MULTICRITERIA DECISION MAKING WITH SAS SOFTWARE Mary G. Crissey, US Air Force

Transcription

1 MULTICRITERIA DECISION MAKING WITH SAS SOFTWARE Mary G. Crissey, US Air Force INTRODUCTION This study will show how Multicriteria decisions can be modeled by expanding the traditional linear programming approach available with the SAS/oR product. Most Operations Research analysts are familiar with modeling real life situations mathematically by specifying required constraints and then searching for an objecti~e solution which maximizes an overall goal. The SAS procedure LP is designed to find the best answer to a confusing decision involving a single objective goal and several constraining conditions. But what can be done to support decisions involving multicriteria dimensions in both the domain and objective functions? What if there is not one single goal but rather several functions which all need to be optimized together? Linear programming techniques can not handle these multidimensional problems. A common practice in this situation is to apply Goal Progr~ing. Several new theories and user friendly application packages have recently been developed. Both the analysts and the high level decision makers are finding acceptable balances between their goals by satisfactorily trading off between the conflicting objectives. My study presents a few of these new Operation Research interactive algorithms and illustrates how SAS software may be applied to real world multicriteria problems today. ABSTRACT In today's world, many problems can not be optimized via a traditional linear program such as provided with SAS/oR software via PROC LP. Recently in the area of multi-criteria and multi-objective modeling many new algorithms have been designed to aid decision makers who deal with real world problems. The unique Interactive Sequential Goal Programming (ISGP) procedure was implemented in SAS to help military enlisted force mangers face many competing goals across multi-year time horizons. This paper shows how the basic ISGP algorithm can be applied in a generic sense with other SAS databases. A short summary of the ISGP method will be presented along with an interesting comparison with two other popular decision support software products; AHP and GAMS. Using a simple diet problem to illustrate the strengths and weakness of the ISGP algorithm the problem will also be solved via 'the Analytic Hierarchy Process and then with the Generalized Algebraic Modeling System. These new techniques promise to improve the decision makers capability for making wiser trade-offs among the conflicting objectives in the real world today. THE SAMPLE PROBLEM The decision to be made using the three different Operations Research approaches is " How to choose a balanced diet from among a group of six foods: milk meat, eggs, bread, lettuce salad and ~range juice?" There is a limit on the amount of each food that can be included in the diet, and the diet must meet certain minimum standards for number of calories, milligrams (mg) of iron, units of Vitamin A, and grams (g) of protein. In this diet problem, three separate goals are considered: 1. Maximize carbohydrates in the diet. 2. Minimize the cholesterol content 3. Minimize the cost The use of the words maximize and minimize must be explained further. They are used here to indicate the direction of preference for these goals. For example, given the constraints of meeting the minimum nutritional standards, the diet should contain as much carbohydrates ad possible, as little cholesterol as possible, and should be as inexpensive as possible. Table 1 presents the data for this problem. The first six columns are the food types, while the rows give the characteristics of those foods -- the nutritional content, cost, and maximum availability. The last column gives the minimum daily nutritional allowances. We have therefore identified three goals and four daily allowance constraints. The mathematical summary of the problem is given in Exhibit

2 EXHIBIT 1 DATA FOR SAMPLE DIET PROBLEM Lettuce Orange Dally MIlk Meat Eggs Bread Salad Juice Allowance Cpt) Ubi (dozen) (oz) (oz) Iptl!Adult! VltamlnA S (I.u.) Calories S SOO Cholesterol (units). Protein (g) IS lsi 7S 2.S Carbohydrate (g) IS 1.1 S2 Iron (mg) S O.IS S Cost ($) O.S 0.1 O.OS 0.26 Availability S EXHIBIT 2 MATHEMATICAL FORMULATION OF DIET PROBLEM (Carbohydrate Intake) (1) min f2~ = «lx, + 20x" + 120xa min fa~ = 0.22x, + 2.2>< x ,,< + O.OS", x,; (Cholesterol Intake) (2) (Cost) (3) subject to: 72Ox, + 107"" xa + 134x,; + loo<lxs> SOOO (VItamin A requirement) (4) 0.2x, "" xa x x,; + 1.2x,; > 12.S (Iron requirement) (S) 344x, + 146Ox 2 + lo4ox a + 7S,,< x,; + 24% > 2SOO 18x, + lsi"" + 7Sxa + 2.5" x5 + 4x,; > 63 (Calories) (6) (Protein) (7) 0< l!: < ( S ) (Maxinlum Intake) (S). 472

