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DecisionSupportSystem

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DecisionSupportSystem

  1. 1. Decision Support System for Locating Facilities: A Prototype GGR1922 Topics in Geographical Information Science Winter 2008 April 18, 2008 Name: John Frederick Chionglo Email: john.chionglo@utoronto.ca Student No.: 870974680 Locker No.: N/A
  2. 2. Introduction There are several models for locating facilities (Daskin 1995). However, there are situations where not all requirements can be included in a model. For example, information may be unavailable. The source for demand may be estimated using available data. When candidates for facilities are considered in the model, a solution to the model may have selected a facility to service the demand (left picture in Figure 1). If more accurate information were available (Fotheringham and others 1995), another facility may have been selected (right picture in Figure 1). By giving the user alternative solutions to inspect, the user may have the opportunity to evaluate the solutions based on information unavailable to the model. Candidate facilityDemand source (model) Demand source (actual) Figure 1 Known and Unknown Demand Source Furthermore, the feasibility of locating a facility may also be difficult to assess. A person may consider an area a likely candidate for locating the facility while another may say the opposite. This ambiguity may stem from uncertainties propagated from raw data (Brown and Heuvelink, 2008) or the inadequacy of a classification system (Foody 2002 as cited in Brown and Heuvelink 2008, p. 98; Stehman and Czaplewski 2003 as cited in Brown and Heuvelink 2008, p. 98). Reprocessing the raw data or taking a second look may change a person’s opinion of a location’s suitability for a facility. The uncertainty may also affect other parts of a model like cost. For example, the area may border between ground that is stable and one that is unstable. Thus, if a facility were built in the area, additional costs may have to be incurred if the area turns out to be unstable. Again, the user should have the opportunity to inspect alternative solutions, and evaluate the solutions based on information unavailable to the model (Figure 2). 1
  3. 3. Unstable regionCandidate facilityDemand source (model) Figure 2 Stable and Unstable Regions Some requirements are too complicated to specify. For example, when the problem of locating a facility is simultaneously considered with the problem of finding routes for delivery (this is known as the vehicle routing problem), there may be too many constraints in the problem to be implemented in a computer. The problem of finding a route is also known as a traveling salesman problem. Here is a formulation of the traveling salesman problem developed by Dantzig, Fulkerson, and Johnson (as cited in Garfinkel, 1959, p. 25-26). Definitions: V is the set of cities considered in the problem. i and j are cities in the problem ( ). is a 0-1 variable – it is 1 if the route goes from city to cityVjVi ∈∈ ; ijX i j , and 0 otherwise. is the cost of making at trip from city i to city ijc j . is a subset of V ,S VS ⊂∀ means for all subsets of V , and S is the number of cities in the set .S Objective: Min ∑∑∈ ∈Vi Vj ijij Xc Constraints: ∑≠ = ij ijX 1 Vi∈ ∑≠ = ji ijX 1 Vj ∈ ∑∑∈ ∈ −≤ Si Sj ij SX 1 VS ⊂∀ The objective is to minimize the cost of the route. The first set of constraints specifies that every city in the problem connects to another city. The second set of constraints makes sure that each city connects from another city. The third set of constraints makes sure that sub-tours are not allowed. In Figure 3, Route 1 has two sub-tours while Route 2 has no sub-tours. 2
  4. 4. Candidate facilityDemand source (model) Route 2 Route 1 Figure 3 Routes and Sub-tours An instance of this model can be stored in a matrix or table (Figure 4). The objective occupies one row. The first set of constraints occupies n rows (where n is the number of cities). The second set of constraints occupies n rows too. The third set of constraints occupies 2n -2 rows. Although models exist that limit the number of constraints that eliminate sub-tours to O(n2 ) or to the square of the number of cities such as the model by Miller, Tucker and Zemlin (as cited in Garfinkel, 1959, p. 27), it may be worthwhile to relax the sub-tour constraints and present several solutions to the user for inspection even if the solutions have sub-tours. Objective: Min ∑∑∈ ∈Vi Vj ijij Xc Constraints: ∑≠ = ij ijX 1 Vi ∈ ∑≠ = ji ijX 1 Vj ∈ ∑∑∈ ∈ −≤ Si Sj ij SX 1 VS ⊂∀ 3 Figure 4 TSP Model and Matrix Therefore, an interactive decision support system for