A PROCEDURE FOR OPTIMIZING TACTICAL RESPONSE IN OIL SPILL CLEAN UP OPERATIONS Anand V. Srinivasa i2 Technologies 909 East Las Colinas Blvd., 16th Floor Irving, TX 75039 and Wilbert E. Wilhelm Department of Industrial Engineering Texas A & M University College Station, TX 77843-3131 Phone: (409) 845 - 5493 Fax: (409) 847 - 9005 email: [email protected]May 2, 1995 Revised: June 13, 1996 The material in this paper should not be disseminated without the written permission of the authors.
Transcript
A PROCEDURE FOR OPTIMIZING TACTICAL RESPONSE
IN
OIL SPILL CLEAN UP OPERATIONS
Anand V. Srinivasai2 Technologies
909 East Las Colinas Blvd., 16th FloorIrving, TX 75039
and
Wilbert E. WilhelmDepartment of Industrial Engineering
Texas A & M UniversityCollege Station, TX 77843-3131
The material in this paper should not be disseminated without the written permission of the authors.
ABSTRACT
The Tactical Decision Problem (TDP) associated with oil spill clean up operations allocates
available components to compose response systems so that the clean up requirement for each time
period over the entire planning horizon is met. The objective is to minimize total response time to
allow for the most effective clean up possible. The TDP is formulated as a general integer program, a
problem that is difficult to solve due to its combinatorial nature. In this paper, we develop an
optimization procedure that is based on an aggregation scheme and strong cutting plane methods. The
solution of the resulting Aggregated Tactical Model is used in reformulating the TDP, in generating a
family of facets for the TDP, and in several pre-processing methods. Computational experience in also
reported in application to a realistic scenario representing the Galveston Bay Area.
ACKNOWLEDGEMENTS
This material is based on work supported by the Texas Advanced Technology Program under
Grant Number 999903-282. We are indebted to a number of individuals and organizations who have
helped us assure the relevancy of this research. In particular, we acknowledge Commander John
Salvesen, Port Operations Chief, and Lieutenant C. David Weimer, both of the U.S. Coast Guard
Marine Safety Office in Galveston, Texas, for their interest in this work and for making it possible to
gather data describing an application in the Galveston Bay Area. We would also like to express our
appreciation to Mr. Tim McKenna, Director of the Oil Spill Prevention and Response Program, and
Mr. Ronald Brinkley, Regional Manager of Oil Spill Prevention and Response, who are both with the
Texas General Land Office. In addition, this research benefitted from discussions with a number of
individuals, including Mr. Theo Camlin, Program Coordinator: Texas A&M Oil Spill Program at
Galveston, Dr. Bela M. James, Environmental Specialist: Shell Oil Company, and Mr. Raymond G.
Meyer, Operations Manager: Clean Channel Association. Foremost, however, has been the advice and
experience offered by Dr. Richard A. Geyer of the Offshore Technology Research Program at Texas
A&M University. We acknowledge the able efforts of Dr. Sangho Joo, who helped to gather and
formulate the data used in our test problems. Finally, we acknowledge the comments of two
anonymous referees whose comments allowed us to strengthen an earlier version of this paper.
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1 INTRODUCTION
The number of catastrophic oil spills in recent years has demonstrated the need for effective
response management. Among these were a number of widely publicized tanker spills including the
1989 Exxon Valdez in Prince William Sound, Alaska; the 1991 Megaborg in the Galveston Bay Area;
the 1992 La Coruna spill off the Spanish coast; and the spill near the Shetland Islands off the coast of
Ireland in 1993. Indeed, 1991 and 1992 were the two worst years for oil spills since 1983. Table 1
gives a partial list of major oil spill disasters in recent years. In addition, spills from tanker loading and
unloading operations, pipeline ruptures and other sources pose serious threats to the environment,
including fisheries and wildlife preserves. This experience has generated a heightened awareness and
concern about the risks associated with oil spills. While preventive measures may reduce the frequency
of spills, it is impossible to avoid all accidents. Thus, effective oil spill response capability is mandatory.
----------------------------------------------Table 1: A List of Major Oil Spill Disasters.
----------------------------------------------Oil spill response planning prescribes actions that will be performed under emergency
conditions and attempts to minimize damage to the ecology and to the quality of human life. Spill
response occurs within a complex environment that requires time-phased deployment and must deal
with legal constraints and the interests of various political entities. The Oil Pollution Act (OPA) of
1990 designates the Department of Transportation as the lead government agency for the United States
(U.S.), with the U.S. Coast Guard having authority to make final on-site decisions regarding the
acceptability of response. The party responsible for the spill is obligated by law to effect a clean up that
satisfies all requirements and meets with Coast Guard approval.
The objective of this paper is to present an optimizing approach for the Tactical Decision
Problem. In the remainder of the Introduction, we describe the problem setting and clean up
operations. A succinct statement of the Tactical Decision Problem follows this discussion. In addition,
we review relevant literature.
The systems approach to oil spill response identifies three levels of decision making - strategic,
tactical and operational. At the strategic level, resources (i.e., equipments, materials, and personnel)
must be pre-positioned to assure a timely response. Strategic planning involves determining locations
for storing resources and the quantities and types of resources to be stockpiled at each location so that
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adequate capability is provided to deal with the full range of oil spills that might occur over a specified
planning horizon. Decisions made at the strategic level thus impose constraints on those that must be
made at the tactical and operational levels.
The tactical level involves prescribing response systems for a specific oil spill that has occurred
and involves decisions such as which components to dispatch; what equipment systems to compose,
how many of each, and when. This paper addresses the tactical issues involved in oil spill response.
The operational level deals with effective clean up of an oil spill over time. Operational
decisions determine exactly how to utilize the equipment systems prescribed by the tactical level.
The tactical problem assumes that the strategic problem has been solved, since decisions at the
tactical level have to be predicated on equipments made available by the strategic plan. Thus, we
assume that the problem is deterministic and that an oil spill of known type and quantity has occurred
at a known location. Our formulation of the TDP implicitly considers the movement of oil over time as
it disperses in the water. We assume that cumulative response requirements are based on the volume
and rate of oil spilled at the site as well as clean up needs that result from the particular spill conditions
and spill trajectory. The tactical model can be invoked periodically to compensate for unexpected
changes or to take into consideration any improved estimates spill volume or conditions that may
become available.
Oil spill response is dictated by three factors: the type of oil (for example, heavy crude, light,
etc.), the amount, and the spill conditions (e.g., including temperature; prevailing wind and weather
conditions, which affect wave height and current direction; and proximity to ecologically sensitive
areas). To respond to a given spill, specialized clean up equipment must be deployed in order to
contain and recover the oil. Four common methods are used in clean up operations: (1) mechanical
systems, (2) chemical dispersants, (3) burning, and (4) bioremediation. Often, depending on the
prevailing spill conditions, a combination of these methods must be used to ensure effective clean up.
