Optimal location and capability of oil-spill response facilities for the south coast of Newfoundland

Omega 41 (2013) 856–867

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Optimal location and capability of oil-spill response facilities for the south coast of Newfoundland Manish Verma a,n, Michel Gendreau b, Gilbert Laporte c a

Faculty of Business Administration, Memorial University, St. John’s, Canada ´ cole Polytechnique de Montre´al, Canada CIRRELT and Department of Mathematical and Industrial Engineering, E c CIRRELT and Canada Research Chair in Distribution Management, HEC Montre´al, Canada b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 April 2012 Accepted 23 October 2012 Processed by B Lev Available online 1 November 2012

The south coast of Newfoundland (Canada) includes both open sea and semi-enclosed waterways which collectively account for over 20,000 vessel movements annually. Every such movement poses the risk of an oil spill which can endanger the fragile marine life and tourism locales in the region, and is a source of concern to the communities. In an effort to analyze the problem, we present a two-stage stochastic programming approach which tackles both the location and stockpile of equipment at the emergency response facilities. The proposed optimization program was tested on realistic data collected from publicly available reports and through personal communications with emergency response personnel. These data were then varied to solve a number of scenarios which account for the stochastic nature of the problem parameters. Although only two response facilities seem to be appropriate for almost all scenarios, the size of equipment stockpile is a function of both the societal disutility factor and the trade-off between environmental cost and facility and equipment acquisition cost. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Facility location Marine oil spills Stochastic programming Environmental cost Societal risk

1. Introduction Marine transportation, the primary mode for moving oil in Canada, was also responsible for a country-high 60 million tons of crude oil and petroleum products transiting the Newfoundland maritime infrastructure in 2004 [1]. In fact, Newfoundland, together with Nova Scotia and New Brunswick, accounts for the majority of marine transportation of crude oil and petroleum products in Canada. These numbers have been increasing since 2000, and the trend is likely to continue given the proposed development of additional off-shore petroleum platforms and new refineries in southern Newfoundland. While the region has benefitted tremendously from increased oil-related activities, there is a concern, especially amongst the neighboring communities in Placentia Bay, about the province’s preparedness to deal with potential oil-spill emergencies. Addressing some or all the stakeholder concerns was one of the objectives of a study just concluded by Transport Canada, which assessed the risks of oil pollution off the south coast of Newfoundland [2,3]. It focused on an area roughly stretching from Port aux Basques (aka Port of Basques) to a point located 50 nautical miles to the south by following the coast 50 nautical

miles east of St. John’s (Fig. 1). It is estimated that over 20,000 vessels move through this area annually, with 40% just in Placentia Bay. It is important to note that a significant portion of the vessel movements in Placentia Bay result from the oil refinery and transhipment facilities located at the head of the Bay (i.e., Zone 1). In fact, around 325 million barrels of crude oil and refined products were transported using over a thousand oil tanker movements in Placentia Bay alone, which is almost four times the volume (i.e., 85 million barrels) transiting the south coast for destinations along the St. Lawrence Seaway and the Great Lakes. The International Tanker Owners Pollution Federation (ITOPF) maintains a database of oil spills from all accidents involving tankers, combined carriers and barges. For historical reasons, spills are generally categorized by size ( o 7 t, 7–700 t, and 4700 t),1 although the actual amount of spill is recorded. Of the nearly 10,000 incidents now in the database, almost 85% of them fall into the first category [4,5]. In fact for the very first time since 1974, no major oil spills (i.e., 700 t or more) from tankers were recorded in 2009 and, more importantly, the number of such spills has been declining consistently in recent years, so much so that the average yearly number of major spills in the last 10 years has been only three [6]. This is consistent with the statistic for

n

Corresponding author. Tel.: þ1 709 864 6230; fax: þ1 709 864 7680. E-mail addresses: [email protected] (M. Verma), [email protected] (M. Gendreau), [email protected] (G. Laporte). 0305-0483/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2012.10.007

1 This is roughly equivalent to: o 50 barrels; 50–5000 barrels; and , 45000 barrels, respectively.

M. Verma et al. / Omega 41 (2013) 856–867

857

Fig. 1. South coast of Newfoundland [2].

Table 2 Annual spill frequencies (10–3) in the area of interest.

Table 1 Marine accidents and incidents in Newfoundland [7]. Year

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Shipping accidents

Cargo/tanker /bulk carrier

Ferry/ passenger

Tug/barge

73 69 58 80 73 70 74 87 61 35 31

5 4 6 3 2 2 9 2 4 1 4

5 4 4 5 5 2 3 3 6 10 7

3 0 0 2 2 0 0 2 2 1 0

Fishing/ others 63 64 50 76 64 67 64 84 52 23 22

Newfoundland, which has not experienced a single oil tanker episode since 2006 (Table 1) [7]. It is true that the safety record of crude oil tankers has been very good over the past decade, but numerous other vessels also represent a traffic risk and somewhat of an oil-spill risk given the fuel they carry. The south coast of Newfoundland, which is the area of interest (AOI), includes both open sea and semi-enclosed waterways, the latter being in close proximity to fragile marine life and tourism locales. The AOI has been divided into five zones: zones 1 and 2 respectively comprise the inner and outer Placentia Bay, zone 3 provides transit for marine traffic for all zones, zone 4 contains the links to the eastern seaboard of the continent and the St. Lawrence seaway, and zone 5 is the area surrounding the provincial capital, St. John’s. Every oil handling facility and tanker, in the AOI, has a contractual arrangement with the Eastern Canada Response Corporation (ECRC) to respond to oil spills. Given the varying level of activity, each zone is a likely location for oil or fuel spill and could potentially contain a host of possible spill profiles, which are collectively determined by oil type, weather conditions, and volume spilled. Clearly, such spills need to be contained lest marine life and environment be threatened,

Oil type/spill range

Zone

1st

2nd

3rd

4th

5th

6th

Fuel Crude Refined Fuel Crude Refined Fuel Crude Refined Fuel Crude Refined Fuel Crude Refined

1

70 955 744 200 221 183 190 278 266 640 57.3 71.8 340 1.87 91

190 120 160 580 27.9 39.5 550 35.1 57.4 183 7.22 15.5 980 0.24 19.6

10 37.7 30.4 20 4.99 6.90 20 6.28 10 60 1.29 2.70 30 0.04 3.62

– 8.69 1.16 – 4.99 0.74 – 6.28 1.08 – 1.29 0.29 – 0.04 0.21

– 1.30 1.02 – 0.447 0.390 – 0.563 0.583 – 0.12 0.12 – 0.004 0.014

– 4.49 0.518 – 1.66 0.19 – 2.08 0.27 – 0.43 0.07 – 0.21 0.014

2

3

4

5

which is possible only if emergency response facilities with appropriate equipment packages can respond within a predetermined critical time. This has been a major concern of the communities in Placentia Bay since ECRC is located on the outskirts of St. John’s, a few hours away from Placentia Bay.

