot to a vehicle, we use continuous variables to do it. For this formulation to work, we replicate every depot such that we have a depot for each type of vehicle that can be used. This allow us to know which vehicle type performs a route. Then we add a parameter for each original depot. Using and relating variable and parameter we know which depot served every client and guarantee that each vehicle returns to the depot they departed at the end of the route. This formulation works for any quantity of periods even tough here a six-day week is considered.

The formulations use the following sets:

• C is the set of clients, where .
• D is the set of depots (including replicated depots).
• N is the set of all nodes, conformed by clients and depots ().
• F is the set of possible frequencies of visit.
• CP_f is the set of clients with frequency f, where and .
• K is the set of types of vehicles.
• H is the set of days. As referenced before, the scheduling span is of 6 days in this research.
• Directed complete graph G(A,N) where is the set of arcs.
• P_f is the set of possible patterns of visit of frequency f.
• DD_d is the set of depots replicated from depot d plus itself.
• DK_k is the set of depots assigned to type of vehicle k.

Let us illustrate sets DD_d and DK_k using an example: if the problem has two depots and two types of vehicles , depot d₁ has a replica for vehicle type , and depot d₂ has a replica for vehicle type . Then we have, set , set , set , set , and .

The formulation uses the following parameters:

• q_k is the capacity of the vehicle of type k as a function of the number of vending machines that it can attend daily.
• q is the maximum capacity among available types of vehicles.
• dem_c is the number of vending machines of client c that must be supplied in each visit.
• s_c is the time to serve client c.
• is the time window in which node i must be attended (the upper bound for arriving at a depot is the maximum time duration allowed for a route). All the depots share the same upper bound.
• l_c is the maximum stand-by time allowed in client c.
• t_ij is the time to traverse arc (i,j) with any vehicle.
• R_d is the number of vending machines that depot d can attend daily.
• is the number of vehicles available of type k.
• is the number of visits that must receive a client with frequency f in the planning horizon.
• is an element of matrix A, which relates patterns with the days a client is visited. It takes the value 1 if pattern p forces clients to be visited on day h and 0 otherwise.
• A largely enough value (M) is used. According to the types of restrictions where M is used, it could be taken as the upper bound of time window of the depots.
• is a positive natural number assigned to depot d. It is used in the model’s auxiliary variable to relate the route of some vehicle that departed from d. is different for every element of set D.
• is the maximum value of .
• tmin_c is the minimum distance from client c to any depot.

The complete model is now presented. Let us start with the variables:

• : binary decision variable which takes the value 1 if the arc (i,j) is used in the day h, and 0 otherwise.
• : binary decision variable which takes the value 1 if pattern p is assigned to client c, and zero otherwise.
• : time at which a vehicle arrives to node j coming from node i on day h. Each vehicle can depart from any depot starting after the opening of the time window of such depot.
• : load of the vehicle when it is traversing arc (i,j) on day h.
• : auxiliary decision variable to know if a route is assigned to a depot. It takes the value of if the arc (i,j) is served with a vehicle that departed from d, and zero otherwise.
• : stand-by time of the vehicle visiting client c on day h.

Model (1) to (27) presents the 3IF version 2 (3IFv2).

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

The objective function (1) minimizes the total time of all routes, specifically, it considers traveling times, service times, and stand-by times. As the operating cost of the enterprise is principally based on time spent on a route, all times are given the same importance.

Constraints (2) to (4) define that every client is visited the number of times that their frequency dictates with a feasible pattern: (2) define the number of visits of every client, (3) guarantee that each client has an assigned pattern of visit, (4) force one route to visit a client each day the chosen pattern indicates it must be visited. Constraints (5) ensure that every client on any day is left after being visited. Constraints (6) to (8) condition the arrival time of the vehicles in the time window of every client, and the stand-by time of the vehicles: (6) and (7) require that each visited client respect the time window, considering the service time and the possible stand-by time, while (8) force the vehicles on any day to comply with the maximum stand-by time allowed in every client. Constraints (9) to (12) condition the time of arrival to every node. Constraints (9) and (10) allow to compute, when , the arrival time to node j coming from node c considering arrival time to node c, traveling time, service time and stand-by time. They become inactive when . (11) guarantee that the arrival time to the first client from a depot considers the time required to travel from the depot to this first client. (12) force the arrival time to a client from a depot to be zero if the client is not visited directly from that depot. Note that vehicles are not forced to leave the depot on time zero. Constraints (13) indicate that the difference between load in and load out at a node must be equal to the node’s demand.