A variety of components are available for use: for example, containment boom, which helps
control the spread of oil, skimmers, which recover oil; and barges and vacuum ("vac") trucks, which
transport the recovered oil to disposal sites. However, components by themselves have no clean up
capability. Components must be combined to form an equipment system that does offer clean up
capability. For example, an integrated skimming system could consist of a length of, say, 4000 feet of
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boom, two pumps, a skimmer, a barge and ancillary equipment, personnel and supplies. System clean
up capability, measured in gallons per hour, depends upon factors such as the type of oil spilled and
other spill conditions. In addition, the clean up capability of a system may degrade over time, due, for
example, to oil changing consistency as it ages in water.
Timing is of critical importance in achieving effective clean up. Floating oil spreads rapidly, so
a slow response may allow oil to spread over a large area so that booms could not be effective in
containing it, and the slick would be too thin to permit burning or skimming. Furthermore, floating oil
emulsifies as it mixes with water, forming a chocolate-colored mousse that cannot be treated effectively
with dispersants. Thus, when off shore responses are delayed, as in the case of Exxon Valdez oil spill,
they are likely to prove ineffective. In addition, spills close to shore may quickly reach recreational
beaches, fisheries or wildlife preserves. Thus, timely mobilization and co-ordination of components to
compose response systems is vital so that required response capability can be deployed in time.
A variety of oil spill cleanup components owned by companies, co-operative organizations,
government bodies, or contractors are stored at known sites for dispatching to a staging area where a
set of components can be assembled to form a response system. A response system is an equipment
system that is defined more specifically to include the particular locations where each of its constituent
components is stored and the staging area at which the system is composed.
The Tactical Decision Problem is to prescribe the types of response systems to be deployed in
each time period so that, collectively, they meet the clean up requirements. The type and number (or
amount) of each component used in each system, the location at which each component is stored, and
the staging area where each system is composed must be prescribed. The clean up capability,
measured as a gallons-per-hour rate that must be on scene by each time point, is based on the type of
oil, the spill discharge rate and the spill condition as legislated by OPA 90. Figure 1 depicts one
scenario of cumulative clean up requirements at time points t = 1, .. 5. As shown in the figure, the tth
interval from the start of a spill is the duration from time point (t - 1) to time point t. Figure 1 also
shows the major events related to a spill: start of spill, spill notification, end of spill, and end of clean up
activities. A variety of objective functions could be used, but we consider minimizing response time,
since such a solution would expedite deployment to ameliorate environmental impact as much as
possible.
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--------------------Figure 1 here
--------------------To address this complex problem, prior research has focused on developing strategic and
tactical management capabilities with the goal of effective oil spill clean up operations. Psaraftis et al.
(1986) developed a strategic model that prescribes storage locations for clean up equipment,
accounting for the frequency at which oil spills occur and different possible spill conditions. Their
model minimizes total cost, consisting of the fixed costs related to opening warehouses and acquiring
equipment, and the estimated cost of damage as a function of spill volume and response level. Charnes
et al. (1976) developed a chance constrained goal programming method to assist the U.S. Coast Guard
in formulating response plans for major oil spill disasters. However, their model attempts to
simultaneously consider strategic and lower level decisions, thereby limiting the model to small
problems.
Previous quantitative approaches to prescribe tactical response have been rather limited.
Psaraftis and Ziogas (1985) developed a model for allocating individual components, minimizing a
weighted combination of spill-specific response cost and estimated damage cost. Inputs to their model
include information about the oil discharge, availability and performance of cleanup "equipment sets",
and costs of transporting "equipment sets" and on-scene operation. While the Psaraftis and Ziogas
model has merit, it does not deal with the current legal requirements for oil spill response. Also, while
minimizing damage cost may be a valid goal, it is difficult to quantify damage cost, and requirements
invoked by recent laws give top priority to timely response to assure stipulated response capabilities at
all time points. Furthermore, their model assumes that each "equipment set" is stored at a single
location and does not prescribe how to compose sets by combining components stored at different
locations. This necessitates preassigning each component to one and only one "set", thereby reducing
flexibility in the overall decision making process.
Wilhelm and Srinivasa (1994) formulated a general integer programming model for the Tactical
Problem to address these issues. They used graph theory to develop a column generation scheme for
defining response systems. The column generator defines each response system, including constituent
components, the location where each component is stored, and the staging area in which that system is
composed. They also developed two efficient heuristics to obtain approximate solutions to the TDP.
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The objective of this paper is to develop an exact optimization procedure for the TDP
associated with medium-to-large oil spills (we expect that small spills can be handled easily and need
not entail systemic response). The procedure is based on strong cutting plane methodology. While the
strong cutting plane approach has most often been applied to 0-1 programming problems (Vanderbeck
and Wolsey 1992 is one recent exception), we hope to gain further insight by solving a problem
involving general integers.
The rest of this paper is organized as follows. The next section introduces our notation and the
general integer programming formulation for the TDP. Section 3 describes the aggregation scheme
and some important properties of the resulting Aggregated Tactical Model (ATM). Section 4
describes several preprocessing procedures that are used to facilitate the solution of the TDP. In
section 5 we describe a key family of facets and the optimization approach. Section 6 discusses
computational evaluation on several different test problems that are based on a realistic application in
the Galveston Bay Area. Section 7 concludes the paper. Appendix A presents all proofs, and
Appendix B gives a numerical example to demonstrate our aggregation scheme.
2 THE TACTICAL DECISION PROBLEM
In this section, the Tactical Decision Problem is formulated as a general integer program (see
also Wilhelm and Srinivasa 1994).
2.1 Notation
We first introduce the notation used in the model.
Decision variables:xtq number of response systems of type q deployed on scene in period t
Parameters:Aem number (or amount) of components of type e available at location mbq area (square feet) needed to compose a response systems of type q
Bjt area (square feet) of staging area j in period tCitq clean up capability (in a gallons/hour rate) of response system type q at time point t if deployed
at time point i (i ≤ t)gjq duration required to compose response system q in staging area jNemq number of components type e from location m used to compose response system type qtq earliest time period in which response system type q is available for deploymentrq total elapsed time required to deploy response system type q
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?t minimum clean up response requirement (in gallons/hour) at critical time point t
Sets:E set of all componentsJ set of all staging areasM set of all storage locationsR(t) set of all response systems that can respond by critical time point tT set of all critical time points? set of all equipment systemsQ(p) set of all response systems that incorporate equipment system type p
Indices:e component type e e Ei,t critical time points i,t e Tj staging area j e Jm equipment storage location m e Mq response system type q e Qp equipment system type p e ?