2. Problem statement This study deals with location under uncertainty (see, e.g., [8,9]). It is concerned with both the strategic and tactical aspects of the oil-spill response problem for the south coast of Newfoundland. More specifically, we will answer two questions: first, where to locate adequate emergency response facilities; and second, what types of equipment to stockpile at each facility, and how are they going to be assigned in response to an oil-spill event. Given the strategic (and tactical) focus of this work, it is important to take into consideration the different known and uncertain factors likely to impact the spill event, as well as the consequent response planning. For example, emergency response

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center location decisions should incorporate: the projected marine accidents and vessel traffic over the planning horizon; the critical time associated with each spill location; the estimates of economic and environment damage resulting from various oil spills; and the estimates of different oil-spill volumes. Note that while some of these factors could be reasonably captured via deterministic estimates, others cannot. For instance, the exact location, volume and type of spill, as well as weather conditions are uncertain. Hence, probabilistic estimates of various oil-spill profiles were developed using the extensive data made available through Transport Canada reports [2,3] and personal communications with the environmental response unit of the Canadian Coast Guard. It is important to note that while we have captured all of the above uncertain factors, our work does not consider issues such as weathering and movement of oil slick, time-dependent oil physiochemical properties, etc. Although many techniques have been developed to deal with uncertainty in mathematical programming, stochastic programming with recourse is cited as a general-purpose technique that can deal with uncertainty in model parameters. Stochastic programs with recourse are employed to make decisions prior to the realizations of some random variables such that the total expected costs of possible recourse actions are minimized [10]. The proposed location, equipment stockpiling and allocation problem is formulated as a two-stage stochastic programming with recourse to represent the randomness arising from the location, volume and type of spill. In addition, we conduct several sensitivity and scenario analyses.

3. Literature review We review the relevant literature on oil spills, oil-spill response, and the location of emergency response stations with stochastic inputs. Most of the contributions in the oil-spill domain fall within two streams: quantity and cost estimation of oil spill. The last two decades have seen the introduction of trajectory models to estimate the quantity of oil spilled in an accident. Most of this research has been developed in a local context such as in Price et al. [11] for the Gulf of Mexico, Al-Rabeh et al. [12] for the Arabian Sea, and Cronk et al. [13] for the Ohio river. The estimation of spill related costs has been an active research area in the maritime domain, wherein Etkin [14] and Vanem [15] proposed linear cost estimation techniques, while the works of Yamada [16], Kontovas et al. [17] and Psarros et al. [18] adopted a non-linear approach to estimate oil-spill related costs. On the other hand, oil-spill response, a tactical-level decision, involves prescribing response systems for a specific oil spill and entails decisions about which and how many components to dispatch. Wilhelm and Srinivasa [19] formulated a general integer programming model to address these issues, and used column generation to define the response systems. In a subsequent work, Srinivasa and Wilhelm [20] proposed a cutting plane methodology to solve the tactical planning problems involving medium-to-large oil spills. Finally, Wilhelm and Srinivasa [21] developed a general integer programming model for area-wide contingency planning of oil-spill cleanup operations in the Galveston Bay Area. The issue of responsiveness was again visited in the wake of the Deepwater Horizon oil spill in the Gulf of Mexico. To this end, Zhong and You [22] proposed a multi-objective optimization approach to consider both economic and responsiveness criteria in decision making. In a subsequent work, You and Leyffer [23] described a mixed-integer dynamic optimization model incorporating the oil transport and weathering process. It should be evident that oil-spill and damage assessment literature is very rich, but unfortunately the question of locating emergency response facilities has not yet received much attention.

Stochastic programming with recourse is rooted in the work of Beale [24] and Dantzig [25]. Although the study of stochastic and dynamic aspects of facility location started shortly thereafter, most of the research dedicated to these issues has been published in recent years. A comprehensive treatment of the theory and numerical methods can be found in Birge and Louveaux [26], while Ruszczynski and Shapiro [27] present a rigorous overview of basic models, methods and applications of stochastic programming. Drawing from stochastic programming, Louveaux [28] has applied two-stage stochastic programming with recourse to solve plant location and p-median problems, wherein the first-stage decision was to determine the location and size of the facilities, and the second stage specified allocation of production resources. In addition, stochastic location theory has been applied to an array of realworld problems in warehousing, retailing, locating emergency service vehicles, etc. We refer the reader to Owen and Daskin [29] for an extensive review. Reverting to our problem, we draw from three notable works dealing with installation of emergency response centers to respond to marine oil spills. Belardo et al. [30] have developed a partial covering approach to the siting of response resources for major maritime oil spills, and have applied their model to the case study of a spill in the Long Island Sound. Their model considers a multiple objective function consisting of the probabilities of covering spills of various ‘‘groups’’ (in terms of environmental and economic harm), subject to some relatively simple cleanup resource availability constraints. The model’s basic limitation was that equipment needs were determined on the basis of a single (standard) spill volume, thereby neglecting the large variability of the volume of an individual spill, which is perhaps the most important probabilistic feature of this problem. Psaraftis et al. [31] have formulated a mixed integer programming problem to determine the locations of the appropriate levels and types of cleanup equipment to respond to oil spills. Their model takes into account the frequency of spill occurrence, the variability in spill volumes, equipment efficiency, fixed and variable costs of facilities, as well as equipment, transportation, and damage costs. The proposed framework was applied to an illustrative setting in New England, and insights were provided. Finally, Iakovou et al. [32] have suggested that, since strategic and tactical decision levels interact with one another, an integrated framework must be used to determine facility location and equipment allocation. To this end, a linear integer programming model was first developed and a mixed-integer relaxation of the original problem was solved. Two realistic examples related to the east coast of Florida were presented. This work ignores the probabilistic nature of oil spills, and makes use of historical equipment information to make allocation decisions. To the best of our knowledge, no previous work models the oil-spill and equipment stockpile decision as a two-stage stochastic programming problem, which is what we intend to do for a realworld application related to the south coast of Newfoundland. This modeling framework appears to be very well suited to our case since investment decisions are made in a first stage, and emergency responses to random events are implemented in a second stage.