Constraints (14) guarantee that the number of used vehicles does not exceed the number of vehicles available of each type. Constraints (15) do not allow the load of used vehicles to exceed its capacity. Constraints (16) force the accumulated load departing from every depot to be limited by its capacity. Constraints (17) limit the number of vehicles that depart from a depot to be equal to the number of vehicles that arrive at such depot. Constraints (18) to (21) assign each route to a depot: (18) force the auxiliary variable that relates an arc with a depot to be zero if such arc is not used, (19) guarantee that each route departing from a depot it assigned to it, (20) make each route to arrive to its origin depot, and (21) are the flow constraints.

Constraints (22) to (27) indicate the domain of the decision variables.

The main difference between 3IF and 3IFv2 lies in how routes are assigned to depots and tracked in the model. While 3IF uses binary variables and to explicitly assign nodes and depots to clients and trace vehicle usage, 3IFv2 employs a different approach with continuous auxiliary variables and depot replication. This transformation allows 3IFv2 to naturally capture vehicle-depot assignments through the model’s structure: each depot is replicated for each vehicle type, and the auxiliary variable takes the value when an arc is served by a vehicle from depot d, effectively tracking depot assignments and route continuity. Beyond simplifying the variable structure, this approach reduces the total number of binary variables in the model while maintaining its ability to enforce routing constraints. Additionally, 3IFv2 introduces parameter and its maximum value , which work in conjunction with the auxiliary variables to ensure proper route construction and depot assignments, providing a more streamlined way to model the routing decisions compared to the explicit assignment approach of 3IF.

The inclusion of variables is the most important concept of this paper. As a consequence, an example of its usage is presented in Fig 1. In this example, index h is dropped because the relations of the variable are the same for each day. Let us consider a graph with five clients (1 to 5), two depots (d₁ and d₂ with assigned and respectively), and two types of vehicles ( is represented by discontinuous lines and by continuous lines). Because there are two types of vehicles, each depot have to be replicated once (hence we have depots and for ; and the others for ).

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Fig 1. Example to understand the usage of variables .

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Note that ’s route is . Meanwhile, ’s route is . Variables x_ij and for that particular day are represented as:

Note that lets the formulation know that a vehicle of type departed depot two (), and a vehicle of type departed depot one ().

Dantzig, Fulkerson, and Johnson subtour elimination constraints

The classical VRP has three main constraints that must be taken into account: the routes must start and come back to the depot, all clients must be served, and vehicles must leave any clients that they visit. The problem is that these three conditions do not always result in one Hamiltonian cycle. In Fig 3, a solution considering the mentioned constraints is shown. In this solution, the vehicle starts and returns to the depot, and the vehicle leaves all visited clients. But this solution is not feasible because it does not represent a Hamiltonian cycle.

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Fig 3. Example of a subtour.

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Eliminating subtours efficiently has been an important task for operations research practitioners since the TSP was proposed. There are some alternatives to do so, and they are described in [5]. One of these alternatives could be added as valid inequalities iteratively. Proposed by [6], it is usually abbreviated as the DFJ constraints (as they were proposed by Dantzig, Fulkerson, and Johnson). These are the DFJ constraints:

Note that, if the solution in Fig 3 is no longer feasible as and .

The problem with the DFJ constraints is that adding all of them is not practical for large instances. This is because they are exponential. But, adding them iteratively strengthens the linear relaxation of the MILP, so they are often used as valid inequalities.

Comb inequalities

The comb inequalities were introduced to the TSP by [7]. These inequalities make the problem’s linear relaxation tighter. Let us present an example. The graph in Fig 4 represents the solution of a linear relaxation of a TSP.

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Fig 4. TSP’s graph representing an infeasible solution.

