We can also consider uncertainty in the number of people (=demand) at each village. In this section, we consider a robust version for Model 1. Those for Model 2 and 3 can be expressed similarly.
Let us define the feasible region A for Model 1 as follows: 𝛢𝛢 = {(𝒙𝒙, 𝒚𝒚)|𝑦𝑦𝑖𝑖 ≤ � 𝑥𝑥𝑖𝑖 𝑖𝑖∈𝑆𝑆𝑖𝑖 for 𝑆𝑆𝑖𝑖 = �𝑗𝑗: 𝑑𝑑𝑖𝑖𝑖𝑖 ≤ 𝐷𝐷1, 𝑗𝑗 = 1, … , 𝑛𝑛�, 𝑖𝑖 = 1, … , 𝑛𝑛, � 𝑐𝑐𝑖𝑖𝑥𝑥𝑖𝑖 𝑛𝑛 𝑖𝑖=1 ≤ 𝐶𝐶, � 𝑥𝑥𝑖𝑖 𝑛𝑛 𝑖𝑖=1 ≤ 𝑁𝑁; 𝑥𝑥𝑖𝑖∈{0,1}, 𝑦𝑦𝑖𝑖∈{0,1}, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖 = 1, … , 𝑛𝑛}
Now suppose that 𝑝𝑝̂𝑖𝑖 is an estimate of the population in village i and that 𝑝𝑝̂𝑖𝑖𝛾𝛾𝑖𝑖 is the amount by which the true population 𝑝𝑝𝑖𝑖 differs from this estimated value 𝑝𝑝̂𝑖𝑖, so that 𝑝𝑝𝑖𝑖 − 𝑝𝑝̂𝑖𝑖 = 𝑝𝑝̂𝑖𝑖𝛾𝛾𝑖𝑖, i.e., 𝑝𝑝𝑖𝑖 = 𝑝𝑝̂𝑖𝑖(1 + 𝛾𝛾𝑖𝑖) . Also assume that
∑𝑛𝑛𝑖𝑖=1𝑝𝑝̂𝑖𝑖𝛾𝛾𝑖𝑖 = 0 so that ∑𝑖𝑖=1𝑛𝑛 𝑝𝑝𝑖𝑖 = ∑𝑛𝑛𝑖𝑖=1𝑝𝑝̂𝑖𝑖 = 𝐷𝐷 0 < |𝛾𝛾𝑖𝑖| ≤ 𝛾𝛾̅𝑖𝑖 < 1
In other words we know the total population (D) across all n villages, but the true population 𝑝𝑝𝑖𝑖 at village i could be up to 100𝛾𝛾̅𝑖𝑖% higher or lower than its estimated population 𝑝𝑝̂𝑖𝑖.
Let us define the set 𝐵𝐵 as
𝛣𝛣 = {𝜸𝜸| � 𝑝𝑝̂𝑖𝑖𝛾𝛾𝑖𝑖 𝑛𝑛 𝑖𝑖=1 = 0, |𝛾𝛾𝑖𝑖| ≤ 𝛾𝛾̅𝑖𝑖< 1} where 𝜸𝜸 is a vector of 𝛾𝛾𝑖𝑖. Robust Model: max 𝒙𝒙,𝒚𝒚 �inf𝜸𝜸 � 𝑝𝑝̂𝑖𝑖(1 + 𝛾𝛾𝑖𝑖)𝑦𝑦𝑖𝑖 𝑛𝑛 𝑖𝑖=1 �(𝒙𝒙, 𝒚𝒚) ∈ 𝛢𝛢, 𝜸𝜸 ∈ 𝛣𝛣�
Note that for a given feasible selection of outreach centers (𝒙𝒙) and corresponding set of villages covered (𝒚𝒚), the quantity within the braces represents the smallest value of the true total population covered across all different deviations from the estimates that meet conditions 1-3 above. The objective of the model is to find the vectors 𝒙𝒙 and 𝒚𝒚 that maximize this value.
Proposition
If 𝛾𝛾̅1 = 𝛾𝛾̅2= ⋯ = 𝛾𝛾̅𝑛𝑛 = 𝛾𝛾̅, then the optimal solution to the original formulation (Model 1) is the optimal solution to the robust formulation.