2.2 Model
The Tactical Decision Problem may be formulated as:
Min q tqt T q R(t)
Z = xrε ε∑ ∑ 1 (1)
subject to
q
t
i t q i q tq R(t) i = t
C xε
γ≥∑ ∑ 2 t e T; (2)
e me m q t qt T q Q (e m t)
N x Aε ′∈
≤∑ ∑ 3 e e E; m e M; (3)
j q
t
jq i qq G(j, t) i = t - g
tb x B∈
≤∑ ∑ 4 j e J; t e T''; (4)
t qt T q Q( )
x uππ∈ ∈
≤∑ ∑ 5 p e ?; (5)
t q qt T
x uε
≤∑ 6 q e Q; (6)
t q t q t q , integerx µλ ≤ ≤ 7 t e T; q e R(t); (7)Equation (1) states the objective, which is to minimize total response time. Inequality (2)
incorporates the degradation of response system capability over time and assures that the cumulative
clean up requirement at any critical time point t e T will be satisfied. Constraint (3) assures that
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prescribed response systems utilize no more than the number of components of type e that are available
at location m. In this constraint, set Q'(emt) denotes the set of all response systems that could respond
by critical time point t and use component type e stored at location m. Inequality (4) invokes the
capacity limitations (in terms of available space) at each staging area. Set T'' represents the set of time
points at which the response systems can be staged, and set G(j,t) represents the response systems that
could be composed in staging area j during the time interval t. Constraints (5) and (6) represent
generalized upper bound constraints on equipment system and response system types, respectively.
Constraint (7) requires that decision variables be non-negative, bounded integers. Initially, all ?tq are
assumed to be zero. We use sets in the formulation to present a succinct model.
The Tactical Decision Problem is a general, all-integer programming problem. Its structure is
such that both "≤" and "≥" types of constraints are present. Thus, lower bounds and feasible solutions
cannot be guaranteed by a straightforward rounding of the solution to the Linear Programming (LP)
relaxation.
3 AN AGGREGATION SCHEME
Aggregation and disaggregation techniques offer an attractive potential for solving large scale
optimization models (Rogers et al. 1991). According to Rogers et al., aggregation techniques for
solving problems in optimization consists of the following steps: 1) combining data; 2) using an
aggregated model that is reduced in size and/or complexity with respect to the original problem; and,
3) analyzing the results of the aggregated model in terms of the original model. The key issue is to
devise an aggregation scheme that provides a convenient approximation to the original problem. This
paper describes, to the best of our knowledge, the first scheme that aggregates both columns and rows.
The resulting aggregated problem retains certain critical properties of the original one and exploits
them in solving the original problem.
Aggregation and disaggregation schemes have been successfully employed to solve large
manpower planning models (Kao and Queyranne 1986), part family and machine group identification
problems in cellular manufacturing (Wemmerlov and Hyer 1986), and large-scale linear and mixed
integer programming models in forestry (Barros and Weintraub 1982). A variety of other applications
exist as well. Special structures that have been studied include network flow problems (Zipkin 1980a)
and the generalized transportation problem (Evans 1979). In terms of general theory, Zipkin (1980b)
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explores the effects of aggregating variables in large LPs. Zipkin (1980c) also describes the effect of
row aggregation in LPs.
Three primary reasons motivate developing an aggregate model for solving the TDP. First, the
aggregation technique results in a substantially simpler problem to solve (a 75-85% decrease in number
of variables can be achieved) and hence reduces the computational effort dramatically. Second, the
aggregated tactical model provides invaluable insight into determining the underlying polyhedral
structure of the TDP. Finally, the Aggregated Tactical Model (ATM) can be used to generate facets
for the original problem and to devise pre-processing methods.
Aggregation in integer programming has been primarily confined to the theory of surrogate
constraints, where constraints of the original problem are aggregated to form one "surrogate"
constraint. While there has been considerable advancement in the use of surrogate constraints to solve
linear programming problems, prior empirical experience with constraint aggregation for integer
programming problems has not been promising (Onyekwelu 1986). Hence, constraint aggregation
alone is not sufficient, especially in LP-based combinatorial problem solving where "tight"
representations of the LP relaxation are desired. Hallefjord and Storoy (1990) consider column
aggregation of 0/1 programming problems. However, their aggregation-disaggregation scheme is a
non-iterative procedure in the sense that the given problem is aggregated, solved, and the solution is
desegregated.
In this section, we develop a framework for aggregating the TDP and describe the resulting
Aggregated Tactical Model (ATM). Our procedure involves aggregating both columns and rows of
the original model. The ATM has the advantage of retaining some of the crucial characteristics of the
original problem, while requiring far fewer variables. The ATM is used in the preprocessing and
optimization procedures described in sections 4 and 5. An iterative technique for solving the ATM and
the LP relaxation of the TDP is used to obtain successive improvements in the lower bound for the
objective function value of the TDP and in tightening the original formulation for the TDP. The
remainder of this section describes the development of the ATM and discusses the properties of ATM
that can be exploited in obtaining a solution to the original problem.
In order to aggregate the columns of the TDP, the variables are partitioned and columns in
each group are replaced by a small subset of the variables in the ATM. The aggregation is given by the
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following definition.
Definition (1): Let s = {Qp: p=1, 2, ..., ?} be a partition of the column indices {1,2,...,n} such that
Qp denotes the set of all columns representing response systems that incorporate equipment system
type p. By definition,
Q = {1,2,...,n}ππ∈ΠU 8and
1 2 = Q Qπ π φ∩ 9, for all p1 ≠ p2.
Let tq(t T; q R(t))x ∈ ∈ 10represent the columns of the original problem, where R(t) is the set of all responsesystems that can respond by time point t. The aggregated columns, t (t T; ),X π π∈ ∈ Π 11are then given by thefunction:
t q t qt q Q( )
= ( ) = ; t T; x xX ππ
π∈
Φ ∈ ∈ Π∑ 12.
Here, the decision variables in the aggregated model, t (t T; )X π π∈ ∈ Π 13, represent the number of equipment
systems of type p that are deployed by time point t.
We can now represent the constraints of the TDP in ATM form. Since,
Citp = Citq for all q e Q(p), (8)
we can, using definition (1), write the set of constraints represented by (2) asi = t
i t i t (t) i = t
C Xπ
π ππ
γ∈Π
≥∑ ∑ 14 for all t e T; (9)
where tp represents the first time period for which equipment system type p becomes available for use
in the response. That, is tp = {t : p ∉ ?(t-1), and p e ?(t), t≥1}, where ?(t) is the set of equipment
systems that are available for deployment in time period t.
We now turn our attention to a row aggregation scheme for representing the component
availability constraints. Consider the constraints represented by (3). By aggregating over all locations
at which a component type is stored, we obtain:
e me m q t qt T m L(e) q Q(e m t) m L(e)
N x A∈ ∈ ∈ ∈
≤∑ ∑ ∑ ∑ 15 for all e e E;
where L(e) represents the set of all locations that store component type e. Using Definition (1), we can
write:
e me m t t T m L(e) (e) (t) m L(e)
N AXπ ππ∈ ∈ ∈ Π ∩ Π ∈
≤∑ ∑ ∑ ∑ 16 for all e e E; (10)
where ?(e) denotes the set of all equipment system types that use component type e.