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The weight of arcs in the graph are the optimal values of the x variables. Continues lines represent arcs with value 1, while dashed ones represent arcs which variables are fractional. But as a linear relaxation, it works perfectly as every node receives one unit of flow and delivers one unit of flow. To break this infeasible solution, a comb inequality can be added to the linear relaxation of the problem. A comb consists of a node-set H (which is called the handle) and node sets (the teeth) such that: and odd. Also, and are not empty for all j. In Fig 5 a comb is presented for the example with p = 3. In this case, , S₁ := {2,5}, S₂ := {1,4}, and S₃ := {3,6}.

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Fig 5. Design of a comb over a graph.

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The comb inequalities for directed graphs are formulated as follows:

This inequality breaks the previously presented infeasible solution. Let us break the inequality step-by-step: , , , and . So, the left side of the comb inequality for the non-feasible solution is . The right side with p = 3 is 5, which is greater than 4.5. This comb inequality cuts the infeasible TSP solution from the linear relaxation.

[8] presented how the comb inequalities could be used in a VRP within a directed graph. The comb inequality holds, but the depot must be included in the handle.

Multi-star inequalities

The Multi-star inequalities define a group of valid inequalities that consider the maximum number of arcs that can be assigned to a route. Let us define Q as such a number (Q can be obtained considering the capacity of the vehicles, the demand of the clients or other attributes like vehicle autonomy, route length limit, depot capacity). The multi-star inequalities were proposed by [9] for the undirected VRP. Later, [10] presented the multi-star inequalities for directed VRPs.

The multi-star inequalities require two disjoint sets of nodes (S₁ and S₂) that do not include the depot. A multi-start inequality could be written as follows:

Let us present an example. In Fig 6, a graph with five clients is presented (it represents a partial solution, other nodes are not depicted). Set S₁ has clients one, two, and three, and set S₂ has the remaining clients. Let Q = 3. A fractional solution is presented. Note that the sum of all the used arcs totals three, complying with the value of Q. A multi-star inequality breaks this fractional solution because , , and . This results in the left side of the inequality being , which is greater than the right side ().

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Fig 6. Example of a multi-star inequality.

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Unfortunately, using these valid inequalities were challenging in the previous formulation. The reason is simple: due to variables u_cp conditioning how clients are served in the different days of the problem, we do not have a notion on which arcs can be used. We tried using them only with clients that must be served everyday, but the results indicated no improvement in the performance of the solver. If a matheuristic approach decides when clients must be served first, and then the routing, all of these valid inequalities could be used. As a consequence, in the following section, we present a mix of valid inequalities and equalities, and preprocessing techniques to strengthen the linear relaxation of the MDHFPCVRP-TW.

Valid inequalities and formulations for the PVRP

The PVRP was first studied in 1974 [11], motivated by waste collection challenges. However, the term “period routing problem” was not coined until ten years later [12]. According to [13], prior to 2010, the literature on exact methods applied to the PVRP remained limited. [13] proposed an exact method for solving the PVRP using four distinct classes of lower bounds based on problem reduction through constraint elimination, followed by the incorporation of valid inequalities from the CVRP (particularly generalized capacity constraints and clique cuts).

In [14], the authors presented seven valid inequalities for the PVRP with due dates, adapting these inequalities to three different formulations: flow based, flow based with assignment variables, and load based.

In the load based formulation, variables serve the same purpose as in our formulation. It requires an additional set of variables which indicate the load of a vehicle upon traversing arc (i,j) on day h as in our formulation. As a consequence, we can consider the load based formulation in [14] as a similar formulation to the one presented in this paper. The authors presented seven inequalities which were based on the demand served in a subset , or in the number of used arcs to guarantee that the demand is met. We did not try these type of valid inequalities because in our formulation it is not possible to know when a client is visited beforehand.

The other two formulations proposed in [14] had the following characteristics: the flow based formulation used binary variables to indicate if vehicle k traversed arc (i,j) on day h; it resembles the 4IF that we presented previously on [2]. The flow based formulation with assignment variables included binary variables which indicated if client i was visited by vehicle k on day h; it resembled the 3IF that we presented previously on [2].

