4.1. Problem Description
In this section, we focus on applying the lightly robust max-ordering solution concept to the ambulance location problem to help a decision maker in finding the best location patterns for ambulance placement in the event of an unexpected ambulance shortage. This approach provides a minimum value of the maximum distance while also specifying a solution for ambulance placement at specific locations regarding the maximum distance to demand sites. Moreover, by considering the measurements of the gain in robustness and the price to be paid for robustness, decision makers can see how much they have to sacrifice the nominal quality for obtaining robustness on lightly robust max-ordering solutions in each level of robustness of the solution set. Notice that this approach is different from the time-dependent travel model in [
5]. Indeed, the ambulance location problem was captured in [
5] by applying the covering model and solving the problem using the static location problem in each time period, while the uncertainties are time-dependent variations of travel times during the course of the day. As a result, by using this model, the best solutions change more frequently during the day in each time period, while our model provides all the best solutions in solving uncertain multiobjective optimization as an original problem.
In our study, we consider simulated data for the ambulance location problem for finding the appropriate placement for 5 ambulances among all 15 possible candidate locations, such that the suitable locations must satisfy requirements of the longest distance covering between the closest ambulance and any of the 10 demand sites (
Table 1). The orange squares and the blue points in
Figure 1 are used to indicate the candidate ambulance locations and the potential demand sites that are included in this emergency medical services system, respectively.
Through researching the above problem settings of the emergency demand sites, we then have the objective function,
$f:=({f}_{1},{f}_{2},{f}_{3},{f}_{4},{f}_{5},{f}_{6},{f}_{7},{f}_{8},{f}_{9},{f}_{10}).$ As the purpose of solving this problem is to find the most effective location patterns for placing 5 ambulances from 15 candidate locations, the total number of potential ambulance locations can be established. All the possible alternative location patterns concerning the solving of this problem were computed using the following formula:
In this context, these patterns are considered as feasible solutions. So, the feasible set is $X:=\left\{{a}_{k}\right|k\in {E}_{3003}\}\subset {\mathbb{R}}^{5}$, where ${E}_{3003}$ is the index set of each indice k of each possible alternative candidate location pattern.
Here, we consider the problem of locating the ambulance where the situation of unavailability of the ambulance could occur. In this study, we assume that all ambulances are in the same condition. We consider all the possible events with ambulances simultaneously unavailable. So, all possible events of the considered problem are:
Possible events:
There is no unavailable ambulance $\left({\mathcal{U}}_{0}\right)$.
There is one unavailable ambulance $\left({\mathcal{U}}_{1}\right)$.
There are two unavailable ambulances simultaneously $\left({\mathcal{U}}_{2}\right)$.
There are three unavailable ambulances simultaneously $\left({\mathcal{U}}_{3}\right)$.
There are four unavailable ambulances simultaneously $\left({\mathcal{U}}_{4}\right)$.
Since there are 5 ambulances to allocate in this system, for each
$P\in \{0,1,2,3,4\}$ a set
${\mathcal{U}}_{P}$ of each event is composed of subevents itself. Here, a subevent in the set
${\mathcal{U}}_{P}$ is considered as a scenario. According to the above possible events in this problem, for each candidate location pattern
${a}_{k}\in X$ and
$P\in \{1,2,3,4\}$, the number of scenarios in each
${\mathcal{U}}_{P}^{{a}_{k}}$ can be computed by the following formula:
where the notation
P in the Formulation (15) is denoted by the number of ambulances which are simultaneously unavailable. To present it more clearly, we denote each scenario of ambulance unavailability in this system with respect to each candidate location pattern
${a}_{k}$ by the following notations:
${\mathcal{U}}_{0}=\left\{{s}_{\left\{0\right\}}\right\}$.
${\mathcal{U}}_{1}^{{a}_{k}}=\{{s}_{\left\{1\right\}}^{k},{s}_{\left\{2\right\}}^{k},{s}_{\left\{3\right\}}^{k},{s}_{\left\{4\right\}}^{k},{s}_{\left\{5\right\}}^{k}\}$.