4. Mathematical model We first present the two-stage recourse model for our problem, followed by an equivalent single optimization model. 4.1. A two-stage recourse model In our two-stage model, the first stage focuses on facility location and equipment package acquisition decisions, whereas

M. Verma et al. / Omega 41 (2013) 856–867

the recourse problem solved in the second stage uses information about oil spill to make equipment response and dispatch decisions. We next introduce the variables, parameters and then outline the model for the two stages. First-stage problem

859

subject to X e Nijk r U ei 8iA I, 8e A E

ð6Þ

i A Ijk

XX

C ejk N eijk Zvjk Z jk

ð7Þ

i A Ijk e A E

I: E: Aei : Fi: Yi ¼ U ei :

set of possible facility sites, indexed by i; set of equipment packages, indexed by e; acquisition cost of equipment package type e at site i; fixed cost to open a facility at site i; ( 1 if facility at site i is opened, i A I, 0 otherwise; number of equipment packages of type e stockpiled at i

Minimize X XX e e FiY i þ Ai U i iAI

ð1Þ

iAI eAE

subject to MY i U ei Z0 Y i A f0,1g

8i A I, 8e A E

8i A I

U ei Z 0 integer

ð2Þ ð3Þ

8i A I, 8e A E:

ð4Þ

In this model, (1) is the objective function requiring facility and equipment acquisitions decisions, whereas constraint set (2) specifies that no equipment packages can be acquired unless the corresponding facilities are open. Constraints (3) and (4) define the ranges of the variables. Second-stage problem Random variables associated with an outcome o: x(o)¼(j(o), k(o)), where jðoÞEJ : set of possible spill zones; kðoÞEK : set of possible spill profiles; vjk: volume of oil spill to be contained for a spill of profile k in zone j; C ejk : maximum amount of oil an equipment package of type e can contain for a spill of type k in zone j; t eij : travel time for equipment package e to be transported from site i to zone j; Tjk: critical time to respond to an oil spill of type k in zone j; Iejk ¼ fi A I9t eij oT jk g : set of sites i such that an equipment package of type e can be dispatched soon enough to contain a spill of type k in zone j; ECjk: environmental cost resulting from non-containment of spill of type k in zone j; OC ejk : cost to operate one unit of equipment package type e in zone j for oil spill type k; TC eij : transport cost to move one unit of equipment package type e from site i to zone j; fjk: annual frequency of oil spill of type k in zone j; Neijk : number of equipment packages of type e dispatched from site i for an oil spill of type k in zone j; Z jk : fraction of oil spill of type k in zone j contained; Now for any given scenario (j,k) and first stage solution (Y,U), the second stage recourse problem is Q¼(Y,U,j,k) ¼Q(Y,U,x): Minimize    XX e EC jk 1Z jk þ TC ij þOC ejk Neijk iAI eAE

ð5Þ

Neijk Z 0 integer

8iA I, 8eA E

ð8Þ

0 r Z jk r1

ð9Þ

In this model, (5) contains the equipment package dispatch decisions to be made in response to the oil-spill profile. Constraint set (6) ensures that the number of equipment packages dispatched from any facility does not exceed the number of corresponding packages acquired at that facility. Constraint set (7) ensures that the total number of equipment packages of a specific type dispatched in response to a particular oil-spill profile is at least at large as the requirement. Finally, constraint sets (8) and (9) specify the ranges of the variables. 4.2. A single optimization model It is known that a two-stage stochastic programming problem with recourse can be equivalently stated as a single optimization program [27,33]. We make use of this result to develop a single optimization program that has both deterministic and stochastic parameters. Minimize " #  X XX e e XX   XX e FiY i þ Ai U i þ f jk EC jk 1Z jk þ TC ij þ OC ejk N eijk jEJ kEK

iAI eAE

iAI

iEI eEE

ð10Þ subject to MY i U ei Z0 X

Neijk r U ei

8iA I, 8e A E

ð11Þ

8iA I, 8e A E

ð12Þ

i A Ijk

XX

C ejk N eijk Zvjk Z jk

8j A J, 8k A K

ð13Þ

i A Ijk e A E

Y i A f0,1g

8iA I

U ei Z 0 integer N eijk Z0 integer 0 r Z jk r1

ð14Þ 8i A I, 8e A E 8i A I, 8e A E, 8j A J, 8k A K

8j A J, 8k A K:

ð15Þ ð16Þ ð17Þ

5. Problem instance We now describe a problem instance to which our model was applied. 5.1. Parameter estimation In an effort to take advantage of the extensive information developed through numerous studies and also test our model on realistic data, we have estimated values of relevant parameters from the Transport Canada reports [2,3]. These reports considered over 1000 accident records from 1980 to 2005 and the projected marine activities in the region to propose the annual spill frequencies for the three oil types, which yields a realistic baseline for scenario analysis. Note that since crude oil, refined products and

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M. Verma et al. / Omega 41 (2013) 856–867

fuel oil are the only oil types relevant to the AOI, the resulting analysis will focus on these three types only. Finally, Transport Canada categorizes spills from different oil types by their sizes expressed in barrels. These are 1 to 49, 50 to 999, 1000 to 9999, 10,000 to 99,999, 100,000 to 199,999, and greater than 200,000. For expositional reasons, we use the legends 1st, 2nd, 3rd, 4th, 5th and 6th to refer to the six spill ranges, respectively. Transport Canada [2] report also provides an estimate of the environmental cost associated with the six spill ranges, wherein each cost has two components: long-term impact and spill cleanup (Table 3). The former numbers were reached after taking into consideration the results from a number of studies commissioned to assess the long-term impact on fishing, aquaculture, and tourism. Note that [2] also specifies environmental costs based on oil-viscosity, which is relevant for our problem instance since medium and high-persistence oil correspond to the dissipation characteristics of ‘‘fuel & refined’’ and ‘‘crude oil’’, respectively. Finally, the oil-spill cleanup cost for different volumes was estimated using the spill-cost model proposed in Etkin [14]. 5.2. Scenario development

volume. Although there are two possible wind speeds in the AOI, we learned from a personal communication that all-season equipment is purchased and maintained, even though the operating cost increases during winter months. We next outline the remaining features of the AOI. Given the consistent improvement in the spill statistics worldwide, the excellent record for the AOI, the extremely rare nature of spills exceeding 10,000 barrels, and the very expensive but specialized nature of the response equipment, it may not be worthwhile to store equipment packages just for such large spills. This was also confirmed in a personal communication with the environmental response unit of the Canadian Coast Guard (CCG), who also indicated that most large spills necessitate a response from a number of locations, including those dispersed outside the region of interest. For example, equipment and personnel were dispatched from the CCG in response to the Deepwater Horizon oil-spill episode in the Gulf of Mexico. It is important to note that we still consider equipment packages that are capable of handling up to 10,000 barrels of spill volume and that larger spill volumes, if realized, would be handled using the existing packages.

