VisOpt: A Visual Interactive Optimization Tool for PMedian Problems
Hasan Pirkul1, Rakesh Gupta2, Erik Rolland3.
1
School of Management, The University of Texas at Dallas, Richardson, Texas. email:
[email protected]
2
Department of Management, Oklahoma State University, Stillwater Ok, 74078 email:
[email protected]
3
Department of Accounting and MIS, The Ohio State University, Columbus, Ohio 43210 email:
[email protected]
Abstract. In this paper we describe a visual interactive decision support tool ‘VisOpt’ which is designed to solve P-median problems with capacity constraints. Various design features incorporated in VisOpt are also presented and analyzed. We also present a demonstration of the use of VisOpt by a number of human subjects for various problem instances. The quality of solutions obtained by subjects using VisOpt is compared with that obtained from a standard stand alone heuristic. The visual interactive tool provides encouraging results in this specific problem context.
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It is well known that humans are better at processing graphical rather than numerical information; computers are just the opposite [14]. Human experts possess the ability to recognize complex patterns quickly and call upon a large number of heuristics that would help them solve complex problems. However they are constrained by their limited numerical processing capabilities [20][21]. What is needed therefore, are systems wherein the abilities of humans are complemented by the number crunching capabilities of computers so as to serve as an integrated human processor [4]. Still, there exists a fundamental mismatch in communication that must be resolved by utilizing a mode of communication that is best understood by a human decisionmaker.
In this paper we describe a decision support tool “VisOpt” that aids a user in solving a capacitated P-median problem [18]. We also demonstrate the usefulness of VisOpt by utilizing it to solve a number of problem instances. VisOpt presents the user with a graphical representation of the problem and allows easy manipulation of solution characteristics through a point and click approach. By carrying out most of the combinatorial number manipulation involved, the system extends the decision maker’s information processing capabilities while still providing him/her with a gestalt representation in graphical form. Through the use of different colors, shapes and sizes, the system visually cues the user towards making good decision choices, and allows him/her to come up with sequences of decisions that may provide better global solutions.
The P-median problem we address can be considered a structured one in the following sense: all the problem relevant data is known, and the solution procedure (or procedures) is also known. However, due to the combinatorial complexity of the task, solving a realistic sized
instance of the P-median problem to optimality is relatively difficult and time consuming. We have chosen this one problem due to a number of reasons. Firstly, the P-median problem is conceptually a simple one that has been widely studied in the literature. Secondly, the task of selecting P-facilities in a geographical context offers an intuitive and simple spatial representation that can be easily comprehended visually by a decision-maker. The emphasis here is not on solving this one problem, but rather on demonstrating that such systems utilizing human reasoning and pattern recognition abilities, can be developed to successfully tackle mathematical programming problems that can be mapped to meaningful topologies.
Our contributions from this research are the following: we develop a visual interactive decision support tool, which is designed to solve spatial location problems. We compare the performance of users with our tool to that of a stand-alone heuristic, which performs the same task so as to demonstrate the potential usefulness of our tool. Further, we make some general inferences about using such a visual interactive decision support tool, for example are there any problem specific characteristics that make such an approach appropriate.
This paper is structured as follows: We provide a summary of past literature in section 2. Section 3 introduces the problem and discusses VisOpt. Section 4 analyzes the decision task and evaluates different user heuristics. Sections 5 and 6 describe a demonstration of the system in use and discuss findings. Finally section 7 summarizes and concludes the paper and outlines the scope for future research.
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2. Background
A number of researchers have evaluated the use of computer graphics in decision support tools. In general it has been accepted that a graphical method of representing information or solutions improves the performance, understanding, solution quality for users in a decision making task [25][15][17][3]. In addition, due to the immense capabilities afforded by the computer hardware and software of today, visual interactive interfaces for decision support systems present a rapidly expanding area of research. In a comprehensive review of commercial facility location software, Ballou and Masters [1] emphasize the importance that users place on graphics capabilities and user-friendliness. The same was reported by a more recent survey of decision support applications by Eom et al. [9].
While a large number of visual interactive decision support tools have been presented by researchers in the past. For brevity we focus primarily on systems that either deal with transportation or communication networks/systems or those that address optimization related issues similar to our own.
