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## Chapter 10 Multicriteria Decision-Marking Models

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**Application Context**multiple objectives that cannot be put under a single measure; e.g., distribution: cost and time as a single objective function problem if time can be converted into cost supply chain: customer service and inventory cost 多目標而目標沒有共同的衡量方式。 2**Chapter Summary**10.0 Scoring Model 評點法 10.1 Weighting Method 權重法 10.2 Goal Programming 目標規劃 10.3 AHP (Analytical Hierarchy Process)層級分析法 我們只學每個方法基本的概念。 3**Motivation Problem**dinner, two factors to consider: distance and cost three restaurants A: (2, 3) B: (7, 1) C: (4, 2) which one to choose? (distance from home, cost) (7, $) (2, $$$) home (4, $$) B A C 4**B**A Scoring Model 評點法 a subjective method assign weights to each criterion assign a rating for each decision alternative on each criterion Restaurant Selection Example: Version 1 min w1 (distance) + w2 (cost) (7, $) (2, $$$) home (4, $$) (distance from home, cost) C 1 1 5 8 6 1 3 11 10 10 , B C choice w w A B C (2, $$$ ) (7, $ ) (4, $$ ) 1 2 A 6 Version 1只需要決定各目標的權重。**Scoring Model 評點法**Restaurant Selection Example: Version 2 weights of criteria, and ratings on criteria for alternatives Example: Tom dislikes walking and likes good food (from expensive restaurants) 每個選擇在每一項目標中有點數（ratings, scores），而目標有各自的權重（weight）。 7**Scoring Model 評點法**Restaurant Selection Example, Version 2 weights of walking and price by Tom: 1 to 1 ratings (scores) of each restaurant for walking and price: distance price criterion restaurant w1 = 1 w2 = 1 A (2, $$$) 10 8 B (7, $) 3 2 C (4, $$) 6 5 dislikes walking and likes expensive, good food objective of Tom: max w1 (rating of distance) + w2 (rating of price) 8**Scoring Model 評點法**a subjective method on assigning weights ratings 9**Example 10-5 Product Selection**to expand the product line by adding one of the following: microwave ovens, refrigerators, and stoves decision criteria manufacturing capability/cost market demand profit margin long-term profitability/growth transportation costs useful life assigning weights to the criteria and ratings to the three alternatives for each criterion maximizing the total score 10**Example 10-5 Product Selection**Scoremicro = 4(4)+5(8)+3(6)+5(3)+2(9)+1(1) = 108 Scorerefer = 4(3)+5(4)+3(9)+5(6)+2(2)+1(5) = 98 Scorestove = 4(8)+5(2)+3(5)+5(7)+2(4)+1(6) = 106 any comments on the relative values? 11**Weighting Method 權重法**a form of scoring method transforming a multi- to a single-criterion objective function by finding the weights of the criteria 以目標的權重（weight）將多目標的問題轉化為單目標的問題。 13**Weighting Method 權重法**max Z(x) = [z1(x), z2(x), …, zP(x)] s.t. x S turning into a single-criterion objective function by weighting (with weights) max Z(x) = w1z1(x)+w2z2(x)+… +wpzP(x) s.t. x S 14**Weighting Method 權重法**criteria (i.e., objectives) max z1(x) = 2x1+3x2x3 min z2(x) = 6x1x2 max z3(x) = 2x1+x3 constraints x1+x2+x3 15 x1+2x2+x3 20 x3 2 x1, x2, x3 0 15**Weighting Method 權重法**somehow got: w1 = 1, w2 = 2, w3 = 4 max z1(x)2z2(x)+4z3(x) = (2x1+3x2x3) 2(6x1x2) + 4(2x1+x3) = 18x1+5x2+3x3, s.t. x1+x2+x3 15 ; x1+2x2+x3 20; x3 2; x1, x2, x3 0. max z1(x) = 2x1+3x2x3, min z2(x) = 6x1x2,, max z3(x) = 2x1+x3, s.t.x1+x2+x3 15 ; x1+2x2+x3 20; x3 2; x1, x2, x3 0. negative sign 16**Goal Programming**GP: priority + goal priority of the goals (i.e., of the criteria) (saving) money is most important: B (shortest) distance is most important: A (best) food is the most important: A B (7, $) A (2, $$$) C home (4, $$) 19**Goal Programming**a goal an objective with a desirable quantity no good to be over and under this quantity v() u() over under goal 20**General Idea of Goal Programming**suppose the goals are: 3 units for distance, and 2 units (i.e., $$) for price B (7, $) A distance price (2, $$$) C u(d,) v(d,) u(p,) v(p,) home (4, $$) A(2, 3) 1 0 0 1 B(7, 1) 0 4 1 0 C(4, 2) 0 1 0 0 21**General Idea of Goal Programming**priority P1 > P2 > P3 > … P1up > P2vd > P3ud > P4vp P1up, P2vd, P3ud, P4vp P1up, P2ud, P3vd, P4vp P1vp, P2ud, P3vd, P4up distance price u(d,) v(d,) u(p,) v(p,) A 1 0 0 1 B 0 4 1 0 A C C 0 1 0 0 C B is dominated by C, i.e., C is optimal for any priority that B is optimal. 