1 Chapter 10 Multicriteria Decision-Marking Models

2 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

3 Chapter Summary 10.0 Scoring Model 10.1 Weighting Method 10.2 Goal Programming 10.3 AHP (Analytical Hierarchy Process)

4 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? home (distance from home, cost) (2, $$$) (7, $) (4, $$) A B C

5 Scoring Model

6 a subjective method assign weights to each criterion assign a rating for each decision alternative on each criterion Restaurant Selection Example: Version 1 min w 1 (distance) + w 2 (cost) home ( distance from home, cost) (2, $$$) ( 7, $) (4, $$) A B C w 1 w 2 A (2,$$$) B (7,$) C (4,$$) choice A B, C Version 1

7 Scoring Model 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) ratings, scores weight

8 Scoring Model Restaurant Selection Example, Version 2 weights of walking and price by Tom: 1 to 1 ratings (scores) of each restaurant for walking and price: 56C (4, $$) 23B (7, $) 810A (2, $$$) w 2 = 1w 1 = 1 pricedistance restaurant criterion objective of Tom: max w 1 (rating of distance) + w 2 (rating of price) dislikes walking and likes expensive, good food

9 Scoring Model a subjective method on assigning weights ratings

10 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

11 Example 10-5 Product Selection Score micro = 4(4)+5(8)+3(6)+5(3)+2(9)+1(1) = 108 Score refer = 4(3)+5(4)+3(9)+5(6)+2(2)+1(5) = 98 Score stove = 4(8)+5(2)+3(5)+5(7)+2(4)+1(6) = 106 weight microwaverefersstoves manuf. cap./cost 4438 market demand 5842 profit margin 3695 (long-term) prof./growth 5367 Transp. costs 2924 useful life 1156 any comments on the relative values?

12 Weighting Method

13 Weighting Method a form of scoring method transforming a multi- to a single- criterion objective function by finding the weights of the criteria weight

14 Weighting Method max Z(x) = [z 1 (x), z 2 (x), …, z P (x)] s.t. x S turning into a single-criterion objective function by weighting (with weights) max Z(x) = w 1 z 1 (x)+w 2 z 2 (x)+… +w p z P (x) s.t. x S

15 Weighting Method criteria (i.e., objectives) max z 1 (x) = 2x 1 +3x 2 x 3 min z 2 (x) = 6x 1 x 2 max z 3 (x) = 2x 1 +x 3 constraints x 1 +x 2 +x 3 15 x 1 +2x 2 +x 3 20 x 3 2 x 1, x 2, x 3 0

16 Weighting Method somehow got: w 1 = 1, w 2 = 2, w 3 = 4 max z 1 (x) 2z 2 (x)+4z 3 (x) = (2x 1 +3x 2 x 3 ) 2(6x 1 x 2 ) + 4( 2x 1 +x 3 ) = 18x 1 +5x 2 +3x 3, s.t.x 1 +x 2 +x 3 15 ; x 1 +2x 2 +x 3 20; x 3 2; x 1, x 2, x 3 0. max z 1 (x) = 2x 1 +3x 2 x 3, min z 2 (x) = 6x 1 x 2,, max z 3 (x) = 2x 1 +x 3, s.t. x 1 +x 2 +x 3 15 ; x 1 +2x 2 +x 3 20; x 3 2; x 1, x 2, x 3 0. negative sign

17 Goal Programming

18 1/4: Introducing the Ideas of Goal Programming

19 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 home (2, $$$) (7, $) (4, $$) A B C

20 Goal Programming a goal an objective with a desirable quantity no good to be over and under this quantity goal v ( ) over u ( ) under

21 General Idea of Goal Programming suppose the goals are: 3 units for distance, and 2 units (i.e., $$) for price C(4, 2) B(7, 1) A(2, 3) v (p, ) u (p, ) v (d, ) u (d, ) pricedistance home (2, $$$) ( 7, $) (4, $$) A B C

22 General Idea of Goal Programming priority P 1 > P 2 > P 3 > … 0010 C 0140B 1001A v (p, ) u (p, ) v (d, ) u (d, ) pricedistance P1up,P2vd,P3ud,P4vpP1up,P2vd,P3ud,P4vp A P1up,P2ud,P3vd,P4vpP1up,P2ud,P3vd,P4vp C P1vp,P2ud,P3vd,P4upP1vp,P2ud,P3vd,P4up C B is dominated by C, i.e., C is optimal for any priority that B is optimal. P 1 u p > P 2 v d > P 3 u d > P 4 v p