locating facilities may be useful. The challenge is to describe an interactive system that a user can use to define location problems, to specify data or the sources of data, and to inspect alternative solutions. The requirements of the system should include the types of user interactions, the breadth and depth of location problems supported, computing technologies, and information practices. Methods/Approaches Here is a proposed system architecture (Figure 5). A user can manage models and problem instances through an intermediate structure or configuration. Through this configuration, a user can select a model, specify data or data sources, create instances of a model, and generate solutions. A user should also be able to manage the solutions as well. 1 c− 1 1 1A 2A 3A I 1−S
  5. 5. Model Solutions Model Instance User Manage Solutions Lp_solve Generate Solutions Manage Model Configuration Generate Instance Data Facility Location DSS Figure 5 Facility Location DSS Architecture Part of the challenge in developing an interactive decision support system for facility location is to specify how to generate alternative solutions. There are situations when the solution to a problem does not have an alternate solution, or there may be too many alternate solutions. In the former case, an alternate solution should be defined in terms of a solution from another, acceptable model. In the latter case, a subset of the set of alternate solutions should be created so that the number of solutions is more manageable. In both cases, the alternative solutions should be meaningful. The approach considered here is to provide a generic set of mathematical models as a starting point (also called model templates) for users to create model instances: the maximum covering location problem (Daskin 1995) and the p-centre problem (Daskin 1995). These models were chosen because 1) they are two of the most fundamental models used in facility location (Daskin 1995) and 2) they can be configured to generate instances of other types of models such as the set covering problem (Daskin 1995; Williams 1999). For the model instance that a user will eventually generate, a user should be able to select which objective function to optimize, what type of optimization to perform (minimization or maximization), and which combination of constraints (or objective functions) to include in the model instance to be created. These imply that a user will be able to restrict or multiply the number of potential solutions. Configuring a model template to produce a model instance can be applied to generating alternate solutions. After creating an initial model instance, the model template can be configured to produce another model instance. The challenge in this case would be to generate a model instance that is similar to the previous model instance or acceptable to the user as a substitute without prompting the user for input. 4
  6. 6. 5 If a prompt to the user cannot be avoided, then the number of prompts and amount of information should be minimized. Of course, the issues of what algorithms to use to solve the problem instances, and how to effectively update solutions from a given solution are important considerations. However, these are beyond the scope of this paper. It may be worthwhile to develop a prototype based on the ideas presented here. Because of time constraints on the project, this prototype has been confined to an implementation of the maximum covering location problem (Daskin 1995) that uses ASCII files for its configuration, inputs, and outputs. Conclusions An interactive decision support system for facility location may be a valuable tool. This would be an effective tool if it can suggest alternative solutions that are relevant to the user. However, the definition of a relevant or an acceptable alternative solution may not be that easy to define. A prototype of this system may help to clarify what a relevant solution is to a user. The prototype implements the idea that a model instance can be created from a model class. By configuring the same model class, another model instance can be created whose solution(s) may be a good candidate as an alternative solution. The challenge in this case would be to generate a model instance that is similar to the previous model instance or acceptable to the user as a substitute without prompting the user for input or minimizing the amount of information the user provides. Of course, if the initial model instance has several alternative solutions, they are more likely to be good candidates as alternative solutions. However, if there are too many alternative solutions, a subset of the set of alternative solutions should be created.