Problems in graph theory lend themselves readily to a visual representation. As early as 1986, Dao, Habib, Richard and Tallot [6] developed a system that allows its users to define and solve a number of graph theoretic problems in an object-oriented fashion. In fact, due to the increasing importance of graphical interfaces in OR, a large number of systems have appeared in the recent past that deal with graphical analysis of OR problems. EDINET [23] is a network editor intended for easy display and change of network attributes. NETPAD [7] is another
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network modeler and optimizer that has an intuitive user interface. GIN [27] is another graph based interface for network modeling. Tracey et al. [28] present an interactive computer application for graphically presenting the feed distribution solution for cattle feed ranches. Similarly, Jack, Kai and Shulman [11] present NETCAP, which is an interactive optimization system for solving a large multi-period capacity expansion problem for telephone networks.
Perhaps closest to our own research efforts are those of Hurrion [11], Mak, Srikanth and Morton [19], Scriabin and Vergin [26] and Krolak, Felts and Nelson [16]. Hurrion [11] describes a visual interactive method of improving solutions for the Traveling Salesman Problem (T.S.P). By using a simple visual tool (utilized primarily for displaying solutions), the author reported solutions that were within 4% of those obtained from a heuristic. Mak, Srikanth and Morton [19] provide experimental evidence showing that for the same problem (T.S.P.), visual interactive solutions achieved by humans, are better than even the best computer heuristics. On the task of plant layout design, Scriabin and Vergin [26] showed that human’s without the benefit of any prescriptive help from a computer, can create layouts that are better than those obtained from computer programs. Similar results were reported by Trybus and Hopkins [29] for the same problem. Krolak, Felts and Nelson [16] utilized a man-machine approach to solving the Generalized Truck-Dispatching problem and provided the user with a visual representation of intermediate solutions and scenarios. The user was able to interactively change the solution and apply various heuristics to completely solve the problem. By following this methodology the authors reported that a combined man-machine tool resulted in better solutions than those from computer programs. Similarly Fisher [10], proposed an integrated human machine tool wherein a human could intervene at points during the execution of an optimization algorithm and by utilizing
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his/her own perceptual abilities make changes that the algorithm by itself would not be able to make. Jones [13] in a comprehensive review of visualization and related tools in O.R., points out that problem solving requires more than just algorithm building, it also consists of a process of transforming and understanding various representations or visualizations of the problem in question.
Our efforts are different from the prior research in this area due to a number of reasons. VisOpt is designed to provide visual cues to the decision maker which are aimed at informing the user of what might be possible avenues of action. Further, the system is designed so as to avoid restricting the user in his/her actions. For instance we allow the user to create infeasible solutions (i.e. solutions violating capacity constraints for example) since further modifications of such solutions may result in better feasible solutions that those attained so far. This mitigates the system restrictiveness posited by Chu and Elam [5] and provides a method for the user to escape local minima – a difficulty faced by most heuristic procedures for combinatorial problems.
3. The Capacitated P-median Problem and Visual Support for Solving it:
The Capacitated P-Median problem can be stated as follows: given a graph G=(V, E), V1⊂ V, where V1 is the set of potential facility sites with capacities C, V is the set of demand nodes with a demand vector A, we are asked to find a set of nodes S (S ⊆V1), of cardinality P, such as the weighted sum of assigning the nodes V to the set S is minimized while all demand is satisfied without violating capacity constraints. A mathematical formulation of the problem is provided in the appendix . For the problems considered, the costs on the edges are assumed to be
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proportional to the euclidian distance between sites. Problems with non-Euclidean costs pose additional difficulties (in being mapped to a visual representation) which are beyond the scope of this paper. These issues are currently being studied as an extension of this research.
P-Median problems have applications in plant and warehouse locations, where the objective is to find a configuration of facilities that best serves the demands of a market area. Applications can also be found in public sector location modeling, where it is required to locate schools, hospitals, post offices etc.
According to the description given above, the problem poses the following constraints on a decision-maker: 1. Only a predetermined number (P) of facilities can be opened (or located). 2. Every demand point must be supplied. 3. A facility can supply only a predetermined amount of goods/services.
The decision-maker needs to be informed about each of the above criteria and also any violations to them. Further, based on our discussion in the previous section and our intention to create a visual interactive decision making environment, our design had to include features that:
1. Provide the user with a consistent, visual representation of reality minimizing the use of numbers and textual information. 2. Create a simple dynamic representation that displays various parameters as the user evolves her/his solution.