23 2/4 : A More General Goal Programming Approach

24 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 x 1 10 and x 1 7 in a GP if they are soft constraints with the objective function in LP replaced by the priorities of goals in GP

25 Deviation Variables for a Soft Constraints example: a soft constraint on labor hour x 1 units of product 1, each for 4 labor hours x 2 units of product 2, each for 2 labor hours goal: 100 labor hours a soft constraint: 4x 1 +2x deviation variables u and v: 4x 1 +2x 2 + u v = 100 u: under utilization of labor v: over utilization of labor soft constraints

26 Example 10-1: Formulation of a GP three products, quantities to produce x 1, x 2, and x 3 objectives in order of priority min overtime in assembly min undertime in assembly min sum of undertime and overtime in packaging productx1x1 x2x2 x3x3 availability material (lb/unit) pounds assembly (min. unit) minutes packaging (min/unit) minutes Suppose that the material availability is a hard constraint, i.e., there is no way to get more material.

27 Example 10-1: Formulation of a GP GP min P 1 v 1, P 2 u 1, P 3 (u 2 +v 2 ), s.t. 2x 1 + 4x 2 + 3x ( lb., hard const. ) 9x 1 + 8x 2 + 7x 3 + u 1 v 1 = 900 ( min., soft const. ) 1x 1 + 2x 2 + 3x 3 + u 2 v 2 = 300 ( min., soft const. ) all variables 0

28 3/4 : Solution of a Goal Program

29 Example: Solution of a GP min P 1 u 1, P 2 u 2, P 3 u 3, s.t. 5x 1 + 3x ( hard const. )(A) 2x 1 + 5x 2 + u 1 v 1 = 100 ( soft const. ) (1) 3x 1 + 3x 2 + u 2 v 2 = 180 ( soft const. ) (2) x 1 + u 3 v 3 = 40 ( soft const. )(3) all variables 0

u 1 = 0, v 1 > 0 30 Example: Solution of a GP min P 1 u 1, P 2 u 2, P 3 u 3, s.t. 5x 1 + 3x (A) 2x 1 + 5x 2 + u 1 v 1 = 100 (1) 3x 1 + 3x 2 + u 2 v 2 = 180 (2) x 1 + u 3 v 3 = 40 (3) all variables 0 x1x1 x2x feasible solution space 5x 1 + 3x 2 = 150 x1x1 x2x x 1 + 5x 2 = 100 u 1 > 0, v 1 = 0 direction of improvement in P 1

31 Example: Solution of a GP x1x1 x2x Hard (A) Soft (1) P1P1 region with u 1 = 0 x1x1 x2x Hard (A) Soft (1) P1P1 optimal with (A), (1), and (2) 60 Soft (2) P2P2 x1x1 x2x Hard (A) Soft (1) P1P1 optimal with (A), (1), (2), and (3) 60 Soft (2) P2P2 P3P3 Soft (3) Actually at this point we know that the point is optimal even with the third constraint added and the third goal considered. Why?

32 Example 10-2 min P 1 v 1, P 2 u 2, P 3 v 3, s.t. 5x 1 + 4x 2 + u 1 v 1 = 200 (1) 2x 1 + x 2 + u 2 v 2 = 40 (2) 2x 1 + 2x 2 + u 3 v 3 = 30 (3) all variables 0 x1x1 x2x P1P1 region with v 1 = x1x1 x2x P1P1 P2P2 20 x1x1 x2x P1P1 P2P2 15 P3P3 optimal, with v 1 = u 2 = v 3 = 0

33 4/4 : Another Form of Goal Programming

34 Another Form of GP: Weighted Goals goals with weights u 1 = 30, u 2 = 20, v 2 = 20, u 3 = 20, v 3 = 10 the GP expressed as LP min P 1 u 1, P 2 u 2, P 3 u 3, s.t. 5x 1 + 3x (A) 2x 1 + 5x 2 + u 1 v 1 = 100 (1) 3x 1 + 3x 2 + u 2 v 2 = 180 (2) x 1 + u 3 v 3 = 40 (3) all variables 0 min 30u 1 +20u 2 +20v 3 +20u v 3 s.t. 5x 1 + 3x (A) 2x 1 + 5x 2 + u 1 v 1 = 100 (1) 3x 1 + 3x 2 + u 2 v 2 = 180 (2) x 1 + u 3 v 3 = 40 (3) all variables 0

35 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