each criterion are already determined in stage 1 of the ILP approach, in (2), and are rescaled to a value of one. The minimum value with respect to each objective function is rescaled to zero – these minimum values remain to be determined. However, this process is not overly sophisticated since the values may be found at the corner points not associated with the objective.

Consider Figure 20 once more, where we aim to find a solution between the corners of objective 1 and objective 2. The minimum value that may be achieved with respect to objective 1 is simply found at the corner of objective 2, and the minimum of objective 2 is found at the corner of objective 1 – when considering objectives 1 and 2 only. Using the minimum and maximum values for each objective function, values determined between these values may be scaled to between zero (at the minimum) and one (at the maximum). In this manner, the effect of objective function magnitudes on a weighted-sum may be eliminated and the search for solutions is directed by the weights in the weighted-sum. If we were to search for solutions between all three corners, the minimum of each objective still lies at one of the other corners, and the value at each corner needs to be compared to the others before the minimum may be identified. For example, when only considering objectives 1 and 2, the minimum for objective 2 lies at the corner of objective 1. Now, if objective 3 is also considered, then the minimum of objective 2 may be found at either the corner of objective 1 or objective 3. If we look at the figure and the projections of the corner points onto the obj1/obj2-plane, then the minimum for objective 2 is now found at the corner of objective 3 and this minimum value becomes zero relative to other values.

Let A ⊆{1, . . . , Na} denote the set of criteria for which a weighted objective function is sought, with the weights of the selected criteria adding to 1. Furthermore, let va,max denote the maximum value of accessibility criterion a ∈ A, determined in ILP stage 1. Similarly, let va,min denote the minimum value of accessibility criterion a ∈ A, which is pre-determined as the lowest value found with respect to the objective at the Pareto corner points of the criteria included in A. The normalized objective function is then to maximize The objective in (6) is subject to the same constraints (3) to (5).

# 3.7. Optimization process setup

The accessibility improvement in person-hours was determined for each of the 261 sequences, for each season and with respect to health facilities, financial institutions, and municipal headquarters – therefore 6 accessibility criteria in total. Minimum and maximum budget limits of 6 and 13 billion NPR were specified. We used 13 billion NPR as the upper bound because this was the budget for the roads that were eventually selected by the provincial government. The much lower limit of 6 billion NPR provided a large budget span, which allows us to compare rates of accessibility improvement that may be achieved for different services at different budget ranges, and to compare how improvements between services relate to each other across the budget range (e.g. linear or exponential). The objectives in both the ILP and heuristic approaches were to find solutions so that person-hour accessibility improvements with respect to all 6 criteria are maximized.

# 3.7.1. Heuristic

Cost was implemented as an additional, seventh objective by the heuristic, with the aim of minimizing total cost – implemented as a maximization objective, maximizing 1/cost. By adding the cost minimization objective, the heuristic is forced to not only find solutions that maximize the accessibility objectives, but also to explore the maximization of these objectives at lower cost levels, from 13 down to 6 billion NPR. Because of the stochastic nature of the heuristic’s Pareto-front approximation process, the solutions returned by different optimization runs will generally vary in quality. It is therefore standard practice to repeat the process multiple times (Knowles et al., 2006; Kim et al., 2008; Heyns et al., 2019). The results of all the runs are then combined and a final attainment front – essentially the Pareto front of all the combined Pareto front approximations – is identified. We repeated our process 40 times, from which a final attainment front of 5,000 solutions that are approximately Pareto optimal with respect to all 7 objectives was identified. The authors’ personal optimization code was used for the heuristic runs, executed in MATLAB.

26