Fuzzy multi-criteria emergency service centers selection problem in extreme environment

MACSPro'2019 - Modeling and Analysis of Complex Systems and Processes, Vienna
21 - 23 March 2019

Gia Sirbiladze, Bidzina Midodashvili, Bidzina Matsaberidze

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1. Fuzzy multi-criteria emergency service centers selection problem in extreme environment Gia Sirbiladze, Bidzina Midodashvili and Bidzina Matsaberidze Department of Computer Science, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia March 21-23, 2019
2. In this work we consider one fuzzy multi-objective optimization problem for selection of emergency facility locations. Optimal location planning for service centers and timely servicing from emergency service centers to the demand points are key tasks of the emergency management system. We develop a fuzzy multi-attribute decision making approach for the service centers location selection problem under extreme environment for which a fuzzy probability aggregation operators’ approach is used.
3. The formation of expert’s input data for construction of attributes is an important task of the centers’ selection problem. To decide on the location of service centers, it is assumed that a set of candidate sites already exists. This set is denoted by = {1 , 2 , … , }, where we can locate service centers and let = 1 , 2 , … , be the set of all attributes which define CCs selection. For example: S1 -- ”post disaster access by public and special transport modes to the candidate site”; S2 -- ” post disaster security of the candidate site from accidents, theft and vandalism”; S3 -- ” post disaster connectivity of the location with other modes of transport (highways, railways, seaport, airport etc.)”; S4 -- ”costs in vehicle resources, required products and etc. for the location of CCs in candidate site”;
4. Let us assume that = 1 , 2 , … , is the set of all demand points. Let � = {� 1 , � 2 , … , � } be the fuzzy weights of attributes. For each expert from invited group of experts (emergency service dispatchers) = {1 , 2 , … , } let � be the fuzzy positive rating (see Table 1) in triangular fuzzy numbers (TFN) of his/her evaluation for each candidate site , = 1, … , m, with respect to each attribute 𝑗𝑗 , j = 1, … , n. Table 1. Fuzzy terms and their fuzzy ratings.
5. In fuzzy set theory conversion scales are applied to transform fuzzy terms into TFNs. We use a rating scale of 1 - 9 as a semantic form of fuzzy terms of linguistic variables such as attribute's valuation, importance value of an attribute and interaction index of attributes. Our task is to build aggregation operators’ approach, which for each candidate site , = 1, … , m, aggregates presented objective and subjective data into scalar values – site’s selection ranking index. This aggregation we define by the TFCA (Triangular Fuzzy Choquett Averaging) operator: ̃ ≡ ̃ = 𝑇𝑇(� 𝑖 , � 𝑖 , … , � 𝑖𝑖 ), where (� 𝑖 , � 𝑖 , … , � 𝑖𝑖 ) are aggregated arguments of experts’ evaluations – (� 𝑖 , � 𝑖 , … , � 𝑖𝑖 ).
6. The proposed framework of location planning for candidate sites consists of the following steps (the complete presentation of the scheme is limited here): Step 1: Selection of location attributes. Step 2: Selection of candidate location sites Step 3: Locations evaluation using fuzzy aggregation approach. Step 3.1: Assignment of ratings to the attributes with respect to the candidate sites. Step 3.2: Computation of aggregated fuzzy ratings for the attributes and the candidate sites. Aggregation of fuzzy weights of attributes. Step 3.3: Computation and Normalization of the aggregated fuzzy decision matrix. Step 4: Identification of constructive TFVFM which takes into account attributes aggregated fuzzy importance values and attributes fuzzy interactions. Step 5: Computation of selection ranking index of candidate sites by the TFCA operator.
7. Let us consider the fuzzy multi-objective optimization model for the location set covering problem. Let = 1 , 2 , … , be a Boolean decision vector, which defines some selection from candidate sites = {1 , 2 , … , } for facility location. We can build sites’ selection ranking index as a linear sum of triangular fuzzy values - ̃ 𝑗𝑗 𝑗𝑗 . As a result, we obtain fuzzy objective function for selection ranking index of candidate sites ∑𝑗𝑗=1 ̃ 𝑗𝑗 𝑗𝑗 . Its maximization selects a group of sites with the maximal selection ranking index from admissible covering selections. A classical facility location set covering problem “tries” to minimize the total cost needed for opening of service centers - ∑𝑗𝑗=1 𝑗𝑗 𝑗𝑗 , where 𝑗𝑗 is a cost necessary for opening of the candidate site - 𝑗𝑗 .
8. Assume that we know in advance the approximate number of people needed to make a candidate site operate as a service center. This number is denoted by , = 1, 2, … , . Our goal is to locate service facilities centers with minimal number of agents needed to operate the opened service centers - ∑𝑗𝑗=1 𝑗𝑗 𝑗𝑗 . The problem aims at locating service facilities in minimal travel time from candidate sites. In extreme environment for emergency planning a radius of service center is defined based not on distance but on maximum allowed time for movement, since the rapid help and servicing is crucial for demand points in such situations. Suppose that experts evaluated movement fuzzy times between demand points and candidate sites ̃ 𝑗𝑗 , cc-costs 𝑗𝑗 and number of cc-agents 𝑗𝑗 .
