Vehicle routing problem

There are many algorithms that have been developed for the VRP. Because the VRP is known to be NP-Hard, many of these are heuristics. For this network problem, both an MIP formulation and a heuristic method are used in this research. Since we need to solve many VRPs while applying a looping factor, the MIP formulation is used for smaller problems where the VRPs can be solved efficiently, while the heuristic is used when the VRPs take too much time to solve optimally.

5.6.3.1 MIP Formulation

In this section, we describe a mathematical formulation corresponding to the VRPs that we solve. This formulation considers the vehicle type to use for each loop and its capacity. It uses a binary variable as a vehicle flow variable to show if there is travel between two locations using a specific vehicle.

Notation

𝐶𝐶𝑖𝑖𝑖𝑖𝑡𝑡𝑇𝑇 , 𝑑𝑑𝑖𝑖, 𝐵𝐵𝑖𝑖, 𝑃𝑃𝑡𝑡𝑇𝑇, 𝐶𝐶, 𝐻𝐻, and 𝑇𝑇 follow the same notation as the network MIP. Define 𝑥𝑥𝑖𝑖𝑖𝑖𝑡𝑡 ∈ {0,1}: 1 if 𝑖𝑖 and 𝑗𝑗 are connected using vehicle type 𝑡𝑡; 0 otherwise.

𝑦𝑦𝑖𝑖𝑖𝑖𝑡𝑡 ∶ amount of vaccine transported from 𝑖𝑖 to 𝑗𝑗 using vehicle type 𝑡𝑡. The following MIP is used to solve the VRP:

Min � � 𝐶𝐶_{𝑖𝑖𝑖𝑖𝑡𝑡}𝑇𝑇 _𝑥𝑥 𝑖𝑖𝑖𝑖𝑡𝑡 𝑖𝑖,𝑖𝑖∈𝐶𝐶∪𝐻𝐻 𝑡𝑡∈𝑇𝑇 (81) 𝑠𝑠𝑠𝑠𝑏𝑏𝑗𝑗𝑀𝑀𝑐𝑐𝑡𝑡 𝑡𝑡𝑓𝑓 � � 𝑥𝑥𝑖𝑖𝑖𝑖𝑡𝑡 𝑖𝑖∈𝐶𝐶∪𝐻𝐻 𝑡𝑡∈𝑇𝑇 = 1 _{for ∀𝑗𝑗 ∈ 𝐻𝐻 (82)} � 𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 𝑖𝑖∈𝐶𝐶∪𝐻𝐻 − � 𝑥𝑥𝑖𝑖𝑖𝑖𝑡𝑡 𝑖𝑖∈𝐶𝐶∪𝐻𝐻 = 0 _{for ∀𝑝𝑝 ∈ 𝐻𝐻, ∀𝑡𝑡 ∈ 𝑇𝑇 (83)} � � 𝑦𝑦𝑖𝑖𝑖𝑖𝑡𝑡 𝑖𝑖∈𝐶𝐶∪𝐻𝐻 𝑡𝑡∈𝑇𝑇 − � � 𝑦𝑦𝑖𝑖𝑖𝑖𝑡𝑡 𝑖𝑖∈𝐶𝐶∪𝐻𝐻 𝑡𝑡∈𝑇𝑇 = 𝑑𝑑𝑖𝑖 for ∀𝑗𝑗 ∈ 𝐻𝐻 (84) 𝐵𝐵𝑖𝑖𝑥𝑥𝑖𝑖𝑖𝑖𝑡𝑡 ≤ 𝑦𝑦𝑖𝑖𝑖𝑖𝑡𝑡 ≤ 𝑃𝑃𝑡𝑡𝑇𝑇𝑥𝑥𝑖𝑖𝑖𝑖𝑘𝑘 for ∀𝑖𝑖, 𝑗𝑗 ∈ 𝐶𝐶 ∪ 𝐻𝐻, 𝑖𝑖 ≠ 𝑗𝑗, ∀𝑡𝑡 ∈ 𝑇𝑇 (85) 𝑦𝑦𝑖𝑖𝑖𝑖𝑡𝑡 ≥ 0 for ∀𝑖𝑖, 𝑗𝑗 ∈ 𝐶𝐶 ∪ 𝐻𝐻, 𝑖𝑖 ≠ 𝑗𝑗, ∀𝑡𝑡 ∈ 𝑇𝑇 (86) 𝑥𝑥𝑖𝑖𝑖𝑖𝑡𝑡 ∈ {0, 1} for ∀𝑖𝑖, 𝑗𝑗 ∈ 𝐶𝐶 ∪ 𝐻𝐻, 𝑖𝑖 ≠ 𝑗𝑗, ∀𝑡𝑡 ∈ 𝑇𝑇 (87)

Constraints (82) and (83) ensure that a facility is visited exactly once and that if a vehicle visits a location, it must also depart from it. Constraint (84) specifies that the difference between

the quantity of vaccines a vehicle carries before and after visiting a facility is equal to the demand of that facility. Constraint (85) ensures that the vehicle capacity is never exceeded.