5.3. Potential locations

As indicated earlier, equipment packages must be matched with the spill-profile, i.e., oil type, weather condition, and spill Table 3 Environmental cost (EC) ($ thousands). Oil type

Zone

1st

2nd

3rd

Fuel and refined

1 2 3 4 5

90 166 111 95 64

1568 1676 1237 1574 810

10,740 11,118 8195 10,748 5444

Crude

1 2 3 4 5

140 322 166 147 102

2389 2658 1901 2395 1240

16,356 17,320 12,540 16,357 10,425

4th

5th

6th

89,435 94,479 67,337 89,430 47,003

179,082 184,897 134,795 179,123 93,309

349,135 350,664 261,464 484,713 175,557

136,812 142,010 102,485 135,959 70,976

278,726 280,147 209,907 308,618 141,595

529,426 532,513 401,408 665,029 266,612

A total of eight potential sites for locating oil-spill response facilities were identified, while keeping road and marine accessibility in mind, as well as the physical space for installation. Fig. 2 depicts the geographical location of seven facilities, while Port of Basques can be located in Fig. 1. Furthermore, Table 4 provides the cost and coverage area for each of the eight facilities within the predefined critical time period. In both the Transport Canada [2] report as well as in our personal communication with the CCG, it was indicated that storing expensive equipment in remote locations would adversely impact the cost of maintenance and also render them vulnerable to vandalism. This is why the fixed facility opening cost in St. John’s and Come by Chance (Whiffen Head) is the lowest since these locations have round-the-clock surveillance and personnel presence. Although the next three locations have marine ports with sufficient capabilities to service freight and passenger vessels, the incremental cost is more than for the first

Fig. 2. Potential facility locations.

M. Verma et al. / Omega 41 (2013) 856–867

Table 4 Facility cost scenarios and coverage areas. Location

St. John’s (StJ) Whiffen Head (WHd) Placentia (Pla) Mary’s Town (MTn) Port of Basque (PoB) Fortune (For) Saint Bride’s (StB) Trepassey (Tre)

861

Table 5 Equipment acquisition cost ($ thousands).

Scenarios ($ millions)

Zone coverage

FC1

FC2

FC3

1

2

3

2.0 2.0 3.0 3.0 3.0 4.0 4.0 4.0

1.0 1.0 1.5 1.5 1.5 2.0 2.0 2.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

O O O O

O O O O

O

O

O O

O O O

Relationship 4

Scenarios

5 O

O O

Non-linear (NL) Linear (L) Non-linear (NL) Linear (L) Non-linear (NL) Linear (L)

AC1 AC

2

AC3

Volume of crude oil (tons) 50

100

500

1000

5000

10000

250 250 450 450 650 650

300 500 540 900 780 1300

450 2500 810 4500 1170 6500

540 5000 972 9000 1404 13,000

810 25,000 1166 45,000 2106 65,000

972 50,000 1399 90,000 2527 130,000

O Table 6 Operating and transport cost for an equipment package ($ thousands).

two. The last three locations do not possess adequate facilities, are remote, and hence are the most expensive to open. A base number of $1.5 million emerged during the meeting with CCG, which was adjusted to $2 million to reflect current conditions (i.e., FC1 in Table 4). Given our objective of building multiple scenarios for a better understanding of the problem, we also conduct analyses by varying the facility costs in the FC2 and FC3 columns of Table 4. In an effort to take advantage of the data made available through Transport Canada (Table 2), we assume a zone to be covered only if every point in that zone can be reached within the specified response time of 6 h. If some parts of a zone can be reached within the specified time but others cannot, the entire zone is considered to be uncovered. This is an important assumption since the spill frequencies are given by zones, and not by points inside the zone. But it should be clear that our model is easily amenable to more detailed spill frequency values, and in fact will yield more precise results if such data become available. 5.4. Response equipment As indicated earlier, we are interested in three types of oil: fuel oil, refined products such as gasoline and diesel, and crude oil. Both the existing literature on marine oil spills [34] and personal communications with CCG suggest that refined products and other light crude dissipate or dissolve rather quickly. They pose some immediate short-term threat, but evaporate within a couple of days, and hence are almost never subjected to any skimming efforts, but are nevertheless monitored for shoreline oiling. Although this implies that only surveillance and monitoring equipment would be required for spills involving this oil type, in an effort to be conservative, we assume that a realistic cost to monitor or respond to spills involving refined products is 20% of that associated with crude oil. On the other hand, some fuel oil and most crude oil would require a cleaning and containment effort. Four common methods are used in cleanup operations: mechanical systems to contain and recover the oil; chemical dispersants; burning; and bioremediation. Depending on the prevailing spill conditions, a combination of these methods may be used [19]. However, due to the existence of semi-enclosed areas and proximity of environmentally sensitive regions, mechanical systems are mostly used in the AOI, and are hence considered in our analysis. Based on the publicly available information [4,35], and a personal communication with CCG, we assume that an equipment package (or mechanical system) consisting of a skimmer, a boom, a storage tank, a boat, and some personnel is sufficient to contain a 50-t oil spill. In addition, a response vessel fitted with communication devices is required for coordination, monitoring and evaluation. Although it was a challenge to put an exact price tag on this configuration, we were provided with a rough estimate of $450 K as the acquisition cost of an equipment package capable of handling a 50-t crude oil spill. Since we could not obtain any information on larger spills, we have