In order to aggregate the staging area constraints, consider the constraints represented by (4).
By aggregating over all the time points (t e T), we get:
10
j tq t qt T q G(j) t T
b x B∈ ∈ ∈
≤∑ ∑ ∑ 17 for all j e J
where G(j) represents the set of all response systems that are composed at staging area j. From
Definition (1), we can write:
jt t T (j) t T
tb BXπ ππ∈ ∈ Π ∈
≤∑ ∑ ∑ 18 for all j e J (11)
where ?(j) represents the set of all equipment systems whose associated response systems can be
composed at staging area j.
Finally, the GUB type constraints given by (5) are incorporated using the aggregated variables
in a straightforward manner using Definition (1). We then have
t t T
UX ππ∈
≤∑ 19 for all p e ?. (12)
Constraints (6) of TDP dealing with the GUBs for response system types are ignored in the
aggregate model, since the aggregated model does not deal with response systems.
Now, we only need to define the objective function coefficients to complete description of the
ATM.
Definition (2): For p e ?, define min qq Q( )
= { }r rππ∈
20.
Definition 2 considers only one variable (the response system with minimum response time)
from the set of response systems that are based on a particular equipment system type. Use of these
coefficients guarantees that the solution of the ATM will provide a lower bound for the optimum value
of the original problem.
The ATM is then given by:(ATM): min t
t T
Z = r Xπ ππ∈ ∈Π
∑ ∑ 21
subject to (9) - (12) andtq 0,X ≥ 22integer.
A numerical example that demonstrates this aggregation scheme is presented in Appendix B.Properties of the ATM
Property (1): If DCS 23represents the set of all feasible solutions satisfying (2), and A
CS 24denotes the
set of all feasible solutions satisfying (9), then D AC C( ) = S SΦ 25where (x)Φ 26is the function given by Definition
(1).Property (1) says that a solution that is feasible with respect to constraints (2) in the TDP can
be recovered from a solution that is feasible with respect to (9) in the ATM. This is true, since (9) of
the ATM is obtained by aggregating columns only (and not the rows), which agrees with Zipkin's
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(1980c) statement that aggregating only columns allows the aggregated problem to yield a feasible
solution to the original problem.
Property (2): If AES 27is the set of all feasible solutions to (10), and D
ES 28is the set of all feasiblesolutions satisfying (3), then
D AE E ( )S S⊆ Φ 29.
Property (3): If AJS 30is the set of all feasible solutions to (11), and D
JS 31is the set of all feasiblesolutions satisfying (4), then
D AJ J ( )S S⊆ Φ 32.
Properties (1), (2) and (3) indicate that the ATM is a relaxation of the TDP due to the aggregation
of the component availability and the staging area constraints. Thus, Property (1) allows us to optimize
over the requirements polytope in the TDP (represented by constraints (2)) by optimizing over the
requirements polytope in the ATM (represented by (9)). The requirements polytope is the dominant
sub-structure in the TDP, since the TDP involves minimizing a linear objective function and only the
requirements constraints are the "≥" type. This is also reflected by the continuous solutions of our test
problems in which all constraints describing the requirements were active while less than 10% of the
component availability constraints (3), and none of the staging area constraints were active. Thus,
these properties of the ATM can be very useful in preprocessing procedures and in generating facets
for the TDP.
Property (4): If AIPZ 33is the optimum objective function value for the ATM, and if D
IPZ 34is theoptimum objective function value for the TDP, then
A DIP IP .Z Z≤ 35
However, note that nothing can said about the relationship between AIPZ 36and D
LPZ 37, the optimum
objective function value for the LP relaxation of the TDP.
Physical Interpretation
The ATM represents an 'equipment system view' of the tactical planning problem while the
TDP represents a 'response system view'. The difference is due to the distinction between an
equipment system and a response system. Recall that the former is completely defined by the type and
number of its constituent components, while the latter includes, in addition, the locations that store the
constituent components and the staging area where the system is composed. The iterative method
used in the preprocessing and optimization procedures reflects the attempt to 'reconcile' the allocation
of components in the TDP with the clean up requirements of the ATM.
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Optimization of the ATM
In this section, we describe a branch and bound based procedure for optimizing the ATM. The
branch and bound algorithm exploits a special sub-structure in the ATM formulation. A branching
strategy that exploits this underlying structure can have a significant impact on the performance of the
algorithm. Development of a good branching strategy involves knowledge of the variables that will
have a major impact on feasibility. In the rest of the section, we focus on the requirement constraints
and identify variables that impact feasibility significantly.
Consider the set of T requirement constraints in the ATM in which the decision maker must
prescribe response over a horizon of T time periods. Equipment systems from set ?(t), each with a
specified clean up capability, can be deployed in period t, combining with systems deployed in earlier
periods to satisfy the response requirement for period t. Any system deployed in period t is assumed to
stay on scene, contributing to clean up capability in periods t, ...,|T|. Because the clean up capability of
a system may degrade over time, technological coefficients for successive rows in the column
representing the deployment of a particular system in period t are related by
t - 2, t, t - 1, t, t, t, ... C C Cπ π π≤ ≤ 38. Thus, the sub-matrix corresponding to the requirement constraints has the
following time-staged structure
1 1 1
1 2 1 2 2 1 2 2 2 2 2 3
1 3 1 2 3 1 2 3 2 2 3 3 3 1 3 3 2 3 3 3 3 3 4
0 0 0 0 0 0 0C0 0 0 0C C C C
C C C C C C C C′
′′ ′ ′ ′
39 (13)
where a column has a zero element in the row for each time period for which this system cannot respond in time
(due to the remoteness of its storage location) and t, t, t - 1, t, t - 2, t, GE GE C C Cπ π π′ ′′ 40, representing the
degradation of system clean up capability over time.
In (13), we have assumed T = 3; ?= 4 and that ?(1) = {1}; ?(2) = {1, 2, 3}; ?(3) = {1,
2, 3, 4}. Thus, t1 = 1; t2 = 2; t3 = 2; and t4 = 3. Furthermore, due to the degradation of system
1, 1 1 1 2 2 1 3 3 1 1 2 1 2 3 1 1 3 1 = = = ;C C C C C C′ ′ ′′≥ ≥ 41for system 2, 2 2 2 3 3 2 2 3 2 = ;C C C ′≥ 42and for system
3, 2 2 3 3 3 3 2 3 3 = .C C C ′≥ 43It is clear from (13) that the constraint sub-matrix has a "lower triangular" form with
the equipment systems (i.e., columns) with entries at the top left of the matrix having the maximum opportunity to
impact feasibility (i.e., by affecting more time periods). Now, assume that equipment systems are sorted so that
those that are deployed for the first time in the same period are arranged in a non-increasing order of their clean
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up capabilities. That is, in (13) C222 ≥ C223 corresponding to time period 2.