${\mathcal{U}}_{2}^{{a}_{k}}=\{{s}_{\{1,2\}}^{k},{s}_{\{1,3\}}^{k},{s}_{\{1,4\}}^{k},{s}_{\{1,5\}}^{k},{s}_{\{2,3\}}^{k},{s}_{\{2,4\}}^{k},{s}_{\{2,5\}}^{k},{s}_{\{3,4\}}^{k},{s}_{\{3,5\}}^{k},{s}_{\{4,5\}}^{k}\}$.
${\mathcal{U}}_{3}^{{a}_{k}}=\{{s}_{\{1,2,3\}}^{k},{s}_{\{1,2,4\}}^{k},{s}_{\{1,2,5\}}^{k},{s}_{\{1,3,4\}}^{k},{s}_{\{1,3,5\}}^{k},{s}_{\{1,4,5\}}^{k},{s}_{\{2,3,4\}}^{k},{s}_{\{2,3,5\}}^{k},{s}_{\{2,4,5\}}^{k},{s}_{\{3,4,5\}}^{k}\}$.
${\mathcal{U}}_{4}^{{a}_{k}}=\{{s}_{\{1,2,3,4\}}^{k},{s}_{\{1,2,3,5\}}^{k},{s}_{\{1,3,4,5\}}^{k},{s}_{\{1,2,4,5\}}^{k},{s}_{\{2,3,4,5\}}^{k}\}$.
Note that each scenario’s subscription refers to the unavailable ambulance labels. For example, the notation ${s}_{\left\{0\right\}}$ refers to there being no unavailable ambulance in this system, the notation ${s}_{\left\{1\right\}}^{k}$ refers to the 1^{st} label of ambulance being unavailable with respect to the location pattern ${a}_{k}$, and the notation ${s}_{\{1,2\}}^{k}$ refers to the $1\mathrm{st}$ label and the $2\mathrm{nd}$ label of ambulances being unavailable with respect to the location pattern ${a}_{k}$ in this system.
As the possible candidate location patterns in this problem are 3003 patterns, the number of all possible scenarios is:
For convenience, we denote the set of all possible scenarios for this problem by
Here, the ambulance location problem is formulated as an uncertain multiobjective optimization problem
$\mathcal{MP}\left(\mathcal{U}\right)$, where
$\mathcal{MP}\left(\mathcal{U}\right)$ is given as a family of
$\left\{\mathcal{MP}\right(s\left)\right|s\in \mathcal{U}\}$ of deterministic multiobjective optimization problem as
and for each
$i\in {I}_{10}$, the component function
${f}_{i}:X\times \mathcal{U}\u27f6\mathbb{R}$ is defined as
and
where
$\parallel \xb7\parallel $ is a norm on
${\mathbb{R}}^{2}.$ This means
${f}_{i}({a}_{k},{s}_{\square}^{j})$ is defined as the shortest distance of ambulance pattern
${a}_{k}$ to demand site
${D}_{i}$ under scenario
${s}_{\square}^{j}$. We note that the objective function values of the Formulations (
17) and (
18) were generated and computed according to the problem setting as
Figure 1. Notice that the objective function values of the Formulations (
17) and (
18) not only depend on the distance between the closest ambulance and the demand site but also the weight
${d}_{i}$. In practice, the value of weight may be correlated with the statistical importance of a demand site.
Here, the robust counterpart
$\mathcal{LRMOP}(\widehat{s},\epsilon )$ as in the Formulation (
5) of the ambulance location problem (
16) with respect to the relaxation
$\epsilon $ is expressed as follows:
where
${X}_{\mathcal{LRMOP}(\widehat{s},\epsilon )}:=\{{a}_{k}\in X|{\displaystyle \underset{i\in {I}_{10}}{max}}{f}_{i}({a}_{k},\widehat{s})\u2a7d{\displaystyle \underset{i\in {I}_{10}}{max}}{f}_{i}({\widehat{a}}_{k},\widehat{s})+\epsilon \}$ and
$\widehat{s}$ is the nominal scenario. Note that the notations
${\widehat{a}}_{k}$ and
${f}_{i}({\widehat{a}}_{k},\widehat{s})$ indicate the optimal location pattern in the nominal problem and the distance between the closest ambulance of the optimal location pattern
${\widehat{a}}_{k}$ and the demand site
${D}_{i}$ in the nominal scenario, respectively.
