We formulated a solution procedure for vehicle routing problems with uncertainty (VRPU for short) with regard to future demand and transportation cost. Unlike E-SDROA (expectation semideviation robust optimisation approach) for solving the proposed problem, the formulation focuses on robust optimisation considering situations possibly related to bidding and capital budgets. Besides, numerical experiments showed significant increments in the robustness of the solutions without much loss in solution quality. The differences and similarities of the robust optimisation model and existing robust optimisation approaches were also compared.
Nowadays, vehicle routing problem (VRP for short) is a crucial and critical issue in industrial and systems engineering [
The sum of all demands in such a scenario exceeds the capacities that the vehicles can serve by assuming that the demand is known before vehicles start their routes: such a scenario is considered as a “failure.” In contrast, if vehicles in the depot satisfy the demands of all customers when it is assumed that the demands are known before vehicles start their routes, such scenario is considered as a successful one.
Contrary to existing robust optimisation models for VRPU, we summarise the contributions of this research as follows. Supposing the coexistence of failure and successful scenarios, we proposed a robust optimisation approach to minimise the sum of the expected total transportation cost in all failure scenarios and its variation multiplied by a weighted coefficient on the condition of coexistence of failure and successful scenarios. Unlike robustness measure of E-SDROA, our approach can deal with some situations involving bidding or capital budget decisions. Solving for the robust optimal solution of our approach for VRPU is no more difficult than solving a single deterministic SDVRP (split delivery vehicle routing problem) while satisfying all demands in a given bounded uncertainty set.
The rest of this paper is organised as follows: Section
Stochastic optimisation approaches for solving VRPU were designed to optimise the expected value of all possible scenarios [
However, robust optimisation approaches available for VRPU fail to solve situations involving either bidding or capital budget decisions [
Generally speaking, there are two types of robust optimisation approach used for VRPU.
(1) Robust optimisation based on the definition of model robustness [
Lee et al. investigated a vehicle routing problem with deadlines, wherein the objective is to satisfy the requirements of a given number of customers with minimum travel distances while regarding both the customers’ deadlines and vehicle capacities [
Yao et al. designed a robust linear programming model on the basis of a robust optimisation approach wherein hard constraints are guaranteed within an appropriate uncertainty set [
Najafi et al. presented a robust approach for a multiobjective, multimodal, multicommodity, and multiperiod stochastic model to manage the logistics of both commodities and injured personnel in response to an earthquake. The model was used to ensure that the distribution plan performed well under various chaotic situations arising in the aftermath of an earthquake [
(2) A robust optimisation approach that can measure the trade-off between solution and model robustness [
Naumann et al. proposed a robust optimisation approach to vehicle scheduling in public bus transport [
Moreover, Montemanni et al. presented a new extension for solving the traveling salesman problem, where edge costs were specified as a range of possible values [
Moreover, robust optimisation has been extended to multistage problems where the here-and-now decision is amended by a recourse decision [
In practice, estimating transportation cost (a type of marginal cost) is also a difficult task except for demand uncertainty caused by many unpredictable factors, including traffic conditions, accidents, congestion, and weather conditions. For this reason, the proposed robust optimisation approach focuses on two primary sources of uncertainty: the future demands that are to be served by the vehicle fleet and the transportation cost.
VRP with demand and transportation cost uncertainty is a variant of VRPU. It can be defined by a complete undirected graph
We introduced a cost function
Suppose that the total demand in a scenario is known before the vehicle begins its route; then operators attach primary importance to the expected value of total transportation cost with regard to a failure scenario on the condition of coexistence of failure and successful scenarios. The vehicle routing problem with demand and cost uncertainty consists of finding a set of Each client is assigned to, at least, one route. The demand Each route must begin and end at the depot and visits at least one customer. The total demand serviced by any vehicle does not exceed its capacity.
The total transportation cost of a solution
To simplify the process, let
The potential risk of the total transportation cost caused by demand and transportation cost uncertainty is defined as follows.
Potential risk with respect to a solution
The robustness measure of CE-SDROA is defined by
In (
Combining (
The CE-SDROA for the proposed problem can now be formulated as follows.
(CE-SDROA) Minimise
Constraints (
Let
Given an instance
We use Nazif’s algorithm [
According to the definition of
If all realisations of the total demand exceeded
If all realisations of the total demand exceeded
Then,
CE-SDROA becomes
Hence, Proposition
Proposition
We generated some new test instances derived from classical instances of SDVRP [
The first field of the name,
It was obtained from
Simulations were carried out on a PC (Windows 7 operating system, Pentium CPU B950 running at 210 GHz, with 4 GHz internal memory) using CPLEX 12.1 software.
Demand and transportation uncertainty sets are defined as follows.