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3. Let the system carry out tedious tasks like sorting/searching etc. in a fast and transparent (to the user) manner, while still allowing the user to make any changes using his/her own heuristics. 4. Visually indicate relevant features of the problem instance (e.g. Cost, Demand needed, Capacity available etc.). Also, indicate to the user, any violations in constraints and possible ways to remove them. 5. Provide an environment in which the user can explore “what if “ solutions, through intuitive and obvious steps (ex. pointing and clicking on graphic elements). 6. Display appropriate feedback to the user in a consistent form at all times.
3.1 Design Features:
The following design principles were followed in developing the decision support system:
1. Provide a representation that is consistent with reality, preserving a mapping from real life to its visual model. To accomplish this, the two dimensional coordinates of each facility and demand site are mapped on to the screen. Since costs on arcs are proportional to transportation costs, a demand node close to a facility would be cheaper to supply as opposed to one that is far away. 2. The size of each node is displayed proportional to the number of units demanded by it. Therefore, a large demand node as shown in Figure 1 indicates a high demand. This seems consistent with the intuitive notion of “big” meaning “more”.
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3. Networks are displayed without any connecting arcs unless the arcs are demanded by the user. Unless required, identifying numbers for nodes are also not displayed, instead different colors are utilized to identify and group together nodes connected to facilities. 4. Demand nodes are displayed as circles, facilities have a square superimposed over this circle and are colored red.
Insert Figure 1. about here.
5. Since each facility has a preset capacity, usage is indicated by a thermometer like object displayed at the bottom right corner of the node. As capacity usage increases, the "level of mercury" in this thermometer rises. If capacity is exceeded the color of the “mercury” which is ordinarily green (while capacity limit is not exceeded), turns red. An instance of a facility exceeding its capacity is seen in Figure 2. Once a facility exceeds its preset capacity, some of the demand nodes it supplies need to be re-assigned. A node, which if reassigned would bring the facility to within capacity (a critical node), is displayed with a dark border.
Insert Figure 2. about here.
6. To provide a dynamic representation, the system performs a simple heuristic to assign demand nodes to facilities. This heuristic is executed each time a facility is opened or closed and is described in the next section. 7. Feedback on the cost, feasibility and other related statistics are provided to the user in a multiline status/message window at the bottom of the screen.
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8. What-if questions can be answered by simply clicking on various visual elements. For instance by clicking on a demand point and then on a facility – the system responds with the cost of supplying the demand node with the facility, the new value of the total cost and infeasibilities (if any) that result. Similarly, by clicking on two facility sites (the first open and the second closed) the system responds with the resulting objective function value if the first facility were closed and the second opened.
3.2 Heuristic used to assign demand nodes to facilities
VisOpt utilizes a simple heuristic to connect demand nodes to a facility that has been opened. It must be kept in mind that this heuristic has been chosen so as to be (1) fast, to minimize waiting time for a user, (2) transparent to the user. The second point is important so as to avoid a “ gulf of execution” [22] i.e. so that the user fully understands what would happen each time he/she opens or closes a facility. Through the use of such a heuristic the system provides a fast and intuitive way of connecting facility nodes to demand points. Thus the user need not tediously connect every demand point to a facility. Further, such a heuristic provides the user with a convenient means of getting to a feasible solution (a starting point) to the problem. The heuristic (called DemConnect) proceeds as follows: We create a list of demand weighted transportation costs of connecting each demand point with every facility that is open. This list is sorted in ascending order of cost. Next the cheapest elements from this list are assigned to the facilities until either the available capacity at the facilities is exhausted or no unsupplied demand nodes remain.
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4. Task Analysis:
An analysis of the major steps needed to be carried out in solving the P-median problem is presented in this section. A description of the support provided by the computer system in this context is also included.
4.1 Problem Solving Methodology:
The major steps that need to be carried out to get a feasible solution are as follows: 1.
Evaluate and open facilities,
2.
Evaluate and modify the connections (demand nodes to facilities) provided by the system (i.e. provided by heuristic DemConnect),
3.
Evaluate and drop (close) facilities in favor of cheaper ones,
4.
If a satisfactory solution is not attained then if more facilities can be opened go to 1 else go to 2. Else stop.