9. Correspondingly, a set of candidate sites , covering customer ∈ = 1 , 2 , … , , is defined as = {𝑗𝑗 ∈ : 𝑃𝑃𝑃𝑃𝑃𝑃( ̃ 𝑗𝑗 <T)≥ }, 0 < < 1, where is a minimal possibilistic level (selected by administrators) which covering condition ̃ 𝑗𝑗 <T is true. Thus we can state fuzzy multi-objective facility location set covering problem: max 1 = ∑𝑗𝑗=1 ̃ 𝑗𝑗 𝑗𝑗 , min 2 = ∑𝑗𝑗=1 𝑗𝑗 𝑗𝑗 , (1) min 3 = ∑𝑗𝑗=1 𝑗𝑗 𝑗𝑗 , ∑∈ 𝑗𝑗 ≥ 1, (2) = 1, 2, … , ; = 1, 2, … , ; 𝑗𝑗 ∈ 0, 1 .
10. We illustrate the effectiveness of the constructed fuzzy optimization model by the numerical example. Let us consider an emergency management administration of Tbilisi city (Georgia) that wishes to locate some fire stations with respect to timely servicing of critical infrastructure objects. Assume that there are 30 demand points (critical infrastructure objects) and 8 candidate facility sites (fire stations) in the urban area. Let us have 4 experts from Emergency Management Agency (EMA) of the city of Tbilisi for evaluation of the travel times and ratings values of candidate sites, interaction indexes and importance values of attributes.
11. The travel times between demand points and candidate sites are evaluated in triangular fuzzy numbers (evaluated in minutes, omitted because the matrix of fuzzy travel times has large dimensions). According to the standards of the EMA, let the principle of location fire stations be that the fire station can reach the area edge within 5 minutes after receiving the dispatched instruction. Therefore, we set covering radius = 5 minutes. Minimal possibilistic level is equal to 0.9. Based on the semantic form (Table 1) experts’ , (k = 1, 2, 3, 4), ratings � for each candidate site , (i = 1,…,8), with respect to each attribute 𝑗𝑗 , (j = 1,…,10), interaction indexes between attributes and weights for each attribute are aggregated by the steps 1- 5 (all data are omitted).
12. A software is developed, in which the steps 1-5, mathematical programing problem (1)-(2) and -constraint approach are realized. Movement fuzzy times between demand points and candidate sites ̃ 𝑗𝑗 , i=1, 2, …, 30; j = 1, 2, …, 8, evaluated by experts, define the subsets of service demand points 𝑗𝑗 , j = 1, 2, …, 8. At the end, the Fuzzy Multi-objective Emergency Service Facility Location Set Covering Problem (1) – (2) is constructed, where the matrix of covering constraints B is a concatenation of vectors 𝑗𝑗 in which the covering of demand point is presented by “1” and opposite by “0” (omitted here because of large dimensions).
13. The problem takes the following form 1 = ̃ 1 1 + ̃ 2 2 + ̃ 3 3 + ̃ 4 4 + ̃ 5 5 + ̃ 6 6 + ̃ 7 7 + ̃ 8 8 ⇒ max , 2 = 351 + 472 + 553 + 394 + 705 + 626 + 467 + 578 ⇒ min , 3 = 271 + 192 + 313 + 184 + 235 + 296 + 257 + 208 ⇒ min , ≥ 1,1,1,1,1,1,1,1 , ≡ 1 , … , 8 , 𝑗𝑗 ∈ 0, 1 , = 1, … , 8, where fuzzy selection ranking indexes ̃ of candidate sites are given by Table 2.
14. Table 2: Fuzzy selection ranking indexes of candidate sites 1 δ 2 δ 3 δ 4 δ 5 δ 6 δ 7 δ 8 δ 0,0845 0,0822 0,1181 0,0868 0,0951 0,0653 0,1127 0,1169 0,4926 0,4903 0,4991 0,5513 0,4772 0,4497 0,5386 0,4777 1,4506 1,4386 1,4275 1,4752 1,4045 1,3892 1,5820 1,3238
15. After checking all 28 = 256 possible variants of constraints of the problem we receive the following twelve coverings: : {1 , 5 , 8 }, : {2 , 3 , 7 }, : {3 , 4 , 7 }, 𝐶𝐶: 3 , 5 , 8 , 𝐶𝐶: 3 , 6 , 7 , : 3 , 6 , 8 , : 3 , 7 , 8 , : 4 , 6 , 7 , 𝐶𝐶: 1 , 2 , 5 , 7 , 𝐶𝐶: 1 , 5 , 6 , 7 , 𝐶𝐶: 2 , 4 , 5 , 8 , 𝐶𝐶: 4 , 5 , 6 , 8 .
16. In the TFCA aggregation example there exist 6 Pareto solutions - {Cov1, Cov2, Cov3, Cov6, Cov7, Cov8}. It is clear that increasing the service centers selection ranking index of covering in Pareto solutions gives more worse level of the second objective function - the total cost needed for opening of service centers or of the third objective function - number of agents needed for operating the opened service centers. The decision on the choice of fire stations as service centers depends on the decision making person’s preferences with respect to risks of administrative actions. Of course, fire elimination is an extreme process and the reliability of choice of service centers is envisaged in Pareto solutions.
17. Notwithstanding, if we had to consider the problem without TFCA aggregations (interacting attributes), i.e. we would have two-criteria (2 , 3 ) classical FLSC problem, then we would have only one Pareto solution Cov3, having minimal cost and number of agents. This covering also enters into Pareto solution of fuzzy- FLSC problem, but its selection ranking index is rather low. As we see the candidate sites’ selection ranking index aggregates not only direct factors with their interactions, but also experts’ evaluations of such factors as closeness to demand points and central bases, quality of infrastructure, security issues and so on. These factors are decisive in planning distribution networks in extreme environments.
18. Thank you for your attention!