Even though [13] in 2010 opened the door for the usage of more exact-driven methods to solve the PVRP, the bibliography still lacks depth. Additionally, most of the used valid inequalities for the PVRP are adaptations of valid inequalities for the CVRP which require the knowledge of the time period in which each client is served. Our formulation does not comply with this characteristic. As a consequence, we decided to proposed our own set of valid constraints and preprocessing techniques for our version of the problem.

Not considering variables

By design, the proposed formulation does not consider some variables. Computationally, these variables are not even created to produce a more efficient formulation.

The variables that are not defined are:

• : patterns that cannot be assigned to certain clients. (PP1)
• : arc (i,j) cannot be used if it is impossible to meet the time window’s requirement of client j. Note that is the earliest possible arrival from i to j and is the later j can be visited. (PP2)

Pattern assignation to avoid symmetry

Pattern assignation is decided in the model using variable u_cp. For clients on CP₆ that must be visited daily, there is only one pattern available. But for other clients, multiple patterns are available as presented in Table 1.

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Table 1. Possible patterns according to f.

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It is important to remark that patterns are very symmetric because a condition of the enterprise is that the visits to each client are equally spaced in the planning horizon. This can be seen in Table 2. Considering that the other conditions of the problem (time windows, demands, times to traverse arcs, etc.) are not day-dependent, given a set of customers, attending all of them represents the same routing problem independently of the day chosen to serve them.

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Table 2. Visits distribution according to pattern p.

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Let us illustrate this concept with a particular example. Suppose a client distribution where the client 1 has f = 6, clients 2 and 3 have f = 3, clients 4–6 have f = 2, and client 7 has f = 1. A possible pattern distribution to attend the aforementioned clients can be seen in Table 3 where the last column represents the value of the variable u_cp for each client supposing day 1 is Monday. But this is not an obligation as a company can start planning their routes on a Tuesday. So if day 1 is Tuesday the solution would be the same (remember that the routing is not day-dependent) but the values of u_cp change. Now . With the same logic, day 1 could be Wednesday, Thursday, Friday or Saturday (only the values of u_cp change). As a consequence, there are six symmetric solutions that would bare the same result.

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Table 3. Visits distribution of a given instance.

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To eliminate this symmetry, client seven could be forced to be served according to a given pattern (in this case u_7,7 = 1). To generalize this assumption and reduce symmetry in the problem, the following equation is proposed (let us suppose n₁ is the first client with frequency 1).

(28)

This logic could also be applied to the first client with frequency two (n₂), and the first client with frequency three (n₃). Let us present two other assignment equations:

(29)

(30)

Note that these equations cannot be used at the same time as this can eliminate more feasible solutions apart of the symmetrical ones (and even optimal ones)

Valid inequalities

Variables u_cp can be bounded considering the total possible demand the available vehicles can serve in a day. This can be done by comparing the demand served each day due to the assignment of u_cp to every client, with the aggregated capacity of the vehicles. The following constraints must be satisfied in order for the vehicles to meet daily demand.

We can transform this into a valid inequality for a subset of the total of clients each one with a given pattern p_c:

(31)

Another exploitable characteristic of the problem is related with the vehicles’ capacity and the clients’ demand. Let us consider an example: in the dataset presented in [2] (https://github.com/aarenas2/MDHFPCVRP-TW) the vehicles’s capacities are 12 (small) and 16 (medium). If client 1 has a demand of 10 vending machines, and client 2 a demand of 7 vending machines, this clients cannot be attended with the same vehicle. As a consequence, on any given day h, . If we group three clients (1, 2 and 3) instead of two, and the sum of the clients’ demands is greater than 16, now we have . This inequality can be interpreted as: the maximum number of active arcs in the subset of clients is one (more than one implies that the same vehicle visited all the nodes which is infeasible). Let us generalize this valid inequality for any set of clients as:

(32)

Effects of valid constraints and preprocessing on linear relaxation

In this subsection, the effect of preprocessing techniques PP1 and PP2, and constraints (28) to (32) over the linear relaxation of the proposed model are evaluated. To do so, all the datasets used in [2] with total number of visits (comprising 15 datasets) as those were the larger instances. The performance is measured in the average GAP from the best known value of the linear relaxation for each setup, and in the number of times each setup resulted in the best known linear relaxation. As (31) and (32) can be done with multiple sets of clients, for (31) we used every possible combination of three clients, meanwhile, for (32) we used every possible combination of three clients ((32)A) and four clients ((32)B). Let us present the results in Table 4 where BKN represents the number of times the best known linear relaxation was found using the technique on the column.