It is significant that in the above task analysis we have used a generic term “Evaluate” to describe the users choice of : (a) which facility to open, (b) which facility to connect to a demand node and (c) which facility to drop from a set of open facilities and conversely which facility to add to the same. The reason for this is that a human may choose to use one or more, of many “inbuilt” heuristics to carry out each of (a)-(c). Further, the steps listed above are ones which will
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have to be carried out irrespective of which higher level strategy the user may follow (for example greedy in number of facilities, divide and conquer etc.). A fundamental limitation of stand-alone computer based methods is that each algorithm is usually good for problems of a particular structure. There exist few general-purpose algorithms that will do well in all situations. Humans on the other hand, can call upon multiple heuristics to solve a problem, given that there exists some recognizable pattern in its structure. A description of some of the possible “rules of thumb” is presented in appendix 2. It must be stressed that this list is by no means exhaustive. One of the major advantages of this methodology is that a decision-maker can choose any strategy or rule that he/she thinks appropriate. Thus “system restrictiveness” [5] is mitigated.
5. Experimental Demonstration:
The purpose of this demonstration was to attempt to characterize the effectiveness of our visual decision support system. Ideally, the user of a DSS like ours would be an “expert” in that area (or at the very least, a person familiar with the structure of the problem). The participants for the experiment were therefore selected from an undergraduate Decision sciences/Operations Research techniques course in a major Midwestern university. The majority of the students were juniors opting for a major in Business. These students were familiar with the use of linear programming techniques in the context of location problems. As an incentive, each subject was assigned an extra credit score proportional to the quality of the solution obtained by her/him. The scores and scoring scheme was explained to the students before the beginning of the experiment. The experiment was carried out on IBM Compatible microcomputers running MS Windows.
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The solution quality obtained from subjects was compared with that achieved by a previously published heuristic due to Pirkul and Shilling [24]. In addition, time taken to reach a satisficing solution was observed for each of the subjects. Subjects were randomly assigned to one of six experimental sessions over a period of one week. To avoid any biases due to extraneous factors between sessions, subjects within each session were randomly assigned one of the following three problems:
Type 1.
An 88 node problem with actual data, 12 potential facilities, 5 facilities to open.
This problem uses data obtained from the ArcUSA database incorporating information from the US census of 1991. The centroid of each county (in Ohio) was chosen as a demand point. The demand at each site was based on the population of the county. Distances (serving as a proxy for cost on arcs) between each point are actual euclidean distances in miles. Counties with population greater than 200,000 are designated as potential facilities. Type 2.
A random 88 node problem, 12 potential facilities, 5 facilities to open. With data
based on points randomly generated on a square of side 1000 units. Demand at each point was randomly generated over a uniform distribution (0, 200). Costs along arcs were equal to the euclidean distance between the points generated.
Type 3.
A random 49 node problem, 17 potential facilities, 8 to open. This problem was
generated in the same manner as Type 2.
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Each session began with an hour-long training period, during which subjects were asked to solve a randomly generated 68 node problem (the same problem was solved by each subject). Subjects were encouraged to ask any questions regarding the program and experiment with it so that they understood its functionality. At the end of this training period each subject was given 30 minutes to solve one of the problems mentioned above. The objective for each subject was to arrive at a satisficing solution for each of these problems within that time.
As mentioned earlier, the primary variable of interest was the objective function value obtained by each subject. In addition to this value, the system also kept a log of the time taken by each subject to reach this solution. A post-experimental questionnaire was utilized to keep track of demographic information. The questionnaire also required the students to answer some elementary questions about the DSS as well as the P-median problem.
6. Discussion of Demonstration Results:
Results from the experiment are shown in tables I and II. We were interested in answering the following question: Can human subjects, with the aid of a visual problem solving system like ours, effectively solve a complex mathematical problem ? In this section, we compare the solutions attained by humans with those obtained by a previously published heuristic due to Pirkul and Shilling [24].
Insert Figure 3. about here.