We assume that the nominal scenario of this system is ${s}_{\left\{0\right\}}$ because this should be considered as a typical situation (in fact, another scenario can be seen as a nominal scenario depending on which situation we would like to define as the most important event or the frequent event) and consider the distance in ${\mathbb{R}}^{2}$ by computing the Euclidean norm. According to Definition 1 of max-ordering solutions, we obtain that the number of elements in a solution set ${X}_{MO}\left({s}_{\left\{0\right\}}\right)$ are 757, and the longest distance according to these solutions is 193.24 units (a unit of length in this study can be seen as any arbitrary accepted standard for measurement of length).
4.2. Solution Discussions
We now describe the computations of the results which are presented in
Table 2. As we can see from
Table 2, the results of solution sets depend upon a selection of different relaxations
${\epsilon}_{m}$, where
${\epsilon}_{m}\in [0.00,+\infty )$.
For the choice of the relaxation
${\epsilon}_{0}=0.00$, by applying the Definition 2, we obtain that there are 56 optimal location patterns, in which the longest travel distances concerning unavailability of ambulances of these optimal location patterns are
$496.49$ units in the worst-case scenario (see
Appendix A for the explicit information of the solution). Note that all solutions in the set
${X}_{\mathcal{LRMOP}({s}_{\left\{0\right\}},{\epsilon}_{0})}^{*}$ are considered as solutions in the first level of robustness.
By applying the method of computing the relaxation in Theorem 1, the next levels of the robustness of solution set are determined by the relaxations
${\epsilon}_{1}=11.15$ and
${\epsilon}_{2}=32.30$. According to these relaxations, the corresponding optimal location patterns are 56 patterns, and the corresponding longest travel distance of these location patterns are
$496.49$ units in the worst-case scenario. Here, the solution sets corresponding to
${\epsilon}_{1}=11.15$ and
${\epsilon}_{2}=32.30$ are considered as the second level of robustness and the third level of robustness, respectively. We note that the solution set for the second level of robustness is more robust than the solution set for the first level of robustness. Moreover, the solution set for the third level of robustness is more robust than the solution set for the second level of robustness. It is observed that the number of optimal solutions (see
Appendix A for the explicit information of the solution) and the longest distance of these location patterns is the same number as the previous level of the solution set. This indicates that too small a change in the number of relaxations does not produce better results than the previous ones. Indeed, it is not surprising that sacrificing too little of the quality of the nominal scenario does not yield solutions performing better than the solutions in the previous solution set concerning robustness. This is because the small change in the value of relaxation means that the longest distance of feasible solutions is not permissible too far away from the optimal value in the nominal scenario, so that the additional solutions still have limitations and few options.
Continuing with the above idea, the fourth level of robustness of the solution is determined by ${\epsilon}_{3}=49.23$. Here, the corresponding optimal location patterns of this level of robustness are 20 patterns, and the longest travel distances of these optimal location patterns are $471.60$ units in the worst-case scenario. It is observed that the corresponding solution set of a relaxation ${\epsilon}_{3}=49.23$ is disjoint from the solution sets for relaxations ${\epsilon}_{0},{\epsilon}_{1},$ and ${\epsilon}_{2}$.
By applying Theorem 1 again, the fifth level of robustness and the sixth level of robustness are determined by the relaxations ${\epsilon}_{4}=55.43$ and ${\epsilon}_{5}=68.44$, respectively. According to the relaxation ${\epsilon}_{4}=55.43$, the optimal location patterns are 4 patterns, whereas the corresponding longest travel distance of these location patterns is $412.07$ units. Notice that these 4 optimal location patterns are different from elements in a solution set for the relaxation ${\epsilon}_{3}=49.23$. Furthermore, the optimal location patterns correlated with the relaxation ${\epsilon}_{5}=68.44$ are 5 patterns, with the associated longest travel distance of these solutions also being $412.07$ units. There are four elements in this set of solutions that are identical to the solution set for the relaxation ${\epsilon}_{4}=55.43$.