This scheme of ordering the variables results in the variables at the top left of the "triangle"
having lower indices than the ones to their right or below them. Then, by assigning the index value of a
variable as its priority, we can assure that variables with lower indices have higher priorities in
determining the order in which branching is performed. Such a lexicographic ordering of the
equipment systems based on their time of availability and their clean up capabilities provides for a
convenient branching and significantly improves the performance of the branch and bound algorithm.
4 PREPROCESSING PROCEDURESConsider the TDP and denote by D
IP 44the set of all feasible solutions to the TDP:D nI = { x : x satisfies (2), (3), (4), (5), (6), (7)}P Z∈ 45.
The TDP involves prescribing * DI x P∈ 46such that ** = rx ζ 47where *ζ 48is given by
min* DI = { rx : x }Pζ ∈ 49 = the optimum objective function value of the TDP, provided a finite
optimum exists. Our preprocessing procedures seek to reformulate the TDP such that an equivalent set, D
IQ 50is
obtained. DIQ 51is said to be equivalent to D
IP 52if it satisfiesD D
II ,Q P⊆ 53and D *I x : r x = .Q ζ′ ′∋ ∈ 54
Note that DIQ 55contains a subset of the integer points in D
I ,P 56and may even omit some alternate optimal
solutions, if any exist (Hoffmann and Padberg(1991)).
Our preprocessing procedures developed to reformulate the TDP can be classified as (1) column
preprocessing, and (2) row preprocessing. In the former case, we develop procedures for obtaining
tighter bounds on both individual variables and subsets of variables, variable fixing, and variable
reduction. In the latter case, we seek a tighter representation of the TDP by strengthening the right
hand sides of the original constraints.
Bounds on Individual Variables and Subsets of Variables
Tighter upper bounds on individual variables can be obtained by solving a series of linear
programs given by:
ztq = max {xtq: x satisfies (2) - (6) and x≥0}, for t=1,..,T; q e Q(t),and then setting tqtq = u z 57.
In a similar manner, tighter lower and upper bounds for sums of subsets of variables (GLB and
GUB) can be obtained by solving the LPs given by:
14
min t qt T q Q( )
= { : x satisfies(2) - (6), x 0 }xzιπ
π∈ ∈≥∑ ∑ 58 pe? (14)
maxut q
t T q Q( )
= { : x satisfies(2) - (6), x 0 }xzππ∈ ∈
≥∑ ∑ 59 pe? (15)
Note that these expressions also define bounds for the aggregate variables,
since tq tt T q Q( ) t T
= ,x x ππ∈ ∈ ∈
∑ ∑ ∑ 60for all p e ?. Since each of the LPs can be solved in polynomial
time, the above procedures provide an efficient means of tightening the original LP formulation in
polynomial time.
Variable Fixing
Another preprocessing technique uses reduced costs to fix variables at their upper bounds. If
at the optimum solution to the LP relaxation of the TDP any variable has a negative reduced cost and is
at its upper bound, then that variable can be fixed at its upper bound (Crowder, Johnson and Padberg
1983). Thus, if *tq q = ,x u 61for any q, and RCtq is < 0 (where RCtq is the reduced cost of xtq),
then xtq = uq in all optimum solutions.
Variable Reduction
We first state a Proposition that allows us to reduce the number of variables associated with
response systems incorporating a particular equipment system type.
Proposition (1): Let E(p) represent the set of component types necessary to construct equipment system type pand *x 62represent the LP relaxation optimum. Then, if there exists a p' (p' e ? )such that E( ) E( ) = π π φ′∩ 63for any p e ? -{p'}, and if
1 2, Q( )q q π′∈ 64with1 2q q < r r 65then
(i) If1 1
*t q q0 < < ,x u 66for any t e T, then
2
*t qx 67can be fixed at zero and, in addition,
(ii) If2 2
*t q q0 < < ,x u 68for any t e T, then
1 1
*tq q
t T
= x u∈∑ 69.
Intuitively, Proposition (1) permits us to identify the best response systems (in terms of least response
time) that incorporate a particular equipment system type that does not share any of its constituent components
with other equipment system types. Proposition (1) gives rise to a corollary that formalizes a dominance property
that can be used to fix variables a priori.
Corollary (1): Consider the elements in set Q(p) in order1 2 |Q( )|{ , , , }q q q π′K 70such that
1 2 |Q( )|q q q ;r r r π′≤ ≤ ≤K 71then
jt q = 0x 72for all t e T, and j = s+1, ..., |Q(p')|, where s
satisfiesq = s
qq = 1
u uπ′≥∑ 73andq = s - 1
qq = 1
< u uπ′∑ 74.
15
Bounds on DIPZ 75
A lower bound on DIPZ 76is, of course, given by the value of the LP relaxation of the TDP, D
LPZ 77. Another
lower bound, AIPZ 78can be obtained by solving the ATM. The ATM (with integer restrictions) can be solved
efficiently using the branching scheme described in the previous section. By picking max D ALP IP { , }Z Z 79, we
obtain a tighter lower bound for the objective function value of the TDP.
Any feasible solution can be used as a valid upper bound on the objective function value for the TDP.
For example, the heuristics developed by Wilhelm and Srinivasa (1994) can be used to obtain tight upper
bounds.
Bounds on the Number of Equipment Systems
Since the TDP formulation permits one equipment system to substitute for another if both have
the same response times, it is important to have tight bounds on the number of equipment systems as
well. A good lower bound on the number of systems used is easily obtained by solving the LP given
by:
min t qLt T q Q(t)
= { : x satisfies (2) - (6), x 0 }xξ∈ ∈
≥∑ ∑ 80 (16)
and setting the bound on NL so that t q L Lt T q Q(t)
= .x N ξ∈ ∈
≤ ∑ ∑ 81
An upper bound on the number of systems that can be used is given by NU, where NU is such
that the LP defined by
min q t qt T q Q(t)
{ : x satisfies (2) - (6)xr∈ ∈∑ ∑ 82
t q Ut T q R(t)
+ 1, x 0 }x N∈ ∈
≥ ≥∑ ∑ 83 (17)
is either infeasible or has an objective function value greater than the best available upper bound.
Even though this method involves a trial and error procedure to determine NU, we found that
starting the "search" for NU with a good feasible solution typically requires solving the LP only two or
three times.