  7. 7. Appendix A: Facility Location Models Maximum Covering Location Model (MCLM) (Daskin 1995) Definitions: V is the set of demand sources (cities). i is a demand source ( ). W is the set of candidate sites. Vi ∈ j is a candidate site ( ). is the demand source . is a 0-1 variable – it is 1 if demand source is covered, and 0 otherwise. is an indicator coefficient – it is 1 if candidate site Wj ∈ ih i iZ i ija j can cover the demands at source i and 0 otherwise. is a 0-1 variable – it is 1 if a facility is located at candidate site jX j and 0 otherwise. is the number of facilities that must be located.P Objective: Max∑∈Vi ii Zh Constraints: 0≥−∑∈Wj ijij ZXa Vi ∈ ∑∈ ≤ Wj PXj Figure 6 Maximum Covering Location Model (MCLM) Here is a general matrix view of the model. This representation includes the bounds on the decision variables (last 2 rows), and the slack and surplus variables for all inequality constraints (columns 4 to 7). 1 ih− ija -1 -I I 1 P I I 1 I I 1 Figure 7 Matrix View of MCLM Set Covering Facility Location Model (SCFLM) (Daskin 1995) Definitions: V is the set of demand sources (cities). i is a demand source ( ). W is the set of candidate sites. Vi ∈ j is a candidate site ( ). is the cost of locating a facility in candidate siteWj ∈ jf j . is a 0-1 variable – it is 1 if a facility is located at candidate site jX j and 0 otherwise. Objective: Min ∑∈Wj jj Xf Constraints: 1≥∑∈Wj jij Xa Vi ∈ Figure 8 Set Covering Facility Location Model (SCFLM) Here is a general matrix view of the model. This representation includes the bounds on the decision variable (last row), and the slack and surplus variables for all inequality constraints (columns 3 and 4). 1 jf− ija -I 1 I I 1 Figure 9 Matrix View of SCFLM 6
  8. 8. Data Model and Format The data model for the prototype was designed to accommodate both models, and variations of the two models. The semantics of the data models are defined below. However, the prototype does not implement all of the defined properties. Configuration( r , )v The configuration table holds information about how to generate an instance of the Maximum Covering Location Problem, the Set Covering Facility Location Model, or variations of these two models. The information come in pairs: resource name ( r ) and resource value ( ). Here is an example configuration table: v modelInstance [MCLM, SCFLM] multiObj [Y,N] exemptCoverage [Y,N,I] nFacToLocate <integer> demand <filename> facility <filename> demandFacility <filename> modelInstance is the model instance to be generated by the system: MCLM is for Maximum Covering Location Model, and SCFLM is for the Set Covering Facility Location Model. multiObj indicates to the system that the objective functions of the MCLM and the SCFLM models should be used simultaneously; in other words, a multiple objective optimization should be performed. exemptCoverage indicates to the system that 1) it is acceptable (Y) for the solution violate some coverage constraint but not for others – in this case, the decision variable in the demand file should be interpreted properly: some will be specified by the user, and some assigned by the system, 2) it is unacceptable (N) for the solution to violate the coverage constraint – in this case, for all demand source should be set to 1 by the system, or 3) this should be ignore (I). iZ iZ nFacToLocate is the number of facilities to locate, P . Demand( , , )i ih iZ i is the index to the demand source. ih is the demand from source ; ignore this if modelInstance is SCFLM and multiObj is N.i iZ is a decision variable – it is 1 if the source demand has been covered, and 0 otherwise. If exemptCoverage is Y and modelInstance is MCLM, the system should 1) overwrite the value of an attribute with the solution if the original attribute value is -1, and 2) consider the decision variable fixed by the user if the attribute value is 1. If exemptCoverage is N, the attribute should be assigned the value 1 and the model instance generated accordingly. 7
  9. 9. DemandFacility( ,i j , )ija i is the index to the demand source. j is the index to the candidate site. ija is the indicator coefficient for the demand source i and candidate site j - it is 1 if the candidate site can cover the demand of the demand source, and 0 otherwise. Facility( j , , )jf jX j is the index for the candidate site. jf is the cost of locating a facility at the candidate site. jX is a decision variable – it is 1 if the system decides to locate a facility at the candidate site, and 0 otherwise. 8
  10. 10. 9 References Brown JD, Heuvelink BM. 2008. On the identification of uncertainties in spatial data. In Wilson JP, Fotheringham AS, editors. The handbook of Geographic Information Science. Malden (MA); Oxford (Oxford, UK); Carlton (Victoria, Australia): Blackwell Publishing Ltd. p. 94 – 107. Daskin MS. 1995. Network and discrete location: models, algorithms, and applications. New York (NY): John Wiley & Sons, Inc. Fotheringham AS, Densham PJ, Curtis A. 1995. The zone definition problem in location-allocation modeling. Geographical Analysis, 27(1), pp. 60-77. Garfinkel RS. 1985. Motivation and modeling. In: Lawler EL, Lenstra JK, Rinooy Kan AHG, Shmoys DB, editors. The traveling salesman problem. Chichester (UK); New York (NY), Brisbane (Queensland, Australia), Toronto (ON), Singapore: John Wiley & Sons. p. 17-36. Williams HP. 1999. Model building in mathematical programming (4th ed.). New York (NY): John Wiley & Sons, Inc.

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