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Figure 3. shows the average gap obtained from the experiment when subjects are stratified into categories representing the best 20, 40, 60, 80 and 100%. Interestingly enough, the average Zexp obtained for problem type 1 is better (i.e. less) than Zh (for 19 out of 25 subjects). The very opposite occurs in the case of problems of type 2 and 3, for both of these problems, all subjects performed worse than the heuristic. However as shown in Figure 3, on the average subjects achieved objective function values within 4.5% of the heuristic (for type 3) and 2% for problem type 2. Between the random problems (type 2 and 3), there was a difference due to problem size, type 3 subjects perform worse than type 2. Keeping in mind that the type 3 problem is actually a smaller network this result seems at first counter intuitive. A plausible explanation for this result may be that since the number of potential facility sites in type 3 (17 potential facilities) is greater than type 2 (12 potential facilities), subjects had more choices to make and therefore made suboptimal choices.
Insert Figure 4 (a, b and c) about here.
Could there be any specific reasons why subjects consistently perform better for the problem Type 1 vs. 2 and 3 ? An analysis of the problems themselves could help explain this disparity in problem solving abilities. Figure 4(a, b and c) shows the topologies of the three different problems as presented to subjects. From a comparison of Figure 4 (a, b and c) one can easily perceive that the type 1 problem displays a “skewed” distribution in that there are at least three facilities (Cuyahoga, Franklin and Hamilton) that have extremely large demands. Thus a simple application of rule 1 (see appendix 2) would result in choosing these facilities. Further, both Lucas and Franklin counties are lone counties with a large number of small demand nodes
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surrounding them. It seems reasonable to assume, that an informed user would choose to locate facilities at these sites and supply the nodes around them. The other two problems (Figures 4 (b) and (c) ) display no such patterns, and for a good reason: Both problems (type 2 and 3) were generated over a uniform distribution (a square of side 1000). However, the first problem (type 1) represents the actual location of the centroid of each county in Ohio (with demands proportional to the population of each county). Thus, large population centers such as Columbus (Franklin county) and Cleveland (Cuyahoga) would be expected to be surrounded by smaller (in population) counties.
Insert Table II about here.
It seems curious that the heuristic [24] in question seems to perform relatively poorly for problem type 1 as compared to problem type 2. We therefore solved each of the above problems using CPLEX (a Mixed Integer LP package) running on a Sun Sparc-10 (porting SunOS -BSD 4.3.1). Results from this analysis are shown in table III.
Analysis of the time taken to reach the optimal solution (table III) by CPLEX underlines the fact that problem type 1 is a much more difficult one to solve optimally. Specifically it takes over 14 times as much CPU time to solve problem type 1 as compared to problem type 2, even though both have the same number of demand and facility nodes. One of the reasons for this is the fact that the type 1 problem exhibits symmetry, which is known to cause difficulties for traditional optimization procedures like branch and bound [2]. That is, from the topology of the type 1 network one can see that there is very little variance between the size of demand among a large
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number of demand points. For a heuristic which is blind to this fact, choosing to supply one demand point would seem as attractive as choosing to supply another. Therefore a heuristic or a branch and bound procedure (such as that used by CPLEX) would expend significant amounts of computational time comparing such nearly equivalent choices. It is interesting to note, that it is precisely this kind of problem (type 1) that the human subjects solved with relative ease.
Solution times for human subjects ranged from 21-26 minutes as compared to the more than 1 hour taken by CPLEX. From table III it is clear that the comparison heuristic finds relatively good solutions for both of the random problems (Type 2 and 3) and leaves very little room for improvement (gaps of 0.072 and 0.31 % respectively). The only difference between these problems and the Type 1 problem is the network topology, which we emphasized above. Based on this, we conjecture that actual geographic and demographic data contains recognizable (by humans) patterns that can help humans make effective decisions. Further, the presence of symmetry may actually work as an advantage for human decision-makers. While the comparison heuristic lacks the ability to visualize and make pattern matched inferences for the “skewed” distribution in the type 1 problem, humans by combining their own ability to identify and utilize such patterns with those of a computer (in a visual interactive manner) seem to make relatively good decisions. It may be possible that a heuristic other than that due to Pirkul and Shilling may result in better solutions for the type 1. problem, however it is unlikely that a simple (for example add/drop or greedy) heuristic would be able to outperform this particular one.
7. Conclusions and Directions for Future Research:
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In this paper, we have introduced the idea of a computer based decision support tool (VisOpt) that can be used in conjunction with its decision-maker to solve a structured location problem. VisOpt is based on the fundamental notion of visual (as opposed to textual) information being easier to comprehend by a human decision-maker. The importance of problem specific criteria and their mapping into a visual model was also presented. Finally, details of an experimental demonstration of the effectiveness of such a visual interactive decision making paradigm were presented and discussed.