3 ISGP METHOD The Interactive Sequential Goal Programming procedure used in this study is based on the 1988 version of Ching-Lai Hwang and Abu Syed Masud's algorithm. The steps of the ISGP Process is given i'n exhibit 3. Due to the constraint on the length of this paper, I can not explain the theory of ISGP here. Basically I'd like to point out that the key to this process lies in a two phase procedure involving several several batch jobs and interactive choices by the decision makers. The ISGP process was implemented two separate ways. The diet problem was first solved on an IBM MVS mainframe using coding done with SAS~OR software. The second application of the ISGP process to the diet problem was done on a PC running GAMS software. Comparing the two different linear programming environments was helpful in validating the new ISGP process and helped to set benchmarks for the diet solution. Actual SAS coding of the ISGP process is problem specific and involve several PROe LP formulations. For more detailed information on how SAS/OR can be used to model the ISGP algorithm please consult a copy of my previous SUGI 15 paper presentation entitled "A MultiCriteria Decision Support System for Military Force Management". GAMS FORMULATION GAMS, the Generalized Algebraic Modeling System, was run on a Personal Computer. This software provided a high level language to represent the ISGP process with. Since GAMS takes advantage of the problem structure and uses mathematical formulations analysts find their descriptive modeling task is greatly simplified. GAMS was fully suitable for a step by step demonstration of the ISGP routine. The GAMS results for Phase one of the ISGP process were identical to those generated by SA~OR using PROC LP to devise the Positive and Negative solutions for each of the three goals. Five iteratio~s ~ere required to obtain the MAG-MAR solution at 78%. ThE SAS/OR and GAMS solutions are identical until the final iteration. When B=.77, a dominated solution is obtained (the objective value is 0.0). The GAMS solution meets the goals for cholesterol and cost exactly, and uses units of lettuce salad. The goal value of for carbohydrates is exceeded by In contrast, the SAS/OR solution uses only units of lettuce salad and exactly meets the carbohydrate goal. It achieves a cost of only compared to a goal of Either solution meets or exceeds every goal and meets all constraints; thus the solutions are dominated. It is clear that the solutions are not necessarily unique, as was the case with the positive and negative ideal solutions. Phase II of the Interactive Sequential Goal Programming process continues as the decision makers chose their desired diets in this step wise manner. Both the GAMS and the SAS/OR software resulted in the same result. SAMPLE CHOSEN DIET The ISGP process resulted in the following diet as the answer to this multicriteria problem: Milk = 3.00, No meat, No eggs, Bread=10.00, Lettuce = 8.00, and Orange Juice = THE ANALYTIC HIERARCHY PROCESS The Analytic Hierarchy Process (AHP), developed by Dr Thomas L. Saaty provides decision makers with a structural approach to evaluation of alternatives, based on a hierarchical structure of criteria. Expert Choice, the commercially available AHP software was used in this study. The key to AHP is the powerful hierarchical framework which sorts out and organizes complex multi-criteria problems. AHP has proved very versatile and can handle the intagible factors in the decision process while it enables you to use your knowledge, experience, and intuition to make judgements on a problem's criteria and alternatives. 473

4 EXHIBIT:' S STEPS OF THE ISGP IT PROCESS Phase I: Maximum Achievable Goal SOlution Step 1: Positive. Negatlve Ideal Solutions Optimize each goal with respect to the problem constraints. where all other goals are ignored (act as free constiatnts). The resultlng objective values are the Positive Ideal Solutions (PIS) for each goal. Reverse the direction of each goal's objective and repeat the optimization For exampl~. "max" goals (more Is better) should be solved as m1nim1zatlon problems. and vice versa. The resulting solutions are the Negative Ideal Solutions. Step 2: MaxImum Achievable Goal, MaxImum AchIevable Rate Convert each goal into equality constraints where the right-hand sides are a specified percentage of the distance from the NIS toward the PIS. That percentage is the initial estimate of the Maxlmum Achievable Rate (MAR), The equality constraints contain explicit slack. and surplus variables that measure the percent deviation of the goal from the light-hand side value. The objective function contains the weighted sum of the slack (shortfall) Variables. This problem is solved repeatedly. changing the estimate of the MAR until the point is found at which each goal is achieved to at least the MAR. Phase n: Search for Preferred Solution Step 3: Goal Selection Based on infonnatton provided in the Phase I. the decision makers shall choose desired values for each goal. Step 4: FInd PrIncipal Solution The prtnclpal problem is slrnllar to the MAG-MAR formulation, except that the rlghthand sides are the goals specified in step 1. and any non-dominated solution is acceptable. Step 5: FInd AwdIIary Solutions for Each Goal The auxiliary problem treats the goal as a hard constraint. The other goals are treated using the goal equality constraints used in the prtnclpal problem, There is one auxiliary problem for each goal. Steps 3-5 are repeated until one of the solutions is considered acceptable. 474