Transportation cost Uncertainty set
Demand Uncertainty set
Results for the Augerat problems.
|
Time | opt | |
---|---|---|---|
A-n32-k5 | 0, 0 | 3600* | 1041 |
100, 100 | 3600* | 1028 | |
|
|||
B-n34-k5 | 0, 0 | 3600* | 827 |
100, 100 | 3600* | 1023 | |
|
|||
A-n34-k5 | 0, 0 | 3600* | 827 |
100, 100 | 3600* | 1021 | |
|
|||
P-n19-k2 | 0, 0 | 3600* | 261 |
100, 100 | 3600* | 278 | |
|
|||
P-n22-k8 | 0, 0 | 3.727 | 612 |
100, 100 | 29.268 | 653 | |
|
|||
B-n39-k5 | 0, 0 | 3600* | 639 |
100, 100 | 3600* | 842 | |
|
|||
B-n31-k5 | 0, 0 | 3600* | 696 |
100, 100 | 3600* | 812 |
Table
For the given parameters
Table
Effect of
|
opt | dev | |
---|---|---|---|
0.6 | 0.1 | 7736.26 | 3566.85 |
0.7 | 0.2 | 7963.11 | 3419.80 |
0.8 | 0.3 | 8369.23 | 3180.24 |
0.9 | 0.4 | 8923.21 | 2832.48 |
1.0 | 0 | 9136.66 | 2615.06 |
1.1 | 0.6 | 11136.12 | 2165.45 |
1.2 | 0.7 | 10873.69 | 2016.27 |
1.3 | 0.8 | 14273.86 | 827.10 |
1.4 | 0.9 | 16523.82 | 2704.21 |
1.5 | 1 | 19216.39 | 2166.27 |
The results in Tables
Relationship between the optimal objective values of CE-SDROA and stochastic optimisation approach.
Test instances |
|
|
opt |
---|---|---|---|
SD032-60* | 0.6 | 0.1 | 1128.61 (<CE) |
SD032-40* | 0.7 | 0.2 | 876.32 (<CE) |
SD048-60* | 0.8 | 0.3 | 4972.36 (<CE) |
SD064-35* | 0.9 | 0.4 | 2931.27 (<CE) |
SD080-50* | 1.0 | 0.5 | 7527.63 (=CE) |
SD120-40* | 1.2 | 0.7 | 10873.69 (>CE) |
SD144-30* | 1.3 | 0.8 | 14273.86 (>CE) |
SD160-30* | 1.4 | 0.9 | 16523.82 (>CE) |
SD240-30* | 1.5 | 1 | 37216.39 (>CE) |
In order to examine the performance of our solution method, PSO algorithm (determine
The search methods mentioned above have been programmed in MATLAB R2013a and have ran on an Intel(R) Core(TM) i5-3337U CPU@1.80 GHz 8.00 GB-RAM.
It can be observed from Table
The computational results on 7 test instances.
Instance name |
|
PSO | Ant colony algorithm | GA |
---|---|---|---|---|
A-n32-k5 | 150, 50 | 977 | 977 | 977 |
B-n34-k5 | 150, 50 | 1026 | 1045 | 1015 |
A-n34-k5 | 150, 50 | 1216 | 1210 | 1197 |
P-n19-k2 | 150, 50 | 625 | 636 | 597 |
P-n22-k8 | 150, 50 | 756 | 765 | 702 |
B-n39-k5 | 150, 50 | 1256 | 1253 | 1156 |
B-n31-k5 | 150, 50 | 1123 | 1101 | 997 |
Unlike E-SDROA, CE-SDROA made a trade-off between the expected value of total transportation cost in all failure scenarios and its variation under conditions of the coexistence of failure and successful scenarios. Compared with stochastic optimisation approaches for VRPU, CE-SDROA dealt with the expected value of total transportation cost in all failure scenarios under conditions of the coexistence of failure and successful scenarios. The existence of this trade-off was intuitive, but the model provided a means of quantifying the trade-off and determining optimal choices for varying levels of risk acceptance. This provided a basis for improved decision making in a variety of situations.
We draw the following conclusions through theoretical analysis. Compared with stochastic optimisation approaches available, CE-SDROA made a trade-off between its cost threshold and accepting the risk of potentially high costs. E-SDROA failed to solve situations relevant to bidding or capital budget decisions. On the contrary, CE-SDROA considered these situations using a reference level of CE. Solving CE-SDROA for VRPU was no more difficult than solving a single deterministic SDVRP while satisfying all demands in a given bounded uncertainty set.
Although CE-SDROA for VRPU does present some significant improvements over E-SDROA as described above, there remained some problems to be addressed by future research. The optimal solution of CE-SDROA yielded extra inventory costs for some customers in some scenarios.
It is also worth considering the relationship of the current formulation to an explicit multistage dynamic optimization. The current model defines stages by types of decisions (i.e., fleet acquisition versus operational allocation). An alternative modeling approach might consider time stages, with both fleet acquisition and operational variables at each stage. This would make the model more similar to dynamic vehicle allocation models. This is also likely to be a useful effort for future research.
Let
If the values of the optimisation objective for different scenarios were higher than expected, they would then affect the minimisation of the total transportation cost. The robustness measure is given by
By combining (
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions which help improve the quality of the paper.