Kent Business School. Working Paper Series. A Multi-Objective Heuristic Approach for the Casualty Collection Points Location Problam

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1 Kent Business School Working Paper Series ISSN (Online) A Multi-Obective Heuristic Approach for the Casualty Collection Points Location Problam Tammy Drezner and Zvi Drezner California State University-Fullerton Said Salhi Kent Business School 1 Working Paper No. 112 December 2005

Location Problam Tammy Drezner and Zvi Drezner California State

2 A Multi-Obective Heuristic Approach for the Casualty Collection Points Location Problem Tammy Drezner and Zvi Drezner College of Business and Economics California State University-Fullerton Fullerton, CA 92834, U.S.A. Said Salhi Centre for Heuristic Optimisation Kent Business School, The University of Kent Canterbury, Kent, CT 2 7PE, U.K. s.salhi@kent.ac.uk Abstract In this paper we formulate the casualty collection points (CCP) location problem as a multi-obective model. We propose a Minimax Regret Multi-Obective (MRMO) formulation that follows the idea of the minimax regret concept in decision analysis. The proposed multi-obective model is to minimize the maximum percent deviation of individual obectives from their best possible obective function value. This new multiobective formulation can be used in other multi-obective models as well. Our specific CCP model consists of five obectives. A descent heuristic and a tabu search procedure are proposed for its solution. The procedure is illustrated on Orange County, California. Keywords: multi-obective, casualty collection points, facility location, heuristics. 1. Introduction The location of casualty collection points was introduced by Drezner 1. The problem is to locate a set of p facilities that will serve as casualty collection points (CCPs). The CCPs are employed in cases of mass casualty incidents which require delivering emergency medical care to a large number of victims. This preparedness for mass casualty incidents includes maor earthquakes or any other natural or man-made incident over an area. The underlying assumption is that in a catastrophic event such as a Some of this work was carried out at the University of Birmingham, UK. 2

K. e-mail: s.salhi@kent.ac.uk Abstract In this paper we formulate the casualty collection points (CCP) location problem as a multi-obective model.

3 maor earthquake, the area's civil infrastructure (freeways, roads, communications, emergency medical services, etc.) will not be operational. Hospitals may, themselves, become victims or otherwise inundated or inaccessible. Normal emergency medical services may be cut-off from demand points due to collapsed infrastructure. If such a disaster strikes, pre-determined Casualty Collection Points (CCPs) will be operationalized and staffed by medical teams and emergency medical technicians who have been pre-assigned to them. People in need of medical attention will have to get to these CCPs on their own. It can be assumed that they will go to the nearest CCP. These CCPs will be established in public or private facilities, such as colleges, high schools, baseball and football fields, golf courses, and public parks, which are open and large enough to accommodate a large number of people, are relatively free from falling debris, and can accommodate a helicopter pad for air transportation. The CCP problem is formulated as a multi-obective model. The single obective CCP location problem for six obectives was addressed by Drezner 1 and solved optimally for some values of p. Multi-obective optimization is a class of optimization in which there are several obectives, some of which are incompatible or conflicting. It is well known that there is no unique optimal solution for such problems. One can find a set of non-dominated solutions known as efficient points, also called Pareto-optimal or Paretoefficient. In recent years the interest in multi-obectives in location theory has increased considerably dating back to the work of Wendell and Hurter 2. Approaches commonly used in this field include weighted multi-obectives, lexicographical approach and goal programming-based approaches among others (see Steuer 3 for more details). 3

If such a disaster strikes, pre-determined Casualty Collection Points (CCPs) will be operationalized and staffed by medical teams and emergency medical technicians who have been pre-assigned to them.