Based on the above discussion, the following inequality is valid:L t q U
t T q R(t)
N x N∈ ∈
≤ ≤∑ ∑ 84. (18)
Row Preprocessing
Once column preprocessing has been completed, the bounds (i.e., GUB, GLB and bounds on
16
the total number of equipment systems) are used in the ATM. Since the ATM and the TDP share the
same requirements constraint characteristics (e.g. Property (1)), these constraints can be tightened
further by solving the following integer programming problem for each:
AT(t)P 85 : min
t
i t t tt T i = 1
= { C Xπ ππ
γ∈ ∈ Π∑ ∑ ∑ 86
: X satisfies (9) - (12), X 0, integer}≥ 87.While this necessitates solving an IP, our experience has been that the computational effort is
not intensive due to the ease with which the ATM can be solved. For example, a typical TDP in our
test cases involves about 300 variables and 160 constraints while the corresponding ATM consists of
30 variables and 25 rows, so that this preprocessing routine typically takes less than two minutes. We
limit the use of this procedure so that if a two minute limit is exceeded, we forgo it.
The preprocessing methods described in this section were embedded in an iterative scheme that
alternately passes information between the original problem (the TDP) and the aggregated problem
(ATM), and can be used until no improvement in the solution of the original problem can be observed
(Rogers et al. 1991). Figure 2 describes the procedure.
---------------------------Figure 2 comes here
---------------------------5 OPTIMIZATION OF THE TACTICAL DECISION PROBLEM
In this section, we present an optimizing procedure for the TDP that uses the aggregation
scheme described in section 4. Before describing the approach, we define a useful property of the
clean up requirements vector.
Definition (3): The vector (of dimension t x 1; t = 1,2, ..., T) of clean up requirements is said to
be t-minimal if the clean up requirements for the first t time periods are the minimum required to
achieve integer feasibility in the ATM.
Proposition (2): If there exists an optimal solution for the ATM, there exists an optimal solution to
the ATM with the property that the clean up requirements are (T-1) minimal.
The intuitive interpretation of Proposition (2) is as follows: the total clean up capability on
scene at time point t can exceed requirement ?t, but not by an amount more than the capability of the
least capable system included in the response. Proposition (2) permits us to focus our attention on the
optimal solutions that satisfy the (T-1) minimal property. The following observation characterizes
17
solutions with the (T-1) minimal property.
Observation: Since the clean up capabilities of equipment systems degrade over time, in an optimal
solution with the (T-1) minimal property, no equipment system will be deployed earlier than necessary
to meet the (T-1) minimal requirements, ?t (t=1,...,T-1).
This observation emphasizes the fact that additional equipment systems are only deployed to
satisfy the clean up requirements for each time period because no benefit is obtained by deploying an
equipment system that is only necessary to meet minimal clean up requirements,?t, of a later time
period.
We now state a proposition that characterizes the nature of (T-1) minimal solutions and is used
to generate facets for the requirements sub-polytope of the TDP (associated with (2)).
Proposition (3): If X 88minimizes P' where P' is given by
mint
i i t t T i = 1
{ : X satisfies (9) - (11), X 0, X integer },C X πππ∈ ∈Π
≥∑ ∑ ∑ 89 (20)
then the family of inequalities given byt t
i i t i t i ti = 1 i = 1
= C CX Xππ π ππ π
γ∈ Π ∈ Π
≥∑ ∑ ∑ ∑ 90, for t = 1, 2, ...,
T-1; (21)is valid for the ATM.
Note that once a (T-1) minimal solution is at hand, the minimal requirement for the Tth time
period,|T|
,γ 91is easily satisfied by removing excess systems until the minimal requirement |T|
.γ 92 is satisfied.
A system provides excess capability if the solution is still feasible after the decision variable representing its
deployment is fixed at zero.
Thus, we haveProposition (4): If
1 2 |T|-1, , , γ γ γK 93correspond to the response capabilities provided by a (T-1) minimal
solution (and are determined using (21)), and|T|γ 94is obtained as described above, then
t
i t q i q ti=1 q Q(t)
C x γ∈
≥∑ ∑ 95, for t e T (22)
describe facets for the TDP, ift, t+1, q q t+1 t
q R(t+1)C u > - ,γ γ
∈∑ 96 for t e T. (23)
The intuitive interpretation of Proposition (4) is as follows: Proposition (4) shows that (22)
defines facets by first showing that every feasible integer solution has to satisfy the response
18
requirements corresponding to a T-minimal solution. In addition, (23) says that over any two
successive time periods, the total new response capability available in the later time period always
equals or exceeds the additional requirement in the new time period. This assures that no requirement
constraint is dominated by its successor requirement constraint, and hence is necessary in the
description of the requirements sub-polytope.
6 COMPUTATIONAL EVALUATION
This section describes the test problems used to evaluate the solution method. It also discusses
test results.
6.1 Test Problems
Our numerical test problems are based on an actual setting in the Galveston Bay Area. We
defined three "base" cases and four variations of each, representing the combinations of two levels of
each of two factors. To illustrate the base cases, we describe one in some detail.
For the second base case, we identified nine types of equipment systems that we used to
generate a total of 90 response systems by considering the locations at which the constituent
components are stored and the staging area where each response system is composed. The planning
horizon consists five critical time points (i.e., five time periods). Table 2 itemizes some characteristics
of the Galveston Bay Area relative to the second base case. Response requirements are specified in
Table 3. Figure 3 depicts the Galveston Bay Area, including the 6 locations that store components and
the 2 areas that might be used for staging.
-------------------------Tables 2, 3 and Figure 3
-------------------------We now describe the 12 test problems used to evaluate our two heuristics. Three factors were
considered in defining each test problem: size (determined by the number of critical time points and
number of response systems used), response requirements (?t, t e T) and component availability (Aem).
Consequently, problems of three different sizes composed the "base" cases and other problems were
created from each base case by taking different combinations of the response requirement and
component availability. Thus, each test problem was created from a "base" case by fixing each of two
factors at one of two levels.
Table 4 describes the test problems. For example, tdp2_11 represents the second "base" case
19
and level 1 for both the response scenario and component availability factors. tdp2_11 involves 90
response systems, 5 time periods, 30 components stored in 6 locations, and 2 staging areas. Similarly,
tdp3_12 represents the third "base" case with response requirement and component availability at levels
1 and 2, respectively. Each of the test problems is based on the Galveston Bay Area and thus portrays
characteristics that are expected to reflect an actual spill.
--------------------Table 4
--------------------6.2 Test Results
Our solution approach, which combines the cut generation method with a branch and bound
algorithm starts by developing the aggregated model for the problem and then uses the ATM and the
TDP in the iterative scheme for preprocessing described in Section 4. After preprocessing, the
inequalities described by (16) are generated. If the preprocessing procedures and the facets for the
requirements constraints do not yield an integer solution, we resort to branch and bound for the TDP.
The branch and bound algorithm exploits the "lower triangular" structure (Section 4) to define special
branching rules. The preprocessing and cut generation routines are coded in FORTRAN, and we
employ the IBM OSL branch and bound solver.