Cost

Operating Transport

Volume of crude oil (tons) 50

100

500

1000

5000

10,000

25 10

39 19

151 90

302 190

1444 900

2888 1900

made use of both linear and non-linear extrapolations to estimate the equipment package costs for larger spill volumes (Table 5). In addition, we have performed a scenario analysis on the base number of $450 K. In the absence of concrete information, we assume that the appropriate costs of equipment capable of responding to heavy fuel oil spills likely to form clumps is 80% of the corresponding values for crude oil. Finally, we have made use of the ECRC rates schedule [33] to approximate the equipment operating and transport cost for a 50-t spill. This schedule also lists the daily operating cost of all the equipment. The average operating cost for a skimmer is $2000 per day, whereas the booms and the storage tanks would respectively cost around $4000 and $1000 per day. Furthermore the average daily operating cost of a boat is $2000, while it will cost around $11,000 per day to operate a response vessel equipped with communication devices. Finally, an amount of $5000 per day has been earmarked for response personnel, for a total operating cost of $25,000 for a 50-t crude oil spill (Table 6). Since there does not exist sufficient information on the transport cost of equipment, we have made use of the ECRC [35] standby cost as a proxy for the cost to transport. While personnel cost has been estimated at $5000 per day, the remaining amount includes the cost to transport the equipment, i.e., $1000 each for skimmer, boom, boat, storage and response vessel. In addition, the indicated transport costs are from the facility closest to any zone, while the costs for the other zones are proportionately adjusted for distance. For example, the indicated transport costs would be appropriate if the facility in St. John’s dispatches equipment to zone 5, but would be multiplied by 1.25, 1.50 and 2.00 for locations in zones 1, 2 and 3, respectively. In the absence of concrete information tying equipment costs to volume spilled, but in an effort to capture their non-linear behavior ([14]), we assume that for a 100-t spill the number of response vessels will remain the same, while the remaining equipment requirement will approximately be double that for a 50-t spill. The operating cost for each item, except the response vessel, doubles and hence the new amount is $14,000  2¼$28,000, plus $11,000 for a total of $39,000. Finally, transportation cost will roughly double except for that incurred for moving the only response vessel. The remaining operating and transport costs numbers were generated using a similar logic and incorporate the information available. To sum up, we consider six types of equipment for each of oil type for a total of 18 different equipment packages, and cleanup capacities up to: 50; 100; 500; 1000; 5000; and 10,000 t.

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5.5. Spill volumes Recall that the annual spill frequency data are aggregated into six ranges by Transport Canada (Table 2). In an effort to both represent the expanse of the spill range and also to take advantage of the available information, we have decomposed each of the six ranges into three distinct spill volume scenarios (Table 7): SV1 considers the most favorable case and SV3 depicts the other extreme, while SV2 corresponds to the mid-point of the six specified ranges. More explicitly, SV3 represents the worstcase scenario with maximum spill volumes in all the three ranges for fuel-oil, and the six ranges for the other two oil types.

6. Computational results We have used CPLEX 12.4.2 to solve the model of Section 3.2 to optimality for all combinations involving the three different facility costs, three different equipment acquisition costs with

6.1. Scenario results

Table 7 Spill volume scenarios for the three oil types. Scenario

Spill volume (in tons)

SV1 SV2 SV3

1 25 49

50 500 999

1000 5000 9999

10,000 50,000 99,999

100,000 150,000 199,999

200,000 200,000 299,999

Expected EC

AC1

FC1

FC2

FC3

SV1 SV2 SV3 SV1 SV2 SV3 SV1 SV2 SV3

Corresponding to each scenario depicted in Table 8, the complete solution contains details on the facilities, equipment packages, and coverage areas. Table 9 lists the facilities that are opened and the Table 10 Facility and equipment cost versus environmental cost ($ millions). Multiplier

Table 8 Solutions for the 54 scenarios ($ millions). Combinations

both linear and non-linear settings, and three different spill volumes. The problem instances provided in this section consists of 1817 variables and 390 constraints, and the results reported below were obtained in 0.12–0.25 s (170–385 iterations). It is important to note that one disadvantage of scenario-based stochastic programming is that the resulting mathematical models can be very large, therefore, requiring special solution algorithms. However, the size of the model developed for the AOI has not been prohibitively large for commercial optimizers. Table 8 provides a snapshot of the resulting 54 scenarios, which can be identified using the notations introduced in the previous tables. For example, the result from the combination involving facility cost FC1 from Table 4, spill volume SV2 from Table 7, and both linear (L) and non-linear (NL) equipment acquisition cost AC2 from Table 5. Furthermore it indicates that the total costs are $152 million and $125 million, respectively, for linear and non-linear settings. For expositional purposes, we designate this combination as the base case. The results from other combinations can be identified in the same manner.

AC2

Linear

Non-linear

FC & AC

EC

FC & AC

EC

19.31 20.12 24.17 56.48 114.10 153.68 163.85 163.85 163.85 1963.85 1967.45 1971.10

81,904.92 81,904.47 81,901.51 81,800.72 81,723.19 81,685.58 81,678.24 81,678.24 81,678.24 76,348.81 76,346.04 76,344.03

125.72 168.90 193.60 339.80 396.93 397.74 398.91 421.29 464.66 464.66 464.66 470.26

64,657.57 59,311.61 54,097.59 32,143.40 21,551.29 21,547.69 21,547.32 18,638.76 10,645.43 10,645.43 10,645.43 9358.43

AC3

L

NL

L

NL

L

NL

86 148 243 85 145 241 84 145 240

75 113 185 73 110 165 72 109 164

87 152 253 86 151 251 86 150 250

82 125 196 80 122 193 79 122 192

87 154 258 86 153 255 86 153 255

85 136 216 84 134 214 83 133 213

5 8 10 20 30 40 50 60 70 80 90 100

Table 9 Facilities and corresponding coverage for various combinations. AC1

Combinations

Expected EC

FC1

SV1 SV

2

SV3 FC2

SV1 SV2 SV3

FC3

SV1 SV2 SV3

AC2

AC3

L

NL

L

NL

L

NL

StJ 8% StJ 13.3% StJ; PoB 24% StJ; PoB 12% StJ; PoB 17.3% StJ; PoB 24% StJ; for 12% StJ; for 17.3% StJ; for 24%

StJ 21.3% StJ; PoB 35% StJ; PoB 48% StJ; PoB 25.3% StJ; PoB 37.3% StJ; PoB 52% StJ; for 25.3% StJ; for 38.7% StJ; for 52%

StJ 6.7% StJ 10.7% StJ; PoB 14.7% StJ 6.7% StJ; PoB 13.3% StJ; PoB 14.7% StJ 6.7% StJ; for 13.3% StJ; for 14.7%

StJ 18.7% StJ; PoB 29.3% StJ; PoB 34.7% StJ; PoB 21.3% StJ; PoB 29.3% StJ; PoB 34.7% StJ; for 21.3% StJ; for 29.3% StJ; for 34.7%