By continuing this idea, the rest of the level of robustness of the solution set can be obtained by applying the method of computing the relaxation in Theorem 1, as shown in
Table 2.
Remark 5. The number of elements in the solution sets may not be necessarily linked to a rise in relaxation levels. For example, by choice of relaxations ${\epsilon}_{4}$ and ${\epsilon}_{5}$, the number of elements in a solution set correlated with the relaxation ${\epsilon}_{5}$ is greater than the number of elements in a solution set correlated with the relaxation ${\epsilon}_{4}$ (see Appendix A for the explicit information of the solution). 4.3. Trade-Off between the Gain of Robustness and the Price to Be Paid for Robustness
The following table shows a portion of a trade-off between the gain of robustness and the price to be paid for robustness for each solution set.
Remark 6. - (i)
As indicated in Table 3, the gain in robustness and the price to be paid for robustness of two solution sets ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{1})}^{*}$ and ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{2})}^{*}$ are 0. This is because of all solutions in these two sets being identical to the solution set ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{0})}^{*}$ (see Appendix A for the explicit solutions information). - (ii)
For the three relaxations ${\epsilon}_{4}$, ${\epsilon}_{5}$, and ${\epsilon}_{6}$, the associated gain in robustness values is the same number, which is $84.42$. However, it was asserted that the price to be paid for robustness of these three solution sets are different. For the first set ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{4})}^{*}$, the value of the price to be paid for robustness is $55.43$, while the remaining two sets, ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{5})}^{*}$ and ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{6})}^{*}$, are $68.44$. This is due to the fact that the new members in the sets ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{5})}^{*}$ and ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{6})}^{*}$ provided the value of the corresponding longest distance in the nominal problem more than the existing elements in the set ${X}_{\mathcal{LRMOP}(\widehat{s},{\epsilon}_{4})}^{*}$.
Based on the above discussion and information in
Table 3, the question that could be raised to decision makers is which relaxation should be chosen. A direction that can be used for obtaining the answer is considering the trade-off between the gain in robustness and the price to be paid for robustness.
Figure 2 shows the visualization of a trade-off in each level of robustness of the solution set.
Rationally speaking, the ratio of the gain in robustness and the price to be paid for the robustness means the benefits in robustness of solutions which we obtain and the nominal quality of the solutions we lose.
From
Figure 2, we see that the highest ratio value of trade-off is
$1.52$, which is obtained from solutions in the fifth level of robustness of the solution set
${X}_{\mathcal{LRMOP}({s}_{\left\{0\right\}},{\epsilon}_{4})}^{*}$, where
${\epsilon}_{4}=55.43$. This means that the solution set of the fifth level of robustness can be considered the most desirable solution compared with another level of robustness set.
Remark 7. - (i)
An important point to note is that if we choose the optimal location pattern relying on just data on the nominal problem $\mathcal{MP}\left({s}_{\left\{0\right\}}\right)$ and ignore the uncertainty of unavailable ambulances, it is possible that the network components of the location pattern could lose functions when a disaster or crisis occurs in practice. In fact, for example, by choice of location pattern $\{A2,A3,A8,A9,A12\}$, which is an optimal solution in the nominal problem (there is neither disaster nor crisis), the longest distance covering all demand sites with respect to this location pattern is $193.24$ units. However, if there is an unavailability of ambulances once a vehicle is dispatched to a call, then the longest distance covering all demand sites with respect to this location pattern $\{A2,A3,A8,A9,A12\}$ in the worst-case scenario become 644.92 units. Note that the number of the longest distance covering all demand sites by the location pattern $\{A2,A3,A8,A9,A12\}$ is worse than all optimal location patterns, which are computed by the concept of lightly robust max-ordering solution in the worst-case scenario (for more information see Table 2). This means that the benefits of a solution obtained by our proposed solution concept ensure a high performance in serving the longest distance covering all demand sites in uncertain environments. - (ii)
In the general setting on n candidate locations to locate r ambulances, we can calculate all possible scenarios of simultaneously unavailable ambulances by the formula: