- Email: [email protected]
Optimal location and capacity of emergency cleanup equipment for oil spill response
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
EuropeanJournal of OperationalResearch96 (1996) 72-80
Theory and Methodology
Optimal location and capacity of emergency cleanup equipment for oil spill response E l e f t h e r i o s I a k o v o u a, *, C h i M . Ip b, C h r i s t o s D o u l i g e r i s c, A s h u t o s h K o r d e d a Department of Industrial Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA b Alamo Rent A Car, Inc., The 110 Tower - 110 S,E. 6th Street, P.O. Box 22776, Fort Lauderdale, FL 33335, USA c Department of Computer and Electrical Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA a Sea-Land Service, Inc., 6000 Carnegie Boulevard, Charlotte, NC 28209, USA
Received 1 June 1995;accepted24 April 1996
Abstract
Myopic planning and tactical decision-making with regards to oil spill events result in severely inefficient cleanup efforts with high fixed, operating and cleanup/damage costs. We propose an integrated framework that addresses some important strategic issues such as determining the optimal location and capacity of cleanup equipment, taking into account their post event implications. We develop a linear integer programming model, exhibit its complexity and propose a solution method by solving a relaxation of the original problem. Realistic examples applied to the Florida peninsula are presented. Keywords:
Location;Modelling;Strategicplanning; Transportation
1. Introduction
The importance of effective oil spill response strategies for the government as well as for the ocean transportation, oil exploration, and oil production industries is exemplified by the environmental, monetary, and social ramifications of massive disastrous oil spill incidents (e.g. Torrey Canyon spill in Britain, 1967; Amoco Cadiz spill in France, 1978; Exxon Valdez spill in Alaska, 1989; Shetland Islands incident, 1992; and the most recent spill of the Sea Empress off St. Ann's Head at South Wales, February 15, 1996; etc.). Considerable legislative activity has been witnessed since the early 1970s in the United States regarding prevention and response to * Correspondingauthor. Email: [email protected].
major maritime pollution incidents. The Oil Pollution Act of 1990 (OPA 90, Public Law 101-380 [8]) was the response of the U.S. Congress to the public outcry for the large spill that occurred in the Exxon Valdez incident. OPA 90 specifies directions for the response capabilities of state and federal agencies, and of all other parties involved in oil handling and transportation. The United States Coast Guard (USCG) established its own Marine Environmental Protection program (i) to oversee a contingency fund aimed to finance cleanup operations, and (ii) to maintain specialized response equipment and trained response teams responsible for providing support and assistance to major spills. Equipment acquired by the Coast Guard for pollution removal has been distributed to USCG units at major ports and in areas with limited commercial and private resources. Spe-
0377-2217/96/$15.00 Copyright© 1996ElsevierScienceB.V. All rights reserved. PII S0377-2217(96)00106-3
E. lakovou et a L / European Journal of Operational Research 96 (1996) 72-80
cialized equipment (usually very costly) is also maintained by the USCG's National Strike Force Team located on the East, West, and Gulf Coasts. The oil spill cleanup decision-making is structured in three levels of hierarchy ([10]): strategic, tactical, and operational. In the strategic level, issues of investment and long-term planning (5-15 years) strategy against future oil spills are addressed. The tactical level tackles decisions to be made in response to a specific oil spill. The operational level addresses in a detailed fashion, the optimal performance of the cleanup equipment under different weather and operating conditions. Hence at the strategic level, decisions are made regarding the location, number, and type of equipment to be stored at a facility site to respond to future oil spills, while at the tactical level decisions involve the allocation of the appropriate number and type of the equipment, once the occurrence of a spill is made known. Although the complexity of the operational problem is certainly less than that of the strategic and tactical problems, it seems to be the problem of the utmost significance for the responding agencies (especially for the USCG). For major spills the USCG usually reacts by mobilizing every available vessel and equipment to contain a spill, even though a specific vessel may not be appropriate for tackling the particular type of oil spilled (e.g. self-inflatable booms perform very poorly in high currents with velocity greater than 1 knot). This policy often leads to an organizational disarray. It has been observed in many cases that a number of equipment that have been transported to an oil spill site are never used. Furthermore, oil spill equipment are usually very specialized (exhibiting limited flexibility for other uses) and capital intensive. Suboptimal tactical policies can lead to both severely inefficient cleanup efforts and high fixed and operating costs. Therefore it is imperative that the strategic issues of optimal location, number, and type of equipment at various facility sites, as well as the tactical issues of the appropriate capacity levels and post oil spill dispatching policies are addressed adequately. However, it should be recognized that these two echelons of the hierarchical decision-making are highly interconnected. The structure of the paper is as follows. In Section 2 we review the relevant literature and in Sec-
73
tion 3 we provide a description of the problem. In Section 4 we provide the notation, present the problem formulation, and propose solving a relaxation of the original problem. In Section 5 realistic examples applied to the Florida peninsula are presented. Section 6 completes the paper with the conclusion.
2. Literature review
Various mathematical models have been formulated in the past to tackle the strategic and the tactical decision-making problems. Psaraftis et al. [10] formulate a mixed integer programming problem to decide on the location of the appropriate levels and types of cleanup equipment to respond to oil spills. Church and ReVelle [3], Belardo et al. [1] use a partial covering approach to site response resources for major maritime oil spills. Chames et al. [2] propose a chance-constrained goal programming model to evaluate the capacity of the cleanup resources. Daskin and Stern [4], Moore and ReVelle [7] use a hierarchical and multiobjective programming approach to determine the location and allocation of emergency medical service vehicles. The tactical oil spill decision-making problems encountered in the literature address the problems of allocating various available equipment once the occurrence of the spill is known. Psaraftis and Ziogas [9] propose a tactical decision algorithm for the optimal dispatching of oil spill cleanup equipment. Their model decides on the aggregate cleanup capability on scene at a particular time stage to respond to a given spill. The mathematical models described above decide on either the location, number, and type of equipment, or on the allocation of these equipment once the occurrence of the spill is known. All these models address the strategic and the tactical problems independently. However, the two are highly interconnected since strategic decisions impose severe constraints on the post spill decisions. Previous research, while addressing tactical questions, proposes dispatching an equipment set, as a whole, to respond to an oil spill. An equipment set is an integrated, sufficiently equipped and self-contained set capable of cleaning up a prescribed quantity and type of oil in a given period. However, such
74
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
a policy would require each facility site to store a wide range of types of equipment. As discussed in Section 1, some of the equipment in the equipment set are costly and highly dedicated and are used seldomly because of their limited operating capability. It is obvious that a policy of deploying predetermined equipment sets could lead to severely suboptimal solutions and consequently to high fixed, operating, and damage costs.
3. Description of the problem OPA 90 [8] calls for renewed actions to prevent the occurrence of oil spills, to minimize their impact, and improve the efficiency of cleanup efforts. As a direct result of OPA 90, a lot of emphasis has been placed on oil spill prevention. Once an oil spill occurs it is necessary to respond with sufficient cleanup equipment within the shortest possible time in order to protect marine species and minimize cleanup and damage costs. The complex hierarchical nature of the oil spill cleanup problem has resulted in a lack of integrative approaches in response strategies. We intend to address the problem without resorting to solving the strategic problem independently of the tactical problem. We believe that it would be of utmost importance and benefit to the strategic planner of a responding agency to be able to evaluate the merit of alternative high level decisions while taking into account their impact on post oil spill decisions. Given the complexity of the problem under study, these "what-if" analyses are very popular among the planning and responding agencies for two reasons: (1) they can lead to a better understanding of the problem structure (especially when used on a sequential fashion as more information becomes available), and (2) they can be used for the training of planners/responders. We specifically consider the composition of the optimal equipment mixture that should be dispatched from different facility sites to respond to an oil spill. Equipment sets are not predetermined and decisions regarding location of the equipment are made based on the "demand" (imposed by oil spills) that has to be met by the facility site. This demand is forecasted using historical data. As soon as the responding
agency is notified of an oil spill, the available equipment are grouped together and are dispatched from the facility sites which can respond within the critical time. This eliminates the unnecessary storage of equipment with a small probability of usage, at multiple sites. The decisions regarding the optimal location, number, and type of equipment to be stored, are made based on the decisions regarding the optimal allocation of equipment to respond to an oil spill. We study the problem as a deterministic one, since the decisions are made based on the expected volume, type, and weather conditions which are obtained from historical data. Such data are available from various databases of United States' federal and state agencies (e.g. data provided by the Office of Marine Safety, Security, and Environmental Protection of the USCG). The data consist of the frequency, volume, type, location, weather conditions, and sea-state of an oil spill event. The total area under study is partitioned into grids. Statistical analysis is performed on historical data to determine the expected volume, type, and weather and sea conditions for each grid. Using statistical averages the expected demand point for each grid is obtained by analyzing locations and frequencies of oil spills in the past. The above analysis is performed to determine certain input parameters such as the number and type of equipment required to respond to a given spill and the expected travel times for transporting the equipment from a facility site to the demand point (spill site). It should be noted that the travel time for a piece of equipment depends on the distance between the facility site and the spill site, the type of equipment, and on the weather and sea conditions. It is assumed that the fixed costs of opening a facility site and the operating costs of the equipment are known. The fixed costs include the cost of building a facility site and the cost of acquiring the equipment to be located at the site. These costs depend on the location of the site and vary with the geographical location. The operating costs include the cost of maintaining the equipment and the overhead costs at a facility site. The transportation costs take into account implicitly the cleanup costs and the damage costs. The cleanup costs constitute the costs involved in moving the cleanup equipment (travel costs) from a facility site(s) to the demand point, the
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
labor costs, and the overhead. Typically the cleanup costs, as categorized by the U.S. Department of Transportation, consist of: 1. 2. 3. 4. 5.
basic response costs, pollution threat costs, evacuation costs, bridge closing costs, and the costs due to injury or loss of life of a crew member.
The damage costs include, among others, costs due to vessel damage, human deaths, human injuries, cargo damage, navigational aid damage, bridge damage, environmental/marine life loss, and damages to recreational activities and beaches. Another important input parameter is critical time. The Presidential directive passed by Jimmy Carter in 1977 states that the USCG should have sufficient capability to respond adequately to a 100,000 gallons within six hours. There have been various interpretations of the activities that should take place within these six hours (e.g. to reach the spill site or to be able to start cleanup operations within six hours). The directive simply specifies the maximum time within which the spill should be tackled. The critical times for each grid and spill may be different based on the volume, type, location, and weather conditions and are determined by the responder. The responder's position is quite delicate while deciding on the critical times for each grid [5]. The responder has to protect the sensitive marine areas (for which the marine ecologists and the environmental organizations are very concerned), while at the same time he/she should also prevent the oil from impacting the shoreline (an issue of outmost importance for the tourism industry). Based on the above, it is clear that the most important performance measure in the oil spill cleanup effort is the response time since the success and efficiency of all subsequent cleanup efforts rely heavily on it. It is also assumed that all events occur at discrete points in time and that the probability of simultaneous occurrence of two or more spills is negligible. Furthermore, no equipment is subcontracted and all required equipment can be dispatched from USCG facility sites.
75
4. Problem formulation and solution
4.1. Notation The following notation will be used throughout our discussion:
I J K S f~ cij
Finite set of facility sites. Finite set of equipment types. Finite set of spill sites. Finite set of spill types. Fixed cost of opening a facility site i. Operating cost of storing an equipment of type j at facility site i. kj Unit cost/unit time for equipment type j. t,.jk Travel time for equipment type j to be transported from facility site i to spill site k. xi~ Number of type j equipment located at facility site i. zijks Number of type j equipment to be dispatched from facility site i to spill site k for spill type s. y; = 1 if facility site i is opened, 0 otherwise. ~tjks Number of type j equipment required at spill site k to cover a spill of type s. T Critical time to respond to a spill. Fjk = {i : t~jk < T}; the set of all facility sites for which equipment type j can respond to an oil spill at spill site k within the critical time.
4.2. Mathematical formulation We present a formulation that addresses strategic decisions taking into consideration post oil spill event implications. A linear integer programming formulation is developed that decides on the optimum number of facility sites to be opened, the optimum quantities, the type of equipment to be located at a facility site, and the optimal equipment dispatching policy after the occurrence of the spill is made known. The objective function is comprised of the fixed costs of opening a facility site, the operating costs of storing the equipment at facility sites, and the trans-
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
76
portation costs of transporting the equipment from a facility site to the demand point to respond to a spill (which capture cleanup and damage costs, as discussed in Section 3). The constraints address feasibility, capacity, and critical time issues. The linear integer programming formulation for the oil spill cleanup problem is as follows: min E f i Y i -P E E C i j X i j q- E E E EkjtijkZijks
i
i
i
i
j
k
s (1)
subject to: xij~MijYi,
Vi~l,j~J,
~_, Zijks>ct#s,
VjEJ, k~K,s~S,
(2) (3)
i~Fjk
Zi#0, Yi ~ {0,1},
Vi~I,j~J,k~K,s~S, integers,V j ~ J , k ~ K , s ~ S , Vi~l.
(4)
method based on a relaxation of the original problem.
4.3. The relaxed problem Even though the number of integer variables for a realistically sized problem is prohibitively large, the only structural integer variables are the y/'s. Since the number of yi's, III, is typically small, it is relatively easy to solve, using branch and bound, the mixed-integer relaxed problem where only the y;'s are assumed binary (with the integrality requirement of constraint (5) being relaxed). In fact, since the relaxed problem is "transportation-like" its solution in practice is likely to be integral or near-integral (only a few variables are fractional). In the latter case the fractional solution can be rounded to obtain a near-optimal solution.
(5) (6)
Constraints (2) allows the storage of equipment of type j at a facility site i only if the facility site i is opened, where M~j is a known upper bound for xij. Constraints (3) ensures that any demand for equipment type j for spill type s at demand point k has to be covered within the critical time from facility site i. Constraint (4) ensures that the maximum number of equipment of type j for a spill of type s to be dispatched from a facility site i to the demand point k cannot exceed the available capacity of the facility site. Finally, x o. and z;jks are positive integers and Yi is a binary variable. It should be noted that even for small values of III, IJI, Igl, and IsI, the complexity of the problem is very significant. The formulation consists of (1II x IJI X Igl X ISI + III X IJI) general integer variables and III binary variables. Hence, for an example case with I/I = 30, IJI = 100, IKI = 20 and IsI = 10, the number of the general integer variables is 603,000. The number of constraints is (111X IJI + IJI × Igl X ISI + I11X IJI X IKI X ISI). The relaxed problem obtained by fixing the values of Zqk~ is the uncapacitated facility location problem, known to be NP hard [6]. No standard algorithm which can solve such a problem in polynomial time is available in the literature. In the next section, we propose a solution
5. Numerical examples In this section, we present two numerical exampies applied to the Florida peninsula. The first data set consists of six facility sites, nine equipment types, ten grids (demand points), and one type of spill at each demand site. Classification of the data is based on the type of oil spilled, volume of oil spilled, the weather and sea conditions. The second data set differs from the first only by a finer grid size; there are thirty (instead of ten) grids. Oil types are classified based on various physical and chemical properties of the oil. This is necessary because different types of equipment are used to respond to spills of different oil types. Oil is classified into: 1. Light Crude Oil. 2. Heavy Crude Oil. 3. Diesel. The oil spills are categorized into four types based on the volume and type of oil spilled. The classification is based on the capability of the equipment required for the cleanup operations and is as follows: 1. 0-15,000 gallons. 2. 15,000-50,000 gallons.
E. lakovou et a l . / European Journal o f Operational Research 96 (1996) 72-80
Table 1 Expected number and type of equipment
(0/#) required for
77
IKI = 10 grids
Grid number
Equipment Type
J=l ./=2 3=3 ./=4 ./=5 J=6 =7 ,=8 '=9
k=l
k=2
k=3
k=4
k=5
k=6
k=7
k=8
k=9
k=10
3 3 1 4 2 1 4 5 2
5 1 4 3 1 4 2 6 3
2 1 3 3 3 1 0 2 3
1 4 1 5 3 4 2 4 5
5 1 4 2 2 4 2 1 4
3 1 2 3 2 1 1 5 3
1 4 5 4 2 2 4 2 3
3 5 1 4 4 2 4 5 2
2 5 0 4 1 4 2 6 3
2 3 4 2 1 1 4 2 3
3. 50,000-100,000 gallons. 4. 100,000-300,000 gallons.
on the maximum operating significant wave height (this is the standard classification used by the U.S. Coast Guard):
Different types of equipment are required for different weather/sea conditions. The weather/sea conditions are also classified into three categories based Table 2 Expected number and type of equipment required for [KI = 30 grids 0 / t . t - ott.lo ai. ll--al.20 0/1.2t- al.3o Or2,1-- 0~2,10 0/2.11--O/2.20 0/2,2~- 0/2,30 0/3.1 -- O/3.10 Or3,11--/~3,20 0/3,2]- 0/3.30 ~4.1- 0/4.1o oq.i ] - ot4,2o 0/4.21 0t4.30 0/5.]-- aS. j0 0/5, I l - °t5,2o O/5.21 O~5.30 0/6.1 -- 0/6.10 0/6, I 1 016.20 0/6.21 a6.30 0/7,1--0/7.10 0 / 7 : t - ot7.2o 0/7.21 a7.30 ors.i- as.J0 o/s. 1t - crs.20
0 l 0 0 2 0 l 1 2 2 1 2 1 1 0 1 0 0 0 0 0 2 1
ors.23- ots,3o 0/9. i - o/9.1o 0/9. I I - 0/9.20 °/9.21 0/9.30
-
-
-
-
-
-
-
-
- -
-
-
2 0 1 1 2 1 0 0 0 0 2 1 0 0 2 0 2 2 2 0 2 2 1
1 2 2 2 0 2 0 0 1 2 0 I 1 0 1 0 2 0 2 1 2 1 0
2
0
0
l 2 0
0 1 2
1 0 0
I l 0 1 0 2 0 2 0 0 0 2 0 0 1 1 0 0 2 0 0 2 0
2 2 0 0 l 1 2 2 0 1 2 0 1 2 1 1 2 2 0 1 1 2 1
2 0 0 0 0 2 2 2 0 2 0 2 0 2 0 2 0 2 0 I 0 2 2
0 1 2 0 0 2 0 0 0 2 1 2 1 0 0 0 0 0 0 0 1 1 1
2
2
2
2
2 2 0
0 2 1
1 2 2
2 1 0
1 2 0 0 1 2 1 0 1 1 2 0 0 0 0 1 1 1 0 0 2 0 2
1 1 1 1 2 1 2 1 2 0 0 0 2 0 0 0 1 0 0 2 2 1 2
0 0 1 0 2 0 0 2 1 2 2 2 2 2 1 2 1 0 2 2 0 2 1
1
1
0
1 0 2
0 2 1
2 1 0
1. Calm water. 2. Harbor. 3. Offshore.
The maximum operating significant wave height for calm water, harbor, and offshore is 0.3 m, 0.9 m, and 1.8 m respectively. These three categories apply to bays, soft bottom subtidals, seagrass beds, rocky subtidal, intertidal, sandy beaches, coral reefs, mangrove forests, and salt marshes. The oil-spill cleanup equipment types are: 1. 2. 3. 4.
Booms. Skimmers. Chemical dispersants. Sorbents.
5. Earth barriers.
6. 7. 8. 9.
Pumps. Shoreline protection. Storage equipment. In situ burning equipment.
The demand for the first example is given in Table 1. Demand is based on the average calculaTable 3 Fixed cost at each of the facility sites i f,.
I 30,676
2 40,338
3 22,456
4 10,631
5 14,394
6 15,744
78
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
Table 4 Unit cost/unit time for each equipment type j k)
1 35
2 40
3 55
4 40
5 25
6 50
Table 5 Operating costs at each of the six facilities for the nine equipment 7 35
8 50
types
9 60
C=
tions using historical data. Each equipment type is mentioned by its corresponding index as provided above; for example, equipment type 2 is a skimmer.
344 429 454 316 414 461
393 393 300 422 352 411
429 374 461 394 352 489
363 396 275 376 430 427
320 411 363 373 455 461
334 383 317 459 303 378
339 388 497 386 496 476
I[~.i1~1ol414121114I~1KII Daytona
Booth
Ilxs~121al s 4 l a l 4 1 2 1 6 1 0 l l Plerc~ "z-
IIx,~151513121410141614[ ~o~
Palm
Beach
II~Jlal~1415 114141513 ~ Por~ Por~
Fig. 1.
Kverg o~
Miom
ode~
402 348 434 444 324 337
371 323 304 447 476 493
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
The entries of the table are the ajks'S, the required number of type j equipment required at spill site k to contain a spill of type s. The demand for the second example (with the finer grid of Ik[ = 30) is given in Table 2. Table 3 provides the fixed costs f~ for each of the six facilities, while Table 4 exhibits the operational costs for the various types of equipment, c u. Table 5 displays the unit cost/unit time for equipment type j (k j). Table 2, Tables 3 and 4 apply to both examples; namely, the corresponding costs are the same.
It is only the scale of the spill grid sizes that has changed. Fig. 1 displays the schematics of the locations of the potential facilities and the demand sites for the first example as they are applied to the east coast of Florida. The underlying map is the product of ARCINFO (one of the most popular Geographical Informations Systems). Fig. 2 displays the schematics for the second example. In this case each grid of the first example is now divided into three subgrids. Hence the demand requirements for the grids of the
II.~., t 1121 i 121 o 121 ~- 12 I ~- II Do~"t
~r~a
79
E~,gach
I1~Jl2121212121112121211 -I
For
Piorc@
11~,~12121212121212{2t2l[[ West
Palm
Beach
{1~12i21212121212121211 Por~ P o r ~ o#
": :: :.<,.~
Fig. 2.
Ev~rgladQo M,am,
80
E. lakovou et a l . / European Journal of Operational Research 96 (1996) 72-80
two examples are related. More specifically, the demand for each grid of the first example is the sum of the demands of its subgrids from the second case; e.g. a H (of Table 1) = all + alE + a13 (of Table 2) = 3, ¢g12 (of Table 1) = al4 + a15 + ¢g16 (of Table 2) = 5, etc. The mixed LP relaxation is solved using the CPLEX optimizer (Version 3.0). For the first numerical example, the solution opens four facilities, Y3 = Y4 = Y5 =Y6 = 1, (open facilities depicted with an open rectangle in Fig. 1 as opposed to stars for unopened facility sites). The number and type of equipment (x~j) that should be stockpiled at each of these four facilities are also given in Fig. 1 adjacent to each opened facility. For the second numerical example CPLEX suggests opening the same four facilities as in the first example; Y3 = Y4 = Y5 = Y6 = 1 (Fig. 2). However, the x i / s of the solution are now different and are also provided in Fig. 2.
6. Conclusion We have presented an integrative approach that addresses strategic decisions regarding optimum location and capacity of oil spill cleanup equipment, jointly taking into consideration their impact on tactical decision-making. A linear integer programming model was first developed and a mixed-integer relaxation of the original problem is solved. Two realistic examples as applied to the east coast of Florida are presented. Further research needs to be done to test the validity of such models given the lack of consistency and standardization among the databases provided by the plethora of federal and state agencies in the United States.
Acknowledgements The assistance of Lalit Yudhbir in the numerical experimentation is acknowledged. The authors would also like to thank the anonymous referees for their constructive comments.
References [1] Belardo, S., Harrald, J., Wallace, W.A., and Ward, J., " A partial covering approach to siting response resources for major maritime oil spills", Management Science 30 (19841)) 1184-1196. [2] Charnes, A., Cooper, W.W., Karwan, K.R., and Wallace, W.A., " A Chance-constrained goal programming model to evaluate response resources for marine pollution disasters", Journal of Environment Economics and Management 6 (1979) 244-274. [3] Church, R., and ReVelle, C., "The maximal covering location problem", Papers Regional Sci. Appl. 32 (1974) 101118. [4] Daskin, Marks S., and Stern, Edmund H., "'A hierarchical objective set covering model for emergency medical service vehicle deployment", Transportation Science 15 (1981) 137-152. [5] Demis, D.J., "Oil spill management: The damage assessment model and the spatial allocation of cleanup equipment", Unpublished M.S. Thesis, MIT Department of Ocean Engineering, June 1984. [6] Erlenkotter, D., " A dual-based procedure for uncapacitated facility location' ', Operations Research 26 (1978) 992-1009. [7] Moore, George C., and ReVelle, Charles, "The hierarchical service location problem", Management Science 28 (1982) 775-780. [8] Oil Pollution Act, 33 United State Congress, Section 2071 et seq; Pub.L.No. 101-380, 104 Stat. 484. [9] Psamftis, H.N., and Ziogas, B.O., " A tactical decision algorithm for the optimal dispatching of oil spill cleanup equipment", Management Science 31 (1985) 1475-1491. [10] Psamftis, H.N., Tharakan, G.G., and Ceder, A., "Optimal response to oil spills: The strategic decision case", Operations Research 34 (1986) 203-217.
EuropeanJournal of OperationalResearch96 (1996) 72-80
Theory and Methodology
Optimal location and capacity of emergency cleanup equipment for oil spill response E l e f t h e r i o s I a k o v o u a, *, C h i M . Ip b, C h r i s t o s D o u l i g e r i s c, A s h u t o s h K o r d e d a Department of Industrial Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA b Alamo Rent A Car, Inc., The 110 Tower - 110 S,E. 6th Street, P.O. Box 22776, Fort Lauderdale, FL 33335, USA c Department of Computer and Electrical Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA a Sea-Land Service, Inc., 6000 Carnegie Boulevard, Charlotte, NC 28209, USA
Received 1 June 1995;accepted24 April 1996
Abstract
Myopic planning and tactical decision-making with regards to oil spill events result in severely inefficient cleanup efforts with high fixed, operating and cleanup/damage costs. We propose an integrated framework that addresses some important strategic issues such as determining the optimal location and capacity of cleanup equipment, taking into account their post event implications. We develop a linear integer programming model, exhibit its complexity and propose a solution method by solving a relaxation of the original problem. Realistic examples applied to the Florida peninsula are presented. Keywords:
Location;Modelling;Strategicplanning; Transportation
1. Introduction
The importance of effective oil spill response strategies for the government as well as for the ocean transportation, oil exploration, and oil production industries is exemplified by the environmental, monetary, and social ramifications of massive disastrous oil spill incidents (e.g. Torrey Canyon spill in Britain, 1967; Amoco Cadiz spill in France, 1978; Exxon Valdez spill in Alaska, 1989; Shetland Islands incident, 1992; and the most recent spill of the Sea Empress off St. Ann's Head at South Wales, February 15, 1996; etc.). Considerable legislative activity has been witnessed since the early 1970s in the United States regarding prevention and response to * Correspondingauthor. Email: [email protected].
major maritime pollution incidents. The Oil Pollution Act of 1990 (OPA 90, Public Law 101-380 [8]) was the response of the U.S. Congress to the public outcry for the large spill that occurred in the Exxon Valdez incident. OPA 90 specifies directions for the response capabilities of state and federal agencies, and of all other parties involved in oil handling and transportation. The United States Coast Guard (USCG) established its own Marine Environmental Protection program (i) to oversee a contingency fund aimed to finance cleanup operations, and (ii) to maintain specialized response equipment and trained response teams responsible for providing support and assistance to major spills. Equipment acquired by the Coast Guard for pollution removal has been distributed to USCG units at major ports and in areas with limited commercial and private resources. Spe-
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E. lakovou et a L / European Journal of Operational Research 96 (1996) 72-80
cialized equipment (usually very costly) is also maintained by the USCG's National Strike Force Team located on the East, West, and Gulf Coasts. The oil spill cleanup decision-making is structured in three levels of hierarchy ([10]): strategic, tactical, and operational. In the strategic level, issues of investment and long-term planning (5-15 years) strategy against future oil spills are addressed. The tactical level tackles decisions to be made in response to a specific oil spill. The operational level addresses in a detailed fashion, the optimal performance of the cleanup equipment under different weather and operating conditions. Hence at the strategic level, decisions are made regarding the location, number, and type of equipment to be stored at a facility site to respond to future oil spills, while at the tactical level decisions involve the allocation of the appropriate number and type of the equipment, once the occurrence of a spill is made known. Although the complexity of the operational problem is certainly less than that of the strategic and tactical problems, it seems to be the problem of the utmost significance for the responding agencies (especially for the USCG). For major spills the USCG usually reacts by mobilizing every available vessel and equipment to contain a spill, even though a specific vessel may not be appropriate for tackling the particular type of oil spilled (e.g. self-inflatable booms perform very poorly in high currents with velocity greater than 1 knot). This policy often leads to an organizational disarray. It has been observed in many cases that a number of equipment that have been transported to an oil spill site are never used. Furthermore, oil spill equipment are usually very specialized (exhibiting limited flexibility for other uses) and capital intensive. Suboptimal tactical policies can lead to both severely inefficient cleanup efforts and high fixed and operating costs. Therefore it is imperative that the strategic issues of optimal location, number, and type of equipment at various facility sites, as well as the tactical issues of the appropriate capacity levels and post oil spill dispatching policies are addressed adequately. However, it should be recognized that these two echelons of the hierarchical decision-making are highly interconnected. The structure of the paper is as follows. In Section 2 we review the relevant literature and in Sec-
73
tion 3 we provide a description of the problem. In Section 4 we provide the notation, present the problem formulation, and propose solving a relaxation of the original problem. In Section 5 realistic examples applied to the Florida peninsula are presented. Section 6 completes the paper with the conclusion.
2. Literature review
Various mathematical models have been formulated in the past to tackle the strategic and the tactical decision-making problems. Psaraftis et al. [10] formulate a mixed integer programming problem to decide on the location of the appropriate levels and types of cleanup equipment to respond to oil spills. Church and ReVelle [3], Belardo et al. [1] use a partial covering approach to site response resources for major maritime oil spills. Chames et al. [2] propose a chance-constrained goal programming model to evaluate the capacity of the cleanup resources. Daskin and Stern [4], Moore and ReVelle [7] use a hierarchical and multiobjective programming approach to determine the location and allocation of emergency medical service vehicles. The tactical oil spill decision-making problems encountered in the literature address the problems of allocating various available equipment once the occurrence of the spill is known. Psaraftis and Ziogas [9] propose a tactical decision algorithm for the optimal dispatching of oil spill cleanup equipment. Their model decides on the aggregate cleanup capability on scene at a particular time stage to respond to a given spill. The mathematical models described above decide on either the location, number, and type of equipment, or on the allocation of these equipment once the occurrence of the spill is known. All these models address the strategic and the tactical problems independently. However, the two are highly interconnected since strategic decisions impose severe constraints on the post spill decisions. Previous research, while addressing tactical questions, proposes dispatching an equipment set, as a whole, to respond to an oil spill. An equipment set is an integrated, sufficiently equipped and self-contained set capable of cleaning up a prescribed quantity and type of oil in a given period. However, such
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E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
a policy would require each facility site to store a wide range of types of equipment. As discussed in Section 1, some of the equipment in the equipment set are costly and highly dedicated and are used seldomly because of their limited operating capability. It is obvious that a policy of deploying predetermined equipment sets could lead to severely suboptimal solutions and consequently to high fixed, operating, and damage costs.
3. Description of the problem OPA 90 [8] calls for renewed actions to prevent the occurrence of oil spills, to minimize their impact, and improve the efficiency of cleanup efforts. As a direct result of OPA 90, a lot of emphasis has been placed on oil spill prevention. Once an oil spill occurs it is necessary to respond with sufficient cleanup equipment within the shortest possible time in order to protect marine species and minimize cleanup and damage costs. The complex hierarchical nature of the oil spill cleanup problem has resulted in a lack of integrative approaches in response strategies. We intend to address the problem without resorting to solving the strategic problem independently of the tactical problem. We believe that it would be of utmost importance and benefit to the strategic planner of a responding agency to be able to evaluate the merit of alternative high level decisions while taking into account their impact on post oil spill decisions. Given the complexity of the problem under study, these "what-if" analyses are very popular among the planning and responding agencies for two reasons: (1) they can lead to a better understanding of the problem structure (especially when used on a sequential fashion as more information becomes available), and (2) they can be used for the training of planners/responders. We specifically consider the composition of the optimal equipment mixture that should be dispatched from different facility sites to respond to an oil spill. Equipment sets are not predetermined and decisions regarding location of the equipment are made based on the "demand" (imposed by oil spills) that has to be met by the facility site. This demand is forecasted using historical data. As soon as the responding
agency is notified of an oil spill, the available equipment are grouped together and are dispatched from the facility sites which can respond within the critical time. This eliminates the unnecessary storage of equipment with a small probability of usage, at multiple sites. The decisions regarding the optimal location, number, and type of equipment to be stored, are made based on the decisions regarding the optimal allocation of equipment to respond to an oil spill. We study the problem as a deterministic one, since the decisions are made based on the expected volume, type, and weather conditions which are obtained from historical data. Such data are available from various databases of United States' federal and state agencies (e.g. data provided by the Office of Marine Safety, Security, and Environmental Protection of the USCG). The data consist of the frequency, volume, type, location, weather conditions, and sea-state of an oil spill event. The total area under study is partitioned into grids. Statistical analysis is performed on historical data to determine the expected volume, type, and weather and sea conditions for each grid. Using statistical averages the expected demand point for each grid is obtained by analyzing locations and frequencies of oil spills in the past. The above analysis is performed to determine certain input parameters such as the number and type of equipment required to respond to a given spill and the expected travel times for transporting the equipment from a facility site to the demand point (spill site). It should be noted that the travel time for a piece of equipment depends on the distance between the facility site and the spill site, the type of equipment, and on the weather and sea conditions. It is assumed that the fixed costs of opening a facility site and the operating costs of the equipment are known. The fixed costs include the cost of building a facility site and the cost of acquiring the equipment to be located at the site. These costs depend on the location of the site and vary with the geographical location. The operating costs include the cost of maintaining the equipment and the overhead costs at a facility site. The transportation costs take into account implicitly the cleanup costs and the damage costs. The cleanup costs constitute the costs involved in moving the cleanup equipment (travel costs) from a facility site(s) to the demand point, the
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
labor costs, and the overhead. Typically the cleanup costs, as categorized by the U.S. Department of Transportation, consist of: 1. 2. 3. 4. 5.
basic response costs, pollution threat costs, evacuation costs, bridge closing costs, and the costs due to injury or loss of life of a crew member.
The damage costs include, among others, costs due to vessel damage, human deaths, human injuries, cargo damage, navigational aid damage, bridge damage, environmental/marine life loss, and damages to recreational activities and beaches. Another important input parameter is critical time. The Presidential directive passed by Jimmy Carter in 1977 states that the USCG should have sufficient capability to respond adequately to a 100,000 gallons within six hours. There have been various interpretations of the activities that should take place within these six hours (e.g. to reach the spill site or to be able to start cleanup operations within six hours). The directive simply specifies the maximum time within which the spill should be tackled. The critical times for each grid and spill may be different based on the volume, type, location, and weather conditions and are determined by the responder. The responder's position is quite delicate while deciding on the critical times for each grid [5]. The responder has to protect the sensitive marine areas (for which the marine ecologists and the environmental organizations are very concerned), while at the same time he/she should also prevent the oil from impacting the shoreline (an issue of outmost importance for the tourism industry). Based on the above, it is clear that the most important performance measure in the oil spill cleanup effort is the response time since the success and efficiency of all subsequent cleanup efforts rely heavily on it. It is also assumed that all events occur at discrete points in time and that the probability of simultaneous occurrence of two or more spills is negligible. Furthermore, no equipment is subcontracted and all required equipment can be dispatched from USCG facility sites.
75
4. Problem formulation and solution
4.1. Notation The following notation will be used throughout our discussion:
I J K S f~ cij
Finite set of facility sites. Finite set of equipment types. Finite set of spill sites. Finite set of spill types. Fixed cost of opening a facility site i. Operating cost of storing an equipment of type j at facility site i. kj Unit cost/unit time for equipment type j. t,.jk Travel time for equipment type j to be transported from facility site i to spill site k. xi~ Number of type j equipment located at facility site i. zijks Number of type j equipment to be dispatched from facility site i to spill site k for spill type s. y; = 1 if facility site i is opened, 0 otherwise. ~tjks Number of type j equipment required at spill site k to cover a spill of type s. T Critical time to respond to a spill. Fjk = {i : t~jk < T}; the set of all facility sites for which equipment type j can respond to an oil spill at spill site k within the critical time.
4.2. Mathematical formulation We present a formulation that addresses strategic decisions taking into consideration post oil spill event implications. A linear integer programming formulation is developed that decides on the optimum number of facility sites to be opened, the optimum quantities, the type of equipment to be located at a facility site, and the optimal equipment dispatching policy after the occurrence of the spill is made known. The objective function is comprised of the fixed costs of opening a facility site, the operating costs of storing the equipment at facility sites, and the trans-
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
76
portation costs of transporting the equipment from a facility site to the demand point to respond to a spill (which capture cleanup and damage costs, as discussed in Section 3). The constraints address feasibility, capacity, and critical time issues. The linear integer programming formulation for the oil spill cleanup problem is as follows: min E f i Y i -P E E C i j X i j q- E E E EkjtijkZijks
i
i
i
i
j
k
s (1)
subject to: xij~MijYi,
Vi~l,j~J,
~_, Zijks>ct#s,
VjEJ, k~K,s~S,
(2) (3)
i~Fjk
Zi#
Vi~I,j~J,k~K,s~S, integers,V j ~ J , k ~ K , s ~ S , Vi~l.
(4)
method based on a relaxation of the original problem.
4.3. The relaxed problem Even though the number of integer variables for a realistically sized problem is prohibitively large, the only structural integer variables are the y/'s. Since the number of yi's, III, is typically small, it is relatively easy to solve, using branch and bound, the mixed-integer relaxed problem where only the y;'s are assumed binary (with the integrality requirement of constraint (5) being relaxed). In fact, since the relaxed problem is "transportation-like" its solution in practice is likely to be integral or near-integral (only a few variables are fractional). In the latter case the fractional solution can be rounded to obtain a near-optimal solution.
(5) (6)
Constraints (2) allows the storage of equipment of type j at a facility site i only if the facility site i is opened, where M~j is a known upper bound for xij. Constraints (3) ensures that any demand for equipment type j for spill type s at demand point k has to be covered within the critical time from facility site i. Constraint (4) ensures that the maximum number of equipment of type j for a spill of type s to be dispatched from a facility site i to the demand point k cannot exceed the available capacity of the facility site. Finally, x o. and z;jks are positive integers and Yi is a binary variable. It should be noted that even for small values of III, IJI, Igl, and IsI, the complexity of the problem is very significant. The formulation consists of (1II x IJI X Igl X ISI + III X IJI) general integer variables and III binary variables. Hence, for an example case with I/I = 30, IJI = 100, IKI = 20 and IsI = 10, the number of the general integer variables is 603,000. The number of constraints is (111X IJI + IJI × Igl X ISI + I11X IJI X IKI X ISI). The relaxed problem obtained by fixing the values of Zqk~ is the uncapacitated facility location problem, known to be NP hard [6]. No standard algorithm which can solve such a problem in polynomial time is available in the literature. In the next section, we propose a solution
5. Numerical examples In this section, we present two numerical exampies applied to the Florida peninsula. The first data set consists of six facility sites, nine equipment types, ten grids (demand points), and one type of spill at each demand site. Classification of the data is based on the type of oil spilled, volume of oil spilled, the weather and sea conditions. The second data set differs from the first only by a finer grid size; there are thirty (instead of ten) grids. Oil types are classified based on various physical and chemical properties of the oil. This is necessary because different types of equipment are used to respond to spills of different oil types. Oil is classified into: 1. Light Crude Oil. 2. Heavy Crude Oil. 3. Diesel. The oil spills are categorized into four types based on the volume and type of oil spilled. The classification is based on the capability of the equipment required for the cleanup operations and is as follows: 1. 0-15,000 gallons. 2. 15,000-50,000 gallons.
E. lakovou et a l . / European Journal o f Operational Research 96 (1996) 72-80
Table 1 Expected number and type of equipment
(0/#) required for
77
IKI = 10 grids
Grid number
Equipment Type
J=l ./=2 3=3 ./=4 ./=5 J=6 =7 ,=8 '=9
k=l
k=2
k=3
k=4
k=5
k=6
k=7
k=8
k=9
k=10
3 3 1 4 2 1 4 5 2
5 1 4 3 1 4 2 6 3
2 1 3 3 3 1 0 2 3
1 4 1 5 3 4 2 4 5
5 1 4 2 2 4 2 1 4
3 1 2 3 2 1 1 5 3
1 4 5 4 2 2 4 2 3
3 5 1 4 4 2 4 5 2
2 5 0 4 1 4 2 6 3
2 3 4 2 1 1 4 2 3
3. 50,000-100,000 gallons. 4. 100,000-300,000 gallons.
on the maximum operating significant wave height (this is the standard classification used by the U.S. Coast Guard):
Different types of equipment are required for different weather/sea conditions. The weather/sea conditions are also classified into three categories based Table 2 Expected number and type of equipment required for [KI = 30 grids 0 / t . t - ott.lo ai. ll--al.20 0/1.2t- al.3o Or2,1-- 0~2,10 0/2.11--O/2.20 0/2,2~- 0/2,30 0/3.1 -- O/3.10 Or3,11--/~3,20 0/3,2]- 0/3.30 ~4.1- 0/4.1o oq.i ] - ot4,2o 0/4.21 0t4.30 0/5.]-- aS. j0 0/5, I l - °t5,2o O/5.21 O~5.30 0/6.1 -- 0/6.10 0/6, I 1 016.20 0/6.21 a6.30 0/7,1--0/7.10 0 / 7 : t - ot7.2o 0/7.21 a7.30 ors.i- as.J0 o/s. 1t - crs.20
0 l 0 0 2 0 l 1 2 2 1 2 1 1 0 1 0 0 0 0 0 2 1
ors.23- ots,3o 0/9. i - o/9.1o 0/9. I I - 0/9.20 °/9.21 0/9.30
-
-
-
-
-
-
-
-
- -
-
-
2 0 1 1 2 1 0 0 0 0 2 1 0 0 2 0 2 2 2 0 2 2 1
1 2 2 2 0 2 0 0 1 2 0 I 1 0 1 0 2 0 2 1 2 1 0
2
0
0
l 2 0
0 1 2
1 0 0
I l 0 1 0 2 0 2 0 0 0 2 0 0 1 1 0 0 2 0 0 2 0
2 2 0 0 l 1 2 2 0 1 2 0 1 2 1 1 2 2 0 1 1 2 1
2 0 0 0 0 2 2 2 0 2 0 2 0 2 0 2 0 2 0 I 0 2 2
0 1 2 0 0 2 0 0 0 2 1 2 1 0 0 0 0 0 0 0 1 1 1
2
2
2
2
2 2 0
0 2 1
1 2 2
2 1 0
1 2 0 0 1 2 1 0 1 1 2 0 0 0 0 1 1 1 0 0 2 0 2
1 1 1 1 2 1 2 1 2 0 0 0 2 0 0 0 1 0 0 2 2 1 2
0 0 1 0 2 0 0 2 1 2 2 2 2 2 1 2 1 0 2 2 0 2 1
1
1
0
1 0 2
0 2 1
2 1 0
1. Calm water. 2. Harbor. 3. Offshore.
The maximum operating significant wave height for calm water, harbor, and offshore is 0.3 m, 0.9 m, and 1.8 m respectively. These three categories apply to bays, soft bottom subtidals, seagrass beds, rocky subtidal, intertidal, sandy beaches, coral reefs, mangrove forests, and salt marshes. The oil-spill cleanup equipment types are: 1. 2. 3. 4.