Proof:
Let (𝒙𝒙∗, 𝒚𝒚∗) be the optimal solution to Model 1, and consider any feasible (𝒙𝒙, 𝒚𝒚) ∈ 𝐴𝐴 and define C as the index set of villages that are covered and N as the index set of villages that are not covered. Note that C∪N = {1,2,…,n} and the estimated total coverage is ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 while the estimated population not covered is given by ∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖 = 𝐷𝐷 − ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖.
Define 𝜉𝜉(𝒙𝒙, 𝒚𝒚) = �𝑝𝑝�𝑖𝑖 𝑖𝑖∈𝐶𝐶 = �𝑝𝑝�𝑖𝑖𝑦𝑦𝑖𝑖 𝑛𝑛 𝑖𝑖=1 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) = min 𝜸𝜸 ��𝑝𝑝�𝑖𝑖(1 ± 𝛾𝛾𝑖𝑖)𝑦𝑦𝑖𝑖 𝑛𝑛 𝑖𝑖=1 �𝜸𝜸 ∈ 𝛣𝛣�
Note that for the assignment (𝒙𝒙, 𝒚𝒚), 𝜉𝜉(𝒙𝒙, 𝒚𝒚) is the estimated total coverage, while 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) is the smallest actual total coverage possible across all differences from the estimates that satisfy conditions 1-3 described earlier.
Case 1: 𝜉𝜉(𝒙𝒙, 𝒚𝒚) = ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 ≤ 𝐷𝐷/2)
In this case the true total coverage has its minimum value 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) when the true population of each village 𝑖𝑖 ∈ 𝐶𝐶 is 𝑝𝑝̂𝑖𝑖(1 − 𝛾𝛾̅), as long as this minimum can be attained. This minimum is attained as long as the true total population not covered (in the villages indexed by
set N) does not exceed (1 + 𝛾𝛾̅) ∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖, which is the largest possible value that this number can take on.
The true number not covered is given by 𝐷𝐷 − (1 − 𝛾𝛾̅)(∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖) and therefore we need to show that {𝐷𝐷 − (1 − 𝛾𝛾̅)(∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)} ≤ {(1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖)}. This is easily done because
{𝐷𝐷 − (1 − 𝛾𝛾̅)(∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)} − {(1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖)} = {𝐷𝐷 − (1 − 𝛾𝛾̅)(∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)} − {(1 + 𝛾𝛾̅)(𝐷𝐷 − ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)} = 𝛾𝛾̅{2 ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 − 𝐷𝐷} ≤ 0 (because 𝛾𝛾̅ > 0 and ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 ≤ 𝐷𝐷/2) Therefore 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) = �(1 − 𝛾𝛾̅)𝑝𝑝�𝑖𝑖 𝑖𝑖∈𝐶𝐶 = (1 − 𝛾𝛾̅)𝜉𝜉(𝒙𝒙, 𝒚𝒚), and in particular, for (𝒙𝒙∗, 𝒚𝒚∗)
𝜉𝜉̅(𝜸𝜸|𝒙𝒙∗, 𝒚𝒚∗) = (1 − 𝛾𝛾̅)𝜉𝜉(𝒙𝒙∗, 𝒚𝒚∗) .
Since (𝒙𝒙∗, 𝒚𝒚∗) is optimal for Model 1, it follows that 𝜉𝜉(𝒙𝒙, 𝒚𝒚) ≤ 𝜉𝜉(𝒙𝒙∗, 𝒚𝒚∗), and therefore 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) ≤ 𝜉𝜉̅(𝜸𝜸|𝒙𝒙∗, 𝒚𝒚∗).
Therefore (𝒙𝒙∗, 𝒚𝒚∗) is also optimal for Model 2. Case 2: 𝜉𝜉(𝒙𝒙, 𝒚𝒚) = ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 > 𝐷𝐷/2)
Here it is not possible for the true coverage to attain the minimum possible value of (1 − 𝛾𝛾̅) ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 because the actual total number not covered would then exceed its maximum possible value. Instead we make use of the fact that the minimum actual coverage is attained when the actual number not covered is at its maximum of (1 + 𝛾𝛾̅) ∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖. To show this minimum can be attained we need to ensure that the true total number covered is larger than (1 − 𝛾𝛾̅) ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖, which is the smallest value that it can take on.