4 In this paper, we consider five obectives: p-median, p-center, two maximum covering obectives, and the minimum variance obective. The minimum variance obective is included in the multi-obective model because CCPs are a matter of public policy, and some measure of equity should be considered. Since there is no hierarchy among these obectives we treat all of them as equally important. We propose to employ a multi-obective approach that is similar to the minimax regret concept in decision analysis. The obective is to minimize the maximum percent deviation from the optimum of each individual obective. In the next section we introduce the minimax regret multi-obective (MRMO) approach (also discussed in Wηrker 4 ) and formulate the MRMO model for the CCP problem. The solution of the MRMO model requires two phases. In phase 1, as will be discussed in Section 3, we present an optimal as well as heuristic methods for finding the optimal (or a good) solution for each individual obective. Once the optima for the individual obectives are found, the MRMO model is solved in phase 2 and computational experiments for Orange County, California are presented (see Section 4). In Section 5 we summarize our findings and suggest potential avenues for future research. 2. The Minimax Regret Multi-Obective (MRMO) Approach Multiple obective decision making is a branch of multi-criteria decision making which also covers multi-attribute decision making. Multiple obective decision making problems involve selecting the best alternative given a set of multiple, often conflicting, obectives whereas multi attribute decision making refers to making preference decision (i.e., evaluation, prioritization, selection) frm the available alternatives. Criteria can 4

Since there is no hierarchy among these obectives we treat all of them as equally important.

5 incorporate both attributes and obectives. See Hwang and Yoon 5 and also Yoon and Hwang 6 for more details on multiple attribute decision making. One common way to model multi-obective formulations is to use a hierarchical model, assign weights to each obective, and minimize (or maximize) the weighted obective. We propose to apply a minimax regret obective used in decision analysis models. The minimax regret criterion is typically the recommended approach when probabilities for each individual scenario are unknown. In the CCP location case it is not clear how to allocate relative importance to various obectives. We therefore think it is not appropriate to provide a weight for each scenario. Besides, we do not establish a hierarchy between the obectives as they are equally important. Let X = x Lx ) be n variables, and f ( X ), L f ( X ) 0 be m obectives ( 1 n 1 m > (either to be minimized or to be maximized) that constitute the MRMO model. The * * feasible domain for the variables is a set of discrete points in the plane. Let f L 0 1 f m > be the optimal values of the m individual obectives. Define for any vector X: * f ( X ) f if the obective f ( ) is to be minimized * X f ( X ) = * (1) f f ( X ) if the obective f ( X ) is to be maximized * f Then, the MRMO model is to Minimize X { Maximum [ ( X )]} (2) Since all obectives are considered equally important, we aim to stay as close as possible to all of them. Note that weights can be added to the model by multiplying each ( X ) in Equation (2) by a different weight. One can use the obective of minimizing 5

We propose to apply a minimax regret obective used in decision analysis models.

6 p Table 7: Further Experiments with a Random Tabu Tenure Best Known MRMO 100 p iterations 200 p iterations TT in [12, 24] TT in [12, 30] TT in [12, 24] TT in [12, 30] # Time a # Time a # Time a # Time a * * * Minimum Average * Optimal (see Table 4). # Number of times best known solution found. a Total run time in minutes. 1 A solution of obtained 6 times. 25

57 8 26.253 87 96.51 84 96.67 99 193.60 100 193.52 9 28.746 71 109.18 66 109.61 76 218.37 79 217.98 10 26.372 62 122.12 67 122.58 86 244.43 87 243.94 15 18.794 2 177.00 5 179.11 1 354.67 6 354.

7 Figure 1: An example problem 26

8 Figure 2: MRMO Optimal Solutions LA County Riverside County Pacific Ocean San Diego County Potential Sites OC Boundary Demand Points p=4 p=5 27

Ocean San Diego County 0 5 10 15 20 25 30 35 40

9 28

Kent Business School. Working Paper Series. A Multi-Objective Heuristic Approach for the Casualty Collection Points Location Problam

Kent Business School. Working Paper Series. A Multi-Objective Heuristic Approach for the Casualty Collection Points Location Problam Kent Business School Working Paper Series ISSN 1748-7595 (Online) A Multi-Objective Heuristic Approach for the Casualty Collection Points Location Problam Tammy Drezner and Zvi Drezner California State