The main contribution of this research effort lies in the characterization of humans as decision makers in the context of structured problems. Until now, humans have always been considered superior in solving decision problems that are unstructured. On the other hand, structured problems seem to be more effectively solved by stand alone computer programs since for the most part such problems require intense computational capabilities. Our experimental results indicate that the pattern recognition capabilities of humans can be tapped and utilized advantageously for problems that present a spatial mapping of information. Further, the results indicate that actual geographic and demographic data may carry spatial patterns utilizable by humans. Conceptually, the ability to recognize such patterns can be hardcoded into a series of “if - then” rules. However, most humans already carry such heuristics encoded in their minds. Surely, it would be most advantageous if a human decision-maker could take advantage of these rules and use them in conjunction with a DSS to help carry out a problem solving task.
In this effort we have considered a variant of the P-median problem, a problem that is well known and researched. However, the main advantage of an approach of this kind would lie in
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solving problems that have constraints that are difficult to model, or if modeled are difficult to solve. This is because, in a cooperative problem-solving paradigm humans could help enforce the difficult constraints while the computing system limits itself to best satisfying the easier constraints. Thus, this fundamental division of labor could result in better solutions on the whole. Another interesting prospect is utilizing an established heuristic to achieve an initial solution which users could then attempt to improve. Our research therefore represents a small step towards applying a similar problem solving approach to more complex and difficult to solve problems.
Based upon the preceding discussion, a fundamental question arises: Could structured problems always be solved in such a cooperative fashion ? The answer to this question is likely to be no, because as we have stressed before, there needs to exist some convenient mapping from the problem data to a visual model. Further, this visual information must contain recognizable (for humans) patterns so that a human can make effective decisions. The problem we have analyzed does contain such information, but at this point we would hesitate in generalizing these results to any other problem context. More research is needed to effectively answer this question. What we can say is that, we have demonstrated the potential of a combined human-machine problemsolving paradigm, in solving a mathematical programming problem. Thus, human capabilities should not be overlooked in solving such problems.
Decision support systems are fundamentally built upon the concept of human-machine cooperation. However, their use is most effective if the human decision-maker is least restricted in his/her actions. VisOpt was designed with the intent of minimizing such restrictiveness. Within
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the context of a combined human-machine problem-solving paradigm, what is needed is a conceptual framework indicating which tasks to be carried out by which party (human or machine). Similarly, it is important to investigate the best ways to represent information that is non-spatial (financial, temporal) through visual cues. Currently we are engaged in research aimed to provide insights to these and other issues.
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Appendix 1:
Mathematical formulation of P-median problem:
The Capacitated P-Median problem can be stated mathematically as follows:
Cap. P-Median:
∑ ∑
Min
i
∑x
ij = 1
aidijx ij
(1)
j
∀i
(2)
∀i,j
(3)
j
x ij ≤ y j
∑
yj = P
(4)
j
∑ ax
i ij
≤ cj ∀ j
(5)
i
x ij,yj ∈ {0,1} ∀ i,j
(6)
where
ai = demand at node i cj = capacity at facility site j dij = distance from node i to node j P = preset number of facility nodes xij = 1 if node i is assigned to facility j 0 otherwise yj = 1 if facility j is open 0 otherwise The objective function (1) minimizes the weighted sum of assigning demand nodes to facility sites. Constraint set (2) ensures that all demand nodes are assigned to exactly one facility. In constraint (3) we ensure that a node is assigned to a facility only if it is open. The third
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constraint (4) enforces the fact that exactly P facilities are being opened. In constraint set (5) we ensure that the demands of all nodes assigned to a facility are within its capacity. Finally, the integrality of the decision variables are enforced by constraint set (6). A heuristic solution procedure suggested by Pirkul and Schilling [24] was utilized to get a feasible solution.
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Appendix 2:
Sample User Heuristics:
(a) Choosing which facility to open:
Rule 1: “Choose to open a facility that has a large demand over one that has a small demand.”
Insert Figure 5. about here.