22**General Idea of Goal Programming**a goal program parts like a linear program with decisions variables with hard constraints parts unlike a linear program with soft constraints expressed as goals to be achieved co-existence of constraints such as x1 10 and x1 7 in a GP if they are soft constraints with the objective function in LP replaced by the priorities of goals in GP 24**Deviation Variables for a Soft Constraints**example: a soft constraint on labor hour x1 units of product 1, each for 4 labor hours x2 units of product 2, each for 2 labor hours goal: 100 labor hours a soft constraint: 4x1+2x2 100 2 deviation variables u and v: 4x1+2x2 + u v = 100 u: under utilization of labor v: over utilization of labor 人世間有不少soft constraints （可以斟酌的限制式） 25**Example 10-1: Formulation of a GP**three products, quantities to produce x1, x2, and x3 objectives in order of priority min overtime in assembly min undertime in assembly min sum of undertime and overtime in packaging Suppose that the material availability is a hard constraint, i.e., there is no way to get more material. 26**Example 10-1: Formulation of a GP**GP min P1v1, P2u1, P3(u2+v2), s.t. 2x1 + 4x2 + 3x3 600 (lb.,hard const.) 9x1 + 8x2 + 7x3 + u1 v1 = 900 (min.,soft const.) 1x1 + 2x2 + 3x3 + u2 v2 = 300 (min.,soft const.) all variables 0 27**Example: Solution of a GP**min P1u1, P2u2, P3u3, s.t. 5x1 + 3x2 150 (hard const.) (A) 2x1 + 5x2 + u1 v1 = 100 (soft const.) (1) 3x1 + 3x2 + u2 v2 = 180 (soft const.)(2) x1+ u3 v3 = 40 (soft const.) (3) all variables 0 29**Example: Solution of a GP**x2 x2 50 5x1 + 3x2= 150 feasible solution space 2x1 + 5x2= 100 30 x1 x1 min P1u1, P2u2, P3u3, s.t. 5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1) 3x1 + 3x2 + u2 v2 = 180 (2) x1+ u3 v3 = 40 (3) all variables 0 u1 = 0, v1 > 0 direction of improvement in P1 20 u1 > 0, v1 = 0 30 50**Example: Solution of a GP**x2 x2 region with u1 = 0 50 Hard (A) 20 P1 Soft (1) 30 50 x1 x1 optimal with (A), (1), and (2) x2 60 Actually at this point we know that the point is optimal even with the third constraint added and the third goal considered. Why? 50 Hard (A) P2 P1 20 Soft (2) Soft (1) 60 50 60 30 50 x1 Soft (3) Hard (A) P2 P3 optimal with (A), (1), (2), and (3) P1 20 Soft (2) Soft (1) 31 60 30 50**Example 10-2**x2 x2 region with v1 = 0 50 P1 40 x1 x1 min P1v1, P2u2, P3v3, s.t. 5x1 + 4x2 + u1 v1= 200 (1) 2x1 + x2 + u2 v2 = 40 (2) 2x1 + 2x2 + u3 v3 = 30 (3) all variables 0 x2 50 40 50 40 P1 15 P1 P2 P3 P2 20 x1 40 15 optimal, with v1 = u2 = v3 = 0 32 20 40**Another Form of GP: Weighted Goals**goals with weights u1 = 30, u2 = 20, v2 = 20, u3 = 20, v3 = 10 the GP expressed as LP min P1u1, P2u2, P3u3, s.t. 5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1) 3x1 + 3x2 + u2 v2 = 180 (2) x1+ u3 v3 = 40 (3) all variables 0 min 30u1+20u2+20v3 +20u3 +10v3 s.t. 5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1) 3x1 + 3x2 + u2 v2 = 180 (2) x1+ u3 v3 = 40 (3) all variables 0 34**Assignment #4**#1. Chapter 8, Problem 16 (a). Find the maximal flow for this network. Show all the steps. (b). Formulate this problem as a linear program. #2. Chapter 10, Problem 1 #2. Chapter 10, Problem 4 35