The true number covered is given by 𝐷𝐷 − (1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖) and therefore we need to show that {𝐷𝐷 − (1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖)} ≥ {(1 − 𝛾𝛾̅)(∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)}. This is easily done because
{𝐷𝐷 − (1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖)} − {(1 − 𝛾𝛾̅) ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖} = {𝐷𝐷 − (1 + 𝛾𝛾̅)(𝐷𝐷 − ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖)} − {(1 − 𝛾𝛾̅) ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖} = 𝛾𝛾̅{2 ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 − 𝐷𝐷} > 0 Therefore 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) = 𝐷𝐷 − (1 + 𝛾𝛾̅) ∑𝑖𝑖∈𝑁𝑁𝑝𝑝�𝑖𝑖 = 𝐷𝐷 − (1 + 𝛾𝛾̅)�𝐷𝐷 − ∑𝑖𝑖∈𝐶𝐶𝑝𝑝�𝑖𝑖� = (1 + 𝛾𝛾̅)(∑𝑖𝑖∈𝐶𝐶𝑝𝑝�𝑖𝑖)− 𝛾𝛾̅𝐷𝐷 = (1 + 𝛾𝛾̅)𝜉𝜉(𝒙𝒙, 𝒚𝒚) − 𝛾𝛾̅𝐷𝐷,
and in particular, for (𝒙𝒙∗, 𝒚𝒚∗)
𝜉𝜉̅(𝜸𝜸|𝒙𝒙∗, 𝒚𝒚∗) = (1 + 𝛾𝛾̅)𝜉𝜉(𝒙𝒙∗, 𝒚𝒚∗) − 𝛾𝛾̅𝐷𝐷.
Since (𝒙𝒙∗, 𝒚𝒚∗) is optimal for Model 1, it follows that 𝜉𝜉(𝒙𝒙, 𝒚𝒚) ≤ 𝜉𝜉(𝒙𝒙∗, 𝒚𝒚∗), and therefore 𝜉𝜉̅(𝜸𝜸|𝒙𝒙, 𝒚𝒚) ≤ 𝜉𝜉̅(𝜸𝜸|𝒙𝒙∗, 𝒚𝒚∗). ∎
Example:
Suppose we have a total of 100 people in our n villages and the true population in any individual village i could be higher or lower than the estimated value 𝑝𝑝̂𝑖𝑖 by no more 10% (so
𝛾𝛾̅=0.1).
Case 1: Suppose ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 =45 people are estimated to live in villages covered and ∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖 =55 in villages not covered. Then the lowest true coverage possible is 45(0.9) = 40.5 with 59.5 people not being covered.
Case 2: Suppose ∑ 𝑝𝑝̂𝑖𝑖∈𝐶𝐶 𝑖𝑖 =55 people are estimated to live in villages covered and ∑𝑖𝑖∈𝑁𝑁𝑝𝑝̂𝑖𝑖 =45 in villages not covered. Then the lowest true coverage possible is when the actual number not covered is at its maximum of 45(1.1) = 49.5, i.e., with 50.5 people being covered.
According to the previous proposition, the optimal solution to the model without demand uncertainty is the robust solution for uncertain demand. If the assumption about the equality of
the total population is removed, the result is the same because the objective value for all solutions would be decreased by 𝛾𝛾̅ . In addition, even if the assumption of percentage deviation from the estimated population is changed to a fixed amount of deviation from the estimated population, the optimal solution is still the robust solution. The fact that the solution to the robust model is the same as the solution for the original model is because of the following characteristics of the coverage model: 1) it maximizes the number of people who can be covered, 2) the robust model provides the optimum corresponding to the worst-case scenario for the error in the estimated population, and 3) there is no systematic interaction between the populations at different locations. Thus, in order to have the best worst-case performance it is optimal to locate the outreach points at the locations that maximize coverage with the estimated populations. This follows because if the population at each location can either be reduced by a constant percentage or a constant amount then the locations that maximizes coverage in the original problem will still provide the highest coverage for the new problem.