Booms. Skimmers. Chemical dispersants. Sorbents.
5. Earth barriers.
6. 7. 8. 9.
Pumps. Shoreline protection. Storage equipment. In situ burning equipment.
The demand for the first example is given in Table 1. Demand is based on the average calculaTable 3 Fixed cost at each of the facility sites i f,.
I 30,676
2 40,338
3 22,456
4 10,631
5 14,394
6 15,744
78
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
Table 4 Unit cost/unit time for each equipment type j k)
1 35
2 40
3 55
4 40
5 25
6 50
Table 5 Operating costs at each of the six facilities for the nine equipment 7 35
8 50
types
9 60
C=
tions using historical data. Each equipment type is mentioned by its corresponding index as provided above; for example, equipment type 2 is a skimmer.
344 429 454 316 414 461
393 393 300 422 352 411
429 374 461 394 352 489
363 396 275 376 430 427
320 411 363 373 455 461
334 383 317 459 303 378
339 388 497 386 496 476
I[~.i1~1ol414121114I~1KII Daytona
Booth
Ilxs~121al s 4 l a l 4 1 2 1 6 1 0 l l Plerc~ "z-
IIx,~151513121410141614[ ~o~
Palm
Beach
II~Jlal~1415 114141513 ~ Por~ Por~
Fig. 1.
Kverg o~
Miom
ode~
402 348 434 444 324 337
371 323 304 447 476 493
E. lakovou et al. / European Journal of Operational Research 96 (1996) 72-80
The entries of the table are the ajks'S, the required number of type j equipment required at spill site k to contain a spill of type s. The demand for the second example (with the finer grid of Ik[ = 30) is given in Table 2. Table 3 provides the fixed costs f~ for each of the six facilities, while Table 4 exhibits the operational costs for the various types of equipment, c u. Table 5 displays the unit cost/unit time for equipment type j (k j). Table 2, Tables 3 and 4 apply to both examples; namely, the corresponding costs are the same.
It is only the scale of the spill grid sizes that has changed. Fig. 1 displays the schematics of the locations of the potential facilities and the demand sites for the first example as they are applied to the east coast of Florida. The underlying map is the product of ARCINFO (one of the most popular Geographical Informations Systems). Fig. 2 displays the schematics for the second example. In this case each grid of the first example is now divided into three subgrids. Hence the demand requirements for the grids of the
II.~., t 1121 i 121 o 121 ~- 12 I ~- II Do~"t
~r~a
79
E~,gach
I1~Jl2121212121112121211 -I
For
Piorc@
11~,~12121212121212{2t2l[[ West
Palm
Beach
{1~12i21212121212121211 Por~ P o r ~ o#
": :: :.<,.~
Fig. 2.
Ev~rgladQo M,am,
80
E. lakovou et a l . / European Journal of Operational Research 96 (1996) 72-80
two examples are related. More specifically, the demand for each grid of the first example is the sum of the demands of its subgrids from the second case; e.g. a H (of Table 1) = all + alE + a13 (of Table 2) = 3, ¢g12 (of Table 1) = al4 + a15 + ¢g16 (of Table 2) = 5, etc. The mixed LP relaxation is solved using the CPLEX optimizer (Version 3.0). For the first numerical example, the solution opens four facilities, Y3 = Y4 = Y5 =Y6 = 1, (open facilities depicted with an open rectangle in Fig. 1 as opposed to stars for unopened facility sites). The number and type of equipment (x~j) that should be stockpiled at each of these four facilities are also given in Fig. 1 adjacent to each opened facility. For the second numerical example CPLEX suggests opening the same four facilities as in the first example; Y3 = Y4 = Y5 = Y6 = 1 (Fig. 2). However, the x i / s of the solution are now different and are also provided in Fig. 2.
6. Conclusion We have presented an integrative approach that addresses strategic decisions regarding optimum location and capacity of oil spill cleanup equipment, jointly taking into consideration their impact on tactical decision-making. A linear integer programming model was first developed and a mixed-integer relaxation of the original problem is solved. Two realistic examples as applied to the east coast of Florida are presented. Further research needs to be done to test the validity of such models given the lack of consistency and standardization among the databases provided by the plethora of federal and state agencies in the United States.
Acknowledgements The assistance of Lalit Yudhbir in the numerical experimentation is acknowledged. The authors would also like to thank the anonymous referees for their constructive comments.
References [1] Belardo, S., Harrald, J., Wallace, W.A., and Ward, J., " A partial covering approach to siting response resources for major maritime oil spills", Management Science 30 (19841)) 1184-1196. [2] Charnes, A., Cooper, W.W., Karwan, K.R., and Wallace, W.A., " A Chance-constrained goal programming model to evaluate response resources for marine pollution disasters", Journal of Environment Economics and Management 6 (1979) 244-274. [3] Church, R., and ReVelle, C., "The maximal covering location problem", Papers Regional Sci. Appl. 32 (1974) 101118. [4] Daskin, Marks S., and Stern, Edmund H., "'A hierarchical objective set covering model for emergency medical service vehicle deployment", Transportation Science 15 (1981) 137-152. [5] Demis, D.J., "Oil spill management: The damage assessment model and the spatial allocation of cleanup equipment", Unpublished M.S. Thesis, MIT Department of Ocean Engineering, June 1984. [6] Erlenkotter, D., " A dual-based procedure for uncapacitated facility location' ', Operations Research 26 (1978) 992-1009. [7] Moore, George C., and ReVelle, Charles, "The hierarchical service location problem", Management Science 28 (1982) 775-780. [8] Oil Pollution Act, 33 United State Congress, Section 2071 et seq; Pub.L.No. 101-380, 104 Stat. 484. [9] Psamftis, H.N., and Ziogas, B.O., " A tactical decision algorithm for the optimal dispatching of oil spill cleanup equipment", Management Science 31 (1985) 1475-1491. [10] Psamftis, H.N., Tharakan, G.G., and Ceder, A., "Optimal response to oil spills: The strategic decision case", Operations Research 34 (1986) 203-217.