Rationale: In the context of this problem, the costs being minimized are transportation costs, since a facility supplying demand at the same node incurs no transportation costs, it is advantageous to open a facility in a node with a large demand over one with a small demand. Hence as shown in Figure 5 choose facility A over B, C, and D.
Rule 2: “Choose to open a facility that has a large number of demand nodes “clustered” around it, over one that has no such clustering around it. “ Rationale: Once again since transportation costs are being minimized, it may be preferable to locate a facility that is central to a large number of demand nodes. For the configuration shown in Figure 6(a), both Rules 1 and 2 would suggest opening facility A rather than facility B. In Figure 6(b) while one user utilizing rule 1, might choose to open facility B, another user may choose to open facility A because of rule 2.
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Insert Figure 6. about here.
(b) Choosing which facility to supply a demand node from:
Insert Figure 7. about here.
Rule 3: “Keep connecting arcs as short as possible.” Rationale: Keeping with the objective of minimizing transportation costs, a user might wish to connect demand points such that long transportation links are avoided. A look at Figure 7 clearly indicates the effectiveness of such a strategy, following this rule a user might wish to connect the marked demand nodes (enclosed in the box4) to facility A instead of facility B.
Insert Figure 8(a) about here.
Rule 4: “Allow some close (with regards to distance on screen) demand nodes to be connected to far away facilities so that some other demand nodes can be supplied. “ Rationale: An application of this rule can be seen from Figures 8 (a) and (b). After applying rule 3 on the network in Figure 7, the result is an infeasible solution (see Figure 8(a)). Through an application of rule 4 to the nodes enclosed in boxes in Figure 8(a), the result is a feasible and improved (with regards to cost) solution (Figure 8(b)). Note, that the marked demand nodes in
Insert Figure 8(b) about here.
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Figure 8(a) are actually closest to facility A, however the user chose to supply them from facility C so that the overall solution is feasible and improved.
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Legends to Figures: Figure 1. Demand and Facility nodes with progressively large demand requirements Figure 2. Infeasible solution to a 5-median problem Figure 3. Average difference in objective function (experimental vs. heuristic) values for each group stratified by performance within group. Figure 4(a). Topology of problem of type 1. Figure 4(b). Topology of problem of type 2. Figure 4(c). Topology of problem of type 3. Figure 5. Choosing a facility based on its demand Figure 6(a) . Application of Rule 2: Choosing a facility based on demand node “clustering”. Figure 6(b) . Application of Rule 2: Choosing Rule 2 over Rule 1. Figure 7. Rule 3: “Eliminate long arcs” (for nodes in box) Figure 8 (a). Modifications to network in Figure 7: Result of applying Rule 3. to network in Figure 7.; Now, Apply Rule 4 to marked nodes (in boxes). Figure 8 (b). Further modifications to network in Figure 7: After applying Rule 4; Note improvement in cost.
30
Figures:
Facility nodes
Demand nodes Small
Medium
Large
Figure 1.
Indicates capacity available
Critical Nodes
Status Window Exceeding capacity
Figure 2.
31
4 3 (Zexp - Zh)/Zh
Average difference (%) with
solution obtained from heuristic
5
2 1 0 -1
20%
40%
60%
80%
100%
-2 Ty p e 1
-3 -4
Ty p e 2 Su b j e c t C a t e g o rie s ( t o p 2 0 , 4 0 6 0 , 8 0 , 1 0 0 % )
Figure 3.
Figure 4(a).
Figure 4(b).
32
Ty p e 3
Figure 4(c).
D A
C B
Figure 5.
33
B
A
Figure 6(a).
B
A
Figure 6(b).
34
C
B
A
Figure 7.
C
B
A
Figure 8 (a).
35
C
B
A
Figure 8 (b).
36
Tables:
Type
Type 1 Type 2. Type 3.
Number of Nodes
88 88 49
Potential facilities
12 12 17
Number of Subjects
25 22 25
Mean Objective Fn. (Zexp) 458407 1299807 852564
Median Obj. Fn.
456419 1290849 852131
Std. Dev.
Objective
Function
Value
8045 34431 26500
Best 448629 1271799 819432
Worst 472363 1401156 902994
Best 80% Subjects 454984 1286568 845820
Table I. Descriptive statistics of demonstration.
37
Problem
Type 1. Type 2. Type 3.
Number of Subjects 25 22 25
Number of subjects with Zexp