Tables 5, 6, 7 and 8 summarize the results. Table 5 shows the lower and upper bounds for
equipment system types in each test problem obtained by tightening the GLB and GUB constraints
using (14) and (15), respectively. For comparison, we also show the initial upper bounds on the
equipment system types. The initial lower bound for all equipment systems is zero in all the test
problems and hence is not shown in Table 5. Bounds for equipment system 10 are not shown for base
cases 1 and 2, since these do not include equipment system 10. The bound tightening procedures are
effective, closing initial ranges that average around 10 to an average of 2.
Table 6 shows the effectiveness of preprocessing routine (described by (16) and (17)) with
respect to the bounds on the total number of equipment systems necessary [NL, NU] and in terms of the
number of variables fixed by reduced cost fixing and Proposition (1). Table 7 compares the value of
the LP relaxation to that of the original TDP, the value of the LP relaxation after the cuts are added,
and the IP optimal solution. The run times (in seconds) for obtaining IP optimal solutions are also
shown. Table 8 compares the performance of this optimizing approach with that of the two heuristics
20
developed by Wilhelm and Srinivasa (1994) and with the OSL branch and bound solver applied directly
to each problem. The run times in Table 8 are all in seconds.
Results show that the aggregation scheme is successful, not only in improving bounds from the
LP relaxation, but also in finding optimal integral solutions within reasonable times. The reformulation
procedures coupled with the facets for the requirement constraints (described by (22)) succeed in
reducing the integrality gap. Indeed, for some of the smaller test problems, the method fully closes the
integrality gap.
In general, for a base case, the average gap reductions were better for test problems with
component availability (factor 2) at level 1, rather than level 2. However, such a relationship was not
observed relative to the response scenarios (clean up requirements levels (factor 2). This suggests that
the facets for the requirement constraints play a key role in reducing the gap, but to achieve larger gap
reductions they have to be used in conjunction with the reformulation procedures that tighten the
component availability constraints.
------------------------------------------Tables 5, 6, 7 and 8
-------------------------------------------7 SUMMARY AND CONCLUSIONS
In this paper, we formulate the TDP as a general integer program and present an optimizing
approach that is based on a combination of aggregation techniques and strong cutting plane methods.
The aggregation-based iterative procedure exploits special structures in the aggregated model so that
we are able to tighten not only the requirement constraints, but also the component availability, the
GUB, and the GLB types of constraints. We identify a family of facets for the response-requirement
sub-polytope, derive other inequalities based on dominance properties, and compute bounds on the
number of equipment systems necessary. Test results demonstrate that this approach yields an effective
way to obtain optimal solutions to the TDP.
This solution approach is intended for use by managers as a decision support aid in prescribing
optimal, time-phased response to an oil spill. No quantitative methods are currently used to assist
managers in making these important decisions. Managers can use the model to evaluate the
combination of ways in which response can be mobilized, assuring optimal composition and
deployment of response systems. This approach combines components, which are, perhaps, stored at
21
diverse locations, to achieve an effective response, meeting time-phased requirements, which are
intended to minimize environmental impact. The model provides a structure for co-ordinating the
response efforts of contractors, responsible parties, Coast Guard, state government, and other officials.
Managers can also use the model as a planning tool to evaluate the policies by which clean up
is conducted. For example, as demonstrated by the test problems, the model could be used in
(strategic) contingency planning to establish required response scenarios { t : t Tγ ∈ 97}. In addition, test
problems demonstrate application to evaluate system-wide response capability as a function of equipment
availability and could be used, for example, to assess the need for a policy that would require contractors to
provide minimum levels of equipment availability to assure adequate response capability.
The tactical decision model could easily be integrated with models that address the strategic
and operational levels of response. For example, managers could use it to evaluate contingency plans,
assessing the ability of the system prescribed by models that address the strategic level to respond to
certain types of spills at selected risk points. The operational level typically employs a trajectory model
to predict the movement and spread of oil over time. Managers could use trajectory model predictions
as inputs to the tactical decision model, which could be rerun periodically to revise prescribed response
in light of changing conditions. In yet another application, managers could employ the tactical decision
model in training programs to provide decision support for trainees in prescribing response to
simulated spills.
8 REFERENCES
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Charnes, A., W. W. Cooper, J. Harrald, K. R. Karwan and W. A. Wallace, 1976. A Goal IntervalProgramming for Resource Allocation in a Marine Environmental Protection Program. Journal ofEnvironmental Economics and Management. 3, 347-362.
Crowder, H., E. L. Johnson and M. Padberg, 1983. Solving Large Scale Zero-One LinearProgramming Problems. Opns. Res. 31, 803-834.
Evans, J. R. 1979. Aggregation in the Generalized Transportation Problem. Comput. Opns. Res. 6,199-204.
Hallefjord, A., and S. Storoy, 1990. Aggregation and Disaggregation in Integer Programming. Opns.
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Hoffman, K. L. and M. Padberg, 1991. Improved LP-Representations of Zero-One Linear Programsfor Branch-and-Cut. ORSA Jou. on Computing, 3, 121-134.
Hopp, W. J. and M. L. Spearman, 1993. Factory Physics: The Foundations of ManufacturingManagement.
Kao, E.P.C., and M. Queyranne. 1986. Aggregation in a Two-Stage Stochastic Program forManpower Planning in Service Sector. In Delivery of Urban Services. TIMS Studies in ManagementSciences. A.J. Swersey et al (eds). North-Holland, Amsterdam. 22, 205-225.
Onyekwelu, D.C. 1983. Computational Viability of a Constraint Aggregation Scheme for IntegerLinear Programming Problems. Opns. Res. 31, 795-801.
Psaraftis, H. N., G. G. Tharakan and A. Ceder, 1986. Optimal Response to Oil Spills: The StrategicDecision Case," Opns. Res. 34, 203-217.
Psaraftis, H. N. and B. O. Ziogas, 1985. A Tactical Decision Algorithm for the Optimal Dispatching ofOil Spill Clean up Equipment. Mgmt Sci. 31, 1475-1491.
Rogers, D. F., Plante, R. D., Wong, R. T., and J. R. Evans, 1991. Aggregation and DisaggregationTechniques and Methodology in Optimization. Opns. Res. 39, 553-582.
Vanderbeck, F., and L. A. Wolsey, 1992. Valid Inequalities for the Lasdon-Terjung ProductionModel. J. Opl. Res. Soc. 43, 435-441.
Wemmerlov, U., and N. L. Hyer. 1986. Procedures for Part Family/Machine Group IdentificationProblem in Cellular Manufacturing. J. Opns. Mgmt. 6, 125-147.
Wilhelm, W. E., and A. V. Srinivasa. forthcoming. Tactical Response in Oil Spill Clean UpOperations. Management Science.
Zipkin, P. H., 1980a. Bounds for Aggregating Nodes in Network Problems. Math. Prog. 19, 155-177.
Zipkin, P. H., 1980b. Bounds on the effect of Aggregating Variables in Linear Programs. Opns. Res.28, 403-418.