5.6.3.2 Heuristic method

Our heuristic uses a constructive method based on the algorithm of Clark and Wright (1964). In this algorithm, point-to-point routes are combined to form a loop by choosing the routing path that gives the largest transportation cost savings at each iteration until every location is linked. For our network problem, vehicle type is considered when the savings on the route are calculated. For checking if a route is feasible, both vehicle capacity and trip distance are considered.

Modified Clark and Wright algorithm

Label the delivery locations as 1, 2, ..., n and label the origin as 0.

Determine the costs 𝐶𝐶_{𝑖𝑖𝑖𝑖𝑡𝑡}𝑇𝑇 to travel between all pairs of delivery locations and between each delivery location and the origin and for each vehicle type, i.e., for i=0, 1, .., n; j=0, ..., n and j≠i ,

t∈T

1. Calculate the savings 𝑆𝑆_{𝑖𝑖𝑖𝑖𝑘𝑘}=𝐶𝐶_{𝑖𝑖0𝑡𝑡}𝑇𝑇 + 𝐶𝐶_{0𝑖𝑖𝑡𝑡}𝑇𝑇 − 𝐶𝐶_{𝑖𝑖𝑖𝑖𝑡𝑡}𝑇𝑇 for all pairs of delivery -locations i,

j and vehicle types t (i=1, 2...n; j=1, 2...n; i=/j, t∈T).

2. Order the savings, 𝑆𝑆_{𝑖𝑖𝑖𝑖𝑡𝑡}, from largest to smallest. 3. Starting with the largest savings, do the following:

(a) If linking delivery locations i and j results in a feasible route, then add this link to the route; if not, reject the link.

Checking for route feasibility

If the sum of vaccine volumes required at the delivery locations on the route is less than or equal to the capacity of the vehicle and the total travel distance of the vehicle is less than or equal to the maximum travel distance of the vehicle, the route is feasible; otherwise, the route is infeasible.

5.6.4 Numerical example

Table 47 shows results from the Cotonou province of Benin when vehicle routing is considered. This example is small enough that we can use the MIP to solve the network and also use an MIP formulation to solve the VRPs. The original optimal value for the network {N} obtained after solving the problem is 142,543. The corresponding VRPs are then solved and the looping factor for the central distribution center to the hubs is computed as 0.4333 (i.e., 43.33%) while the looping factor for the hub to the clinics is 0.4585 (i.e., 45.85%). There is no hub to hub connection in this original network. The network cost with vehicle routing is estimated as

Z=138,810; this is obtained by multiplying the transportation costs at each route (edge) by its

looping factor. In particular, the transportation costs per km (𝐶𝐶_{𝑖𝑖𝑖𝑖𝑡𝑡}𝑇𝑇 ) from the central distribution center to each hub and from each hub to a clinic are multiplied by 0.4333 and 0.4585, respectively.

Next, the MIP is solved again with transportation costs based on the above looping factors and we obtain a new network {Nnew} with a cost of 138,393. After solving the associated

VRPs this new network yields values of 1.00 and 0.391 respectively for the looping factors for central to hubs and hub to clinics each, and the true cost for this network {Nnew} with routing is

(138,333<138,810 (Table 47)), we perform a second iteration after resetting Z=138,333 and {N}≡{Nnew}.

After the second iteration, the new network {Nnew} is different from {N} and the VRPs

yield new looping factors of 1.00 and 0.316. Since there is improvement in the network cost (Znew= 137,494<138,333=Z (Table 47)), a third iteration is performed after resetting Z=137,494.

After the third iteration, the solution to the network design problem is the same as the one from the previous iteration. Therefore, we stop here and accept this network structure with vehicle routing as the final one. Table 48 also shows the results for the same problem using the heuristic method for the VRPs instead of the MIP formulation. The iterations proceed in a similar fashion but the final network is different with looping factors of 1.00 and 0.3533 and a final cost of

Z=137,831. The final network with the MIP VRP solver is little bit better than with the heuristic

VRP solver (137,474 <137,831), because the MIP VRP solver provided optimal VRP solutions.

Table 47. Results of applying a looping factor for Benin (MIP-MIP)

Initial Network Iteration 1 Iteration 2 Iteration 3

MIP Cost 142,543 138,393 138,171 137,494 Looping Factor (MIP) C-H 43.33% 100.0% 100.0% 100.0% H-H - - - - H-I 45.85% 39.10% 31.60% 31.60% Cost (Z) 138,810 138,333 137,494 137,494

Table 48. Results of applying a looping factor for Benin (MIP-Heuristic)

Initial Network Iteration 1 Iteration 2 Iteration 3

MIP Cost 142,543 138,753 138,419 137,831 Looping factor (Heuristic) C-H 43.33% 100% 100% 100% H-H - - - - H-I 50.64% 41.85% 35.33% 35.33% Cost (Z) 139,106 138,539 137,831 137,831

Figure 23 shows the network structures at each iteration. As the iterations proceed, the number of hubs decreases. This is because allowing vehicle routing reduces the transportation cost substantially and this result is similar to the one obtained while conducting sensitivity analysis on transportation costs.

Figure 23. Network structure at each iteration (MIP-MIP)