5 The first two steps of AHP call for the creation of a criteria hierarchy, while the third step requires the evaluation of alternatives with respect to each of the criteria. For our diet problem the hierarchy is shown in Exhibit 4. The highest level, is the overall goal, at level 0 which will be our optimal diet. Level 1 criteria of taste, cost, nutritional value, and healthfulness. Under healthfulness, level 2 criteria are cholesterol and caloric content. Next the decision makers form importance judgements about the criteria at each level. These judgements are then subjected to a mathematical process that yields weights for each of the criteria. First the decision makers must establish the relative importance of cholesterol content, fattiness and caloric content, and so on. IMPORTANCE SCALE Importance judgements are formed pair-wise using a 1 to 9 ratio scale. For example, the decision makers would be asked, "How much more important is healthfulness than nutrition?" The responses need to be on this scale: 1. Equally important 2. Equally to moderately more important 3. Moderately more important 4. Moderately to strongly more important 5. Strongly more important 6. Strongly to very strongly more important 7. Very strongly more important 8. Very strongly to extremely more Important 9. Extremely more important Exhibit 4 Analytic Hierarchy of Diet Problem Diet 3 475

6 Exhibit 5 shows an example of an importance matrix for the level 1 crite~ia. Healthfulness was specified as very strongly more important than taste as the entry 7 in the matrix for {Healthfulness,Taste) and 1/7 as the entry for (Taste, Healthfulness). Note that the subdiagonal elements are the inverses of the superdiagonal elements. The diagonal elements themselves are all equal to I, as.a criterion must be of equal importance to itself. A ter all criteria have weights assigned to them, the decision makers proceed to the last step of AHP. Each alternative solution must be evaluated with respect to each of the lowest level criteria. Again, the preferences are established using pair-wise comparisons of the alternatives on a 9-point scale. The scale is the same as the importance scale except the word preferred is substituted for the word important. Three alternative diets were evaluated in this stage of AHP. One diet was the MAG-MAR selected diet of the ISGF method, the second was the initial principal solution from the first phase of ISGF, and the third is a preferred diet selected by a hypothetical group of decision makers. Pair-wise comparisons are made only at the lowest levels and within each criteria, preference weights and then finally scores within each subcriterion are calculated and summed. Finally a global preference score for each of the three alternatives is calculated. EXHIBITS LEVEL 1 IMPORTANCE COMPARISONS. DIET PROBLEM ~ ~.!:l!!1.. Nutritton Healthfulnegg Taste 1/2 1/5 1/ Cost 2 1 1/4 1/ Nutrition / Healthfulness Exhibit :f. Criteria Weights and Alternative Preference Scores, Diet Problem Overall Preference Scores: Diet Diet Diet )

7 Exhibit 6 shows the criteria weights and Alternative Preference Scores for our sample diet problem. In this example, Diet 2, the initial principal solution is the most favored diet$ with an overall score of The score can be confirmed by multiplying the preference score for Diet 2 for a particular subcriterion by that criterion's weight and the weight of all subcriteria above it in the hierarchy. The process is repeated for every subcriteria and the results are summed. In our example the AHP process found a different "best" diet than did SAS, GAMS, and ISGP. Note, the insight that AHP added into the diet choice problem by including the taste parameter. It is precisely this larger and richer decision base that has attracted so much success to AHP!!! SUMMARY The ISGP process which is coded with SAS/OR is substantially different from AHP in the three following ways: (1). The AHP builds a decision hierarchy and evaluates alternatives based on those criteria, whereas ISGP is used to construct a single solution directly. (2). As a decision tool, ISGP is holistic, as decision makers are not required to itemize or articulate the basis of their preferences. The AHP breaks the decision down into the critical elements, or criteria. (3). ISGP presents the decision makers with many alternatives along the way toward determining a single, most-desired solution. The decision makers can influence the alternatives by changing goals. In.contrast, the AHP works with a fixed, exhaustive set of unchanging alternatives. The most importance difference between the two methods, though, is that there is no guarantee that the same set of decision makers would reach the same decisions under the two separate methods. RECOMMENDATION In order to enhance the benefits from both the AHP and the ISGP, joint ventures which adopt both approaches together should be pursued. Operation Research Analysts can apply goal programming techniques and use software such as the SAS Linear programming procedure code to generate a set of alternative solutions. Then I suggest the AHP process and software tools be applied to select the BEST option. As the SAS Institute continues to release bigger and better versions of software we see SAS becoming more and more portable. SAS works on numerous platforms and can communicate with several database systems. I challenge other creative Operation Research Analysts and programmers to expand the domain of SAS software and combine SAS with other PC driven recently developed multi criteria packages! CREDITS Special recognition is granted to Mr Lawrence Andrew Arnold of Syllogistics, Inc., for his work as principal investigator and programmer on this study. Further information on AHP and related products is available from Expert Choice, Inc Ellsworth Avenue Pittsburg,PA (412) Further information on GAMS is available from The Scientific Press 651 Gateway Blvd Suite 1100 So San Francisco, CA (415)