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Table 4. Results of the preprocessing techniques and valid constraints over the linear relaxation of the 3IFv2.

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The results in Table 4 demonstrate the impact of different preprocessing techniques and valid constraints on the model’s linear relaxation performance. The inequality using combinations of four clients (32)B proved most effective, achieving a 0.14% average GAP and finding the best known solution in 13 out of 15 instances. PP2 showed moderate effectiveness with a 3.93% average GAP and found the best solution twice, while using combinations of three clients (32)A achieved a 4.48% average GAP. PP1 provided no improvement over the original formulation, both showing 4.87% GAP. This could be due to constraints (3) already forcing those variables to zero. Nevertheless, PP1 is useful reducing the number of considered binary variables in the model. Pattern assignment constraints ((28), (29), (30)) and client combination constraints (31) delivered minimal improvements, with GAPs ranging from 4.78% to 4.87%. These results indicate that capacity-based inequalities incorporating larger client combinations were most successful at tightening the linear relaxation bounds, while simpler preprocessing and symmetry-breaking constraints demonstrated limited effectiveness in improving the model’s performance.

The final formulation used in the following section was the original 3IFv2 ((1) to (27)) plus:

• PP1 and PP2: even though did not improve much the linear relaxation, they reduce the number of binary variables.
• (29): eliminated some symmetry and had the best improvement on the linear relaxation of the pattern assignment constraints.
• (32)B: produced the tighter lower bounds.

Results of the 3IFv2 versus previous formulations

Following the experimental methodology from [2], we evaluated the performance of 3IFv2 against the 4IF and 3IF formulations. The experiments were conducted on the same set of instances, grouped by total visits required ( and 120) and client frequency distribution (dense, balanced, and sparse). Each configuration was tested with five different randomly generated instances using identical parameters, with a 3600-second time limit. Results can be seen in Tables 5 to 8. The code to generate the instances can be found in https://github.com/aarenas2/MDHFPCVRP-TW).

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Table 5. Results with .

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Table 6. Results with .

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Table 7. Results with .

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Table 8. Results with .

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The results demonstrate that 3IFv2 consistently outperforms both previous formulations across all instance sizes and distributions. For small instances (), 3IFv2 achieves optimality in almost all cases (number in parenthesis next to Avg. GAP), with average GAPs of 0% for both dense and sparse distributions, and 0.1% for balanced instances. The time to find first feasible solutions (TFFFS) remains competitive with 3IF while significantly improving upon 4IF.

As instance size increases, 3IFv2’s advantages become more pronounced. For , it maintains single-digit GAPs (0.7%-4.4%) while both 4IF and 3IF show deteriorating performance (3.8%-12.3%). The TFFFS values for 3IFv2 are consistently lower, often by an order of magnitude compared to the other formulations.

The most striking improvements appear in large instances (). For , while 4IF fails to find solutions in most cases and 3IF struggles with feasibility (2-3 failures out of 5), 3IFv2 finds solutions in nearly all instances. The GAPs for 3IFv2 remain reasonable (3.5%-8.6%) compared to 3IF’s 7.2%-15.0%. Most notably, 3IFv2’s TFFFS values stay manageable (81.2-1687.7 seconds) when both other formulations frequently exhaust the time limit.

Finally, the fact that the 3IFv2 was able to find a feasible solution in 58 of 60 datasets demonstrates the strength of the formulation.

The consistent feasibility, lower GAPs, and faster solution times across all instance sizes demonstrate that 3IFv2’s structural improvements and valid constraints effectively strengthen the formulation while maintaining computational tractability.

Unfortunately, in the instance MedellinVending262, the 3IFv2 was not able to find a feasible solution after 12 hours. On the other hand from the 3IF and 4IF which exhausted the local memory of the CPU, the 3IFv2 improved the lower bound until time-limit was reached.