StJ 5.3% StJ 9.3% StJ 12% StJ 5.3% StJ 9.3% StJ; PoB 14.7% StJ 5.3% StJ; for 12% StJ; for 14.7%

StJ 12% StJ; PoB 22.7% StJ; PoB 28% StJ; PoB 14.7% StJ; PoB 22.7% StJ; PoB 28% StJ; for 14.7% StJ; for 22.7% StJ; for 28%

M. Verma et al. / Omega 41 (2013) 856–867

863

Table 11 Solution ($ millions), overall coverage and type (number) of equipment. Multiplier

1 5 8 10 20 30 40 50 60 70 80 90 100

Linear

Non-linear

Solution

Coverage (%)

Equipment

Solution

Coverage (%)

Equipment

152 427 672 833 1620 2400 3150 3990 4650 5390 6030 6530 7040

10.7 14.7 17.3 20.0 29.3 33.3 36.0 42.7 42.7 42.7 44.0 45.3 46.7

3 5 6 7 8 9 10 12 12 12 13 13 13

125 248 307 336 417 440 453 466 477 482 484 485 487

29.3 37.3 49.3 54.7 73.3 78.7 80.0 84.0 86.7 89.3 89.3 89.3 90.7

10 10 12 13 15 16 17 19 20 20 20 20 20

(8) (20) (22) (24) (58) (70) (72) (77) (77) (77) (113) (118) (119)

500

Cost ($ millions)

Cost ($ millions)

8000 6000 4000 2000 0

(66) (142) (179) (202) (322) (368) (371) (374) (392) (427) (427) (427) (432)

0

55

110

375 250 125 0

0

55

Multiplier

110

Multiplier

Fig. 3. Cost as a function of the multiplier. (a) Linear equipment cost. (b) Non-linear equipment cost.

resulting overall coverage for the given scenarios. For example, for the base case with linear cost setting, only the facility at St. John’s (StJ) would be open, and will cover 10.7% of the overall oil-spill situations. The coverage increases to 29.3% under the non-linear setting, although another facility at Port of Basques (PoB) would have to be opened. In general, coverage increases under the non-linear setting since it is cheaper to set up equipment packages capable of tackling larger oil spills. Note that each of the five zones in the area of interest could potentially have 15 oil-spill situations: three for fuel oil, six for crude oil, and six for refined products (Table 2). Hence, an overall coverage of 29.3% implies having the capability to respond to 22 of the 75 oil-spill situations. The overall coverage figure can be further broken down into equipment acquisition and response by each open facility, and proportion of oil-spill situations for the three oil types being covered in the five zones. For example, for the 29.3% coverage, while both the facilities at St. John’s and Port of Basque would acquire two types of equipment to respond to fuel oil spills. The former would also acquire four types of equipment to respond to crude oil spills and two types for refined products (Table A1). Furthermore, we can also tell that only the three fuel oil-spill situation are covered in zones 2, 4 and 5, while some situations involving spills of the other two types are covered in zones 1 and 3 (Tables A2 and A3). For expositional reasons, we have organized the allocation and coverage into three tables and placed them in the Appendix. We invite the reader to consult the appropriate tables for relevant details. It is clear from Table 9 that with higher equipment acquisition costs, fewer equipment packages are stockpiled at the open facilities, thereby reducing the overall coverage. On the other hand, with decreasing facility costs, one notices an increase in the number of open facilities, which in turn has an equivalent impact on coverage. Note, that the maximum coverage over all the combinations is 52%, while a number of spill situations have less than 10% coverage. It should be evident that since coverage results from the trade-off between expected environmental cost with respect to facility and equipment costs, some results may not do much to assuage the

Table 12 Incremental and greenfield solutions. Cost

StJ WHd Pla MTn PoB For StB Tre FC ($ mn) VCNL ($ mn) CoverageNL (%) VCL ($ mn) CoverageL (%)

Number of facilities 1

2

3

4

5

6

7

8

O

O

O O

O O O

O

O

O

O O O O O

O O O O O

O O O O O O*

10.00

13.00

O 17.00

O 21.00

O O O O O O O O 25.00

2.00 25.36 25.3 4.32 10.7

5.00 7.00 27.23 29.3 5.31 13.3

concerns of the general public. This is because the ‘‘expected environmental cost’’ does not capture the impact of intangible attributes such as public disutility or adverse perception of catastrophic events such as an oil spill. In an effort to conduct additional analyses by incorporating such intangibles, we apply a multiplier to the expected environmental cost term in Eq. (10), and then solve the problem instance for different values of the multiplier between 1 and 100.

6.2. Impact of disutility factor We have solved the problem for both the linear and non-linear equipment cost settings. Table 10 provides the facility and equipment cost versus the environmental cost for various values of the disutility multiplier. It is clear from Table 10 that larger values of the multiplier put greater emphasis on the expected environmental cost, thereby

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M. Verma et al. / Omega 41 (2013) 856–867

Table A1 Facility location and equipment packages for all spill situations. AC1

Combinations

Expected EC

FC1

SV1

SV2

SV3

FC2

SV1

SV2

SV3

FC3

SV1

SV2

SV3

AC2

AC3

L

NL

L

NL

L

NL

StJ F:1 C:1 R:1 StJ F: 1,3 C: 1 R: none

StJ F: 1,3 C: 1,4,6 R: 1,4 StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,2 C: 1,4,6 R: 1,4,6 PoB F: 1,2 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,3,5,6 R: 1,3,5 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4 For F: 1,3 C: 6 R: none StJ F: 1,3 C: 1,3,5,6 R: 1,3,5 For F: 1,3 C: 3,5,6 R: 3,5 StJ F: 1,3 C: 1,4,6 R: 1,4,6 For F: 1,3 C: 1,4,6 R: 1,4,6

StJ F: 1 C: none R: 1 StJ F: 1,3 C: 1 R: none

StJ F: 1,3 C: 3,4,6 R: 1,4 StJ F: 1,3 C: 1,3,5,6 R: 3,5 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 4,6 R: 1,4 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,3,5,6 R: 3,5 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 4,6 R: 1,4 For F: 1,3 C: 6 R: none StJ F: 1,3 C: 1,3,5,6 R: 3,5 For F: 1,3 C: 6 R: 3,5 StJ F: 1,3 C: 1,4,6 R: 1,4,6 For F: 1,3 C: 6 R: 4,6