Zipkin, P. H., 1980c. Bounds for Row Aggregation in Linear Programming. Opns. Res. 28, 903-916.
9 APPENDIX A: Proofs
Proof of Property (1): If DCx S∈ 98, from Definition (1) and by (8), it follows that
23
X = (x)Φ 99satisfies (9). Hence, ACX S∈ 100.
To show that we can construct a solution x 101that is feasible relative to (2), given a solution X 102that
is feasible relative to (9), define
i q i = x X π′ 103 for all p e ? , and for some q' e Q(p) and,
iq = 0x 104 for all other q e Q(p) - {q'}.
Then, by (8),
q
i = t i = t
i t i t q i qi t (t) i = q R(t) i = t t
= C C xXπ
π ππ
γ∈ Π ∈
≥∑ ∑ ∑ ∑ 105 for all t e T.
Hence, DCX .S∈ 106 Q.E.D.
Proof of Property (2): The proof is obvious, since each of the constraints in (10) is obtained by taking
a non-negative linear combination of a subset of constraints in (3). Q.E.D.
Proof of Property (3): Again, the proof is straightforward, since each of the constraints in (11) is
obtained by taking a non-negative linear combination of a subset of constraints in (4). Q.E.D.
Proof of Property (4): Follows from Property (2) and Definition (2). Q.E.D.
Proof of Proposition (1): Since both q1 and q2 represent the same equipment system type, the clean
up capabilities of the two response systems are the same (i.e.,1 2(i t q (i t q) ) = C C 107for all t and i).
Furthermore, the number of components of type e required by both response systems is the same. Hence, the
allocation of components between these two response systems is resolved solely according to response times.
Q.E.D.
Proof of Proposition (2): We will prove this by showing that if *X 108is any optimal solution to the ATM with
* * = rZ X 109without the (T-1) minimal property, then we can derive an X 110, such that *r X = Z 111,
and X 112has the (T-1) minimal property.
Since (by assumption) *X 113does not exhibit the (T-1) minimal property, there exists a t (t ≤ T - 1)
such that
minmint
i = t*i i t 1 t 2 t t t t
{ t : > 0 }i = 1 x
- { { , , ..., } }C C C CXπ
ππ π π πππ
γ∈Π
≥∑ ∑ 114 (19)
wheretγ115is the response requirement for t e T.
Let p' (p' e ?(t)) be an equipment system for which relation (19) is satisfied in period t. Then,
24
by postponing the use of p' in the response until time period (t+1), the objective function remains
unchanged, while feasibility is maintained. Repeating the procedure for all p' (p' e ?(t)) that satisfy
relation (19), the optimal solution now exhibits the t-minimal property. By repeating the same
procedure for each of the periods t+1, t+2, ..., T-1, we can achieve a solution that has the (T-1)
minimal property. Q.E.D.
Proof of Proposition (3): First note that the optimum objective function value for problem (20)
represents the minimum total clean up capability necessary to be feasible. And, by Observation (1), no
equipment systems are deployed earlier than necessary. So, 1 2 (T -1), , , γ γ γK 116represent the minimum
clean up requirements in each time period to achieve feasibility. In other words, X 117represents the (T-1)
minimal solution for the ATM. Q.E.D.
Proof of Proposition (4): Proving the validity of the inequalities (22) is straightforward and follows directlyfrom Proposition (3) and Property (2).
To show that inequalities (22) describe facets, note that by definition oftγ118(teT), there does not exist
an ˆ ˆ DIx ( x )P∈ 119such that
ˆt
iq iq ti=1 q Q(t)
< C x γ∈
∑ ∑ 120, for teT.
Furthermore, the condition t+1 ,qt t+1q R(t+1)
> - Cγ γ∈∑ 121assures that each requirement constraint is necessary in
the description of DIP 122. The result follows. Q.E.D.
10 APPENDIX B: A Numerical Example
Notation? set of equipment system types = {p1, p2}Q set of response system types = {q1, q2, q3, q4}T set of time periods = { 1, 2, 3}J set of staging areas = {a, b}Q(j) set of response systems that can be staged in staging area jQ(p) set of response systems that employ equipment system type pM set of storage locations = {m1, m2}E set of equipment component types = {e1, e2}xtq number of response systems of type q deployed in time period t.
25
Description of Response SystemsResponse Equipment (Component- Qty Time StagingResponseSystem System Location) required available Area Time
q1 p1 e1m1 3 1 a 10
q2 p1 e1m2 3 2 b 18
q3 p2 e1m1 4 1 a 12e2m2 5
q4 p2 e1m2 4 3 b 25
e2m1 5
Description of Equipment SystemsEquipment Clean up capability Periods Required ResponseSystem Barrels/day After Deployment Staging Area Time
p1 500 0 250 10450 1405 2
p2 900 0 400 12800 1650 2
26
The Tactical Decision Problem (non-negativity and integer restrictions are omitted for brevity)
(1) This hypothetical example is intended to demonstrate the aggregation scheme, not a realistic case.
(2) Response time for a response system is determined by the maximum time that its constituent components require to respond.
(3) An aggregated requirements constraint results directly from column aggregation.
(4) An Aggregated GUB constraint results from a combination of column and row aggregation.
(5) An Aggregated Component Availability constraint is obtained by aggregating over columns, rows, and all time periods for storage locationm, and, thus, describes the constraint on availability of a component type over the entire planning horizon and at all locations. It is obtained byaggregating the original component availability constraints over applicable storage locations, then applying the column aggregation scheme.
29
(6) An Aggregated Staging Area constraint is obtained by aggregating over columns, rows, and all time periods for staging area j, and, thus,describes the restriction on the availability of space at staging area j over the entire horizon. It is obtained by aggregating constraints forstaging area j in the original model.
30
Table 1: A List of Recent, Major Oil Spill Disasters.
_______________________________________________________________________________Year Name Location Total Volume of Oil
Spilled (in million gallons)______________________________________________________________________________1978 Amoco Cadiz Brittany 681979 Burmah Agate Galveston 111979 Ixtoc Mexico and Texas Coasts 1551984 Alvenus Louisiana and Texas Coasts 31989 Exxon Valdez Alaska 111990 Megaborg Galveston 4.91992 Aegean Sea Spanish Coast 231993 Braer Shetland Islands 251993 Maersk Navigator Indonesia 78______________________________________________________________________________
Table 2: Galveston Bay Area: Characteristics_____________________________________________________________Number of Component Types : 30Number of Equipment System Types : 9Number of Response Systems : 90Number of Storage Locations : 6Number of Critical Time Points : 5Number of Potential Staging Areas : 2_____________________________________________________________
Table 3: Galveston Bay Area: Response Requirements_______________________________________Critical Time Response RequirementPoint (Gallons/hour)_______________________________________1 80002 150003 250004 350005 45000______________________________________