StJ F: 1 C: none R: none StJ F: 1,3 C: 1 R: none

StJ F: 1,3 C: 4,6 R: none StJ F: 1,3 C: 1,3,5 R: 3,5 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 4,6 R: none PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,3,5 R: 3,5 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 4,6 R: none For F: 1,3 C: none R: none StJ F: 1,3 C: 1,3,5 R: 3,5 For F: 1,3 C: none R: none StJ F: 1,3 C: 1,4,6 R: 1,4,6 For F: 1,3 C: none R: 4,6

StJ F: 1,3 C: 1 R: 1,3 PoB F: 1,3 C: none R: none StJ F: 1 C: 1 R: 1 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1,3 PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1,3 PoB F: 1,3 C: none R: none StJ F: 1 C: 1 R: 1 For F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1,3 For F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1,3 For F: 1,3 C: 1 R: 1

StJ F: 1,3 C: 1 R: 1 PoB F: 1,3 C: none R: none StJ F: 1 C: none R: 1

StJ F: 1,3 C: 1 R: none PoB F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1 PoB F: 1,3 C: none R: none StJ F: 1 C: none R: 1

StJ F: 1,3 C: 1 R: none For F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1 For F: 1,3 C: none R: none

StJ F: 1,3 C: 1 R: 1

StJ F: 1 C: none R: none

StJ F: 1,3 C: 1 R: none

StJ F: 1,3 C: 1 R: 1 PoB F: 1,3 C: none R: none StJ F: 1 C: none R: none

StJ F: 1,3 C: 1 R: none For F: 1,3 C: none R: none StJ F: 1,3 C: 1 R: 1 For F: 1,3 C: none R: none

M. Verma et al. / Omega 41 (2013) 856–867

yielding better coverage through increased facility and equipment costs. Given the trade-off between expected environmental cost and facility and equipment cost, it is not surprising to observe an increase in coverage with larger values of the disutility multiplier. Clearly larger values of the multiplier mean that not covering an oil-spill situation is more expensive and results in better coverage. It was interesting to note that both the number and identities of the response facilities did not change for any of the scenarios depicted in Table 11 (and Fig. 3), except for one instance. In each of the cases, the facilities at St. John’s and Port of Basques were open, except under the linear setting for the base case where only the former location was used. This also implies that better coverage essentially results from stockpiling higher and more varied equipment packages. For example, under linear equipment cost setting, the overall coverage increases from 20% to 36% for respective disutility multiplier values of 10 and 40. This was achieved by acquiring three additional types of equipment, and increasing the equipment stockpile from 24 to 72 (Table 11). Based on the above analysis, it is possible to state that locating emergency response centers at both St. John’s and Port of Basques would bring about the most appropriate preparedness since they are a part of all solutions, except for one instance. Finally, it is possible to deduce from Table 11 and Fig. 3 that the total cost curve under the non-linear setting is definitively concave, and somewhat concave under the linear equipment cost setting. This implies that the total cost will increase at a decreasing rate with increasing disutility factor value, and hence it will cost proportionately less to provide better coverage.

865

6.3. Incremental and greenfield solutions Since we do not know the budget of the policy makers nor their precise mandated coverage requirement, we have conducted further analyses to facilitate future decision making with respect to adding new facilities, equipping them and the resulting coverage. To this end, we have generated two types of solutions using the base case instances under both linear and non-linear equipment cost settings. First, we make use of the current location of the response facility in St. John’s (StJ) to generate the incremental solutions. Next, we just specify the number of facilities to be opened and generate the so-called greenfield solutions. Due to the similarity of the final solutions under the two schemes, we report them together in Table 12 and comment on any deviations. Table 12 indicates that, with both linear and non-linear equipment cost settings, and for both the incremental and greenfield settings, the first emergency response facility will be located at St. John’s (StJ), followed by the second one at Port of Basques (PoB). Interestingly, the sequence of facilities to be opened was exactly the same for both the non-linear and linear setting for the incremental solution instance. The respective facility cost (FC), variable cost (VC), and % coverage of oil-spill situations are listed in Table 12. As expected, the non-linear equipment cost scenarios require larger stockpiles of various equipment packages, and hence higher variable cost and coverage. Now, under the greenfield scenarios, the sequence in which eight facilities was added only differ for the last two insertions: the facility at Fortune (For) would be added before the one at St. Brides (StB). In addition, the various cost components

Table A2 Coverage for the three oil types under various combinations. AC1

Combinations

Expected EC

FC

1

1

SV

SV2

SV3

FC2

SV1

SV2

SV3

FC3

SV1

SV2

SV3

AC2

AC3

L

NL

L

NL

L

NL

8% F: 4 C: 1 R: 1 13.3% F: 9 C: 1 R: none 24% F: 11 C: 3 R: 4 12% F: 7 C: 1 R: 1 17.3% F: 10 C: 1 R: 2 24% F: 11 C: 3 R: 4 12% F: 7 C: 1 R: 1 17.3% F: 10 C: 1 R: 2 24% F: 11 C: 3 R: 4

21.33% F: 8 C: 6 R: 2 35% F: 15 C: 7 R: 4 48% F: 12 C: 13 R: 11 25.3% F: 11 C: 6 R: 2 37.3% F: 14 C: 7 R: 7 52% F: 15 C: 13 R: 11 25.3% F: 11 C: 6 R: 2 38.7% F: 14 C: 8 R: 7 52% F: 15 C: 13 R: 11

6.67% F: 4 C: none R: 1 10.7% F: 7 C: 1 R: none 14.7% F: 9 C: 1 R: 1 6.7% F: 4 C: none R: 1 13.3% F: 9 C: 1 R: none 14.7% F: 9 C: 1 R: 1 6.7% F: 4 C: none R: 1 13.3% F: 9 C: 1 R: none 14.7% F: 9 C: 1 R: 1

18.67% F: 8 C: 4 R: 2 29.3% F: 13 C: 5 R: 4 34.7% F: 13 C: 6 R: 7 21.3% F: 10 C: 4 R: 2 29.3% F: 13 C: 5 R: 4 34.7% F: 13 C: 6 R: 7 21.3% F: 10 C: 4 R: 2 29.3% F: 13 C: 5 R: 4 34.7% F: 13 C: 6 R: 7

5.33% F: 4 C: none R: none 9.33% F: 6 C: 1 R: none 12% F: 7 C: 1 R: 1 5.3% F: 4 C: none R: none 9.3% F: 6 C: 1 R: none 14.7% F: 9 C: 1 R: 1 5.3% F: 4 C: none R: none 12% F: 8 C: 1 R: none 14.7% F: 9 C: 1 R: 1

12% F: 7 C: 1 R: 1 22.7% F: 12 C: 3 R: 2 28% F: 13 C: 3 R: 5 14.7% F: 9 C: 2 R: none 22.7% F: 12 C: 3 R: 2 28% F: 13 C: 3 R: 5 14.7% F: 9 C: 2 R: none 22.7% F: 12 C: 3 R: 2 28% F: 13 C: 3 R: 5

866

M. Verma et al. / Omega 41 (2013) 856–867

and coverage values associated with each insertion were exactly the same for both solution instances. It is evident from Table 12 and the above analysis that it does not make sense to locate more than two facilities. This is because the percentage of oil-spill situations covered does not improve even with more facilities. In addition, we observe that the variable cost does not change since opening a new facility just forces a portion of the required equipment package to be bought by the newly added facility, the total number of equipment remaining constant. For example, the 29.3% coverage results from stockpiling a total of 66 equipment packages across any number of open facilities (two, three or more). Hence, it is appropriate to locate the facilities only in St. John’s and in Port of Basques, since this will not only provide better coverage, but also be robust to different values of the disutility multiplier discussed earlier.

7. Conclusion The south coast of Newfoundland accounts for a significant portion of marine transportation of crude oil and petroleum products in Canada, and has also been a source of concern due to the potential for oil-spill emergencies. In this work, we have proposed a two-stage

stochastic programming approach that answers two questions: the location of emergency response facilities, and the appropriate equipment stockpile at each facility. The proposed optimization program was tested on realistic data collected from publicly available reports and through personal communications with a representative of the Eastern Canada Response Center. In an effort to account for the stochastic nature of input parameters, a number of scenarios were generated using the base numbers procured from the above sources. Through extensive computational experiments, we can conclude that in general the number of facilities and equipment stockpiles is a function of the trade-off between expected environmental cost versus facility and equipment cost. Furthermore, the size of the equipment stockpile for the region depends on the societal disutility factor, and increases with higher values of the factor. More specifically, for the AOI, a maximum of two facilities is most appropriate since any additional facility will not improve oil-spill coverage, but will merely result in the redistribution of equipment packages among various open facilities. In addition, linear equipment cost settings will always yield poorer coverage than the non-linear settings. Other directions for future research include, capturing environmental risk explicitly; applying the proposed modeling framework to other regions of the world for facility location and equipment

Table A3 Oil types (F,C,R) and coverage in the five zones. AC1

Combinations

L Expected EC

1

FC

SV

1

SV2

SV3

FC2

SV1

SV2

SV3

FC3

SV1

SV2

SV3

st

1 : 1;1;1 2nd:1;0;0 3rd:1;0;0 4th:0;0;0 5th:1;0;0 2;1;0 2;0;0 1;0;0 2;0;0 2;0;0 2;1;2 2;1;1 2;1;1 3;0;0 2;0;0 1;1;1 1;0;0 1;0;0 3;0;0 1;0;0 1;1;2 2;0;0 2;0;0 3;0;0 2;0;0 2;1;2 2;1;1 2;1;1 3;0;0 2;0;0 1;1;1 1;0;0 1;0;0 3;0;0 1;0;0 1;1;2 2;0;0 2;0;0 3;0;0 2;0;0 2;1;2 2;1;1 2;1;1 3;0;0 2;0;0

AC2

AC3

NL

L

NL

L

NL

2;4;2 2;1;0 2;1;0 0;0;0 2;0;0 3;5;2 3;0;0 3;1;2 3;1;0 3;0;0 2;4;3 2;4;3 2;4;3 3;0;0 3;0;2 2;4;2 2;1;0 2;1;0 3;0;0 2;0;0 2;4;3 3;1;2 3;2;2 3;0;0 3;0;0 3;5;3 3;4;3 3;4;3 3;0;0 3;0;2 2;4;2 2;1;0 2;1;0 3;0;0 2;0;0 2;4;3 3;1;2 3;3;2 3;0;0 3;0;0 3;5;3 3;4;3 3;4;3 3;0;0 3;0;2

1;0;1 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 2;0;0 0;0;0 2;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0 1;0;1 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 2;0;0 2;0;0 2;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0 1;0;0 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 2;0;0 2;0;0 2;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0

2;2;2 2;1;0 2;1;0 0;0;0 2;0;0 1;4;2 3;0;0 3;1;2 3;0;0 3;0;0 1;4;3 3;0;2 3;2;2 3;0;0 3;0;0 2;2;2 2;1;0 2;1;0 2;0;0 2;0;0 1;4;2 3;0;0 3;1;2 3;0;0 3;0;0 1;4;3 3;0;2 3;2;2 3;0;0 3;0;0 2;2;2 2;1;0 2;1;0 2;0;0 2;0;0 1;4;2 3;0;0 3;1;2 3;0;0 3;0;0 1;4;3 3;0;2 3;2;2 3;0;0 3;0;0

1;0;0 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 1;0;0 0;0;0 2;0;0 1;1;1 2;0;0 2;0;0 0;0;0 2;0;0 1;0;0 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 1;0;0 0;0;0 2;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0 1;0;0 1;0;0 1;0;0 0;0;0 1;0;0 1;1;0 2;0;0 1;0;0 2;0;0 2;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0

1;2;0 2;0;0 2;0;0 0;0;0 2;0;0 1;3;2 3;0;0 2;0;0 3;0;0 3;0;0 1;3;3 3;0;0 3;0;2 3;0;0 3;0;0 1;1;1 2;0;0 2;0;0 2;0;0 2;0;0 1;3;2 3;0;0 2;0;0 3;0;0 3;0;0 1;3;3 3;0;0 3;0;2 3;0;0 3;0;0 1;2;0 2;0;0 2;0;0 2;0;0 2;0;0 1;3;2 3;0;0 2;0;0 3;0;0 3;0;0 1;3;3 3;0;0 3;0;2 3;0;0 3;0;0

M. Verma et al. / Omega 41 (2013) 856–867

stockpiling decisions; and investigating the adoption of robust and multi-objective optimization techniques to solve similar problems.

Acknowledgments This research was in part supported by three grants from the National Science and Engineering Research Council of Canada (OGP 312936, 338816-10, and 39682-10). All three authors are members of the Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) and acknowledge the research infrastructure provided by the Centre. In addition, the first author is also a member of the IBM Academic Initiative and acknowledges the access to CPLEX Optimization Package. The comments and suggestions of two anonymous referees and the Area Editor helped improve the paper significantly.

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