Questions for exam preparation...

CONFIDENTIAL Page 3 of 7 DSC3704 October/November 2020 Question 1 To answer Question 1 of Assignment 03, you had to read the article A note on the statistical analysis of point judgment matrices by MG Kabera and LM Haines, published in the journal ORiON , Volume 29, No. 1, 2013, pp.75 − 86. Now answer the following questions by referring to the same article. 1.1 In which society’s journal is the article published? (1) 1.2 What is the aim of the paper? (3) 1.3 Suppose that n objects are to be compared according to a particular criterion and that the pairwise comparisons are assembled into a judgement matrix. Write down the mathematical relationship between the matrix elements that can be used to complete the matrix if 1.3.1 d decision makers state their preference for each pair with no ties permitted. (2) 1.3.2 a single decision maker expresses his/her relative preferences on a ratio scale. (2) 1.4 Name two descriptive characteristics of the elements of a point judgement matrix. (2) 1.5 Name two examples of relative preference scales. (2) 1.6 Who introduced the first statistical approach to the analysis of point judgement matrices? What is this method called? (2) 1.7 Describe how you will verify that the LLSM weights in Table 1 were indeed calculated correctly. (2) 1.8 Which method for weight estimation do the authors prefer? Why? (2) [18] Question 2 A facilitator elicits pairwise comparisons of four items from a participant. According to the word scale, the participant feels that Item 1 is strongly more important than Item 2, and that Item 2 is very strongly more important than Item 3. Item 4 is just as (equally) important as Item 3. 2.1 Allocate numerical values to the participant’s three pairwise comparisons according to Saaty’s fundamental scale, specifying the corresponding three elements a ij of the judge- ment matrix A = [ a ij ], with i = 1 , . . . , 4 and j = 1 , . . . , 4. (3) 2.2 Complete the judgement matrix A = [ a ij ] of the participant’s pairwise comparisons of the four items by allocating numerical values to all its elements through logical deductions according to Saaty’s fundamental scale and by relevant inverse relationships between the elements. (5) 2.3 Is the completed judgement matrix perfectly consistent? Why or why not? (2) [10] TURN OVER

CONFIDENTIAL Page 4 of 7 DSC3704 October/November 2020 Question 3 Five factors are considered crucial in making a certain decision. The relative importance of these factors can be summarised in the following complete matrix of pairwise comparisons: A = [ a ij ] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 4 1 1 2 1 2 1 5 3 1 2 1 4 1 5 1 1 3 1 3 1 1 3 3 1 1 2 2 3 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 3.1 At least how many pairwise comparisons of the five factors were needed to complete this pairwise comparison matrix? (1) 3.2 Use the geometric means method to calculate the implied preference and weights vector. Round off your final answers to two decimals. (5) 3.3 Normalise the third column of the A matrix. Use it as the initial vector v 0 and perform one iteration of the iterative method to determine the weights vector. Work accurately to three decimals. (Round off to two decimals in the last step.) (6) [12] Question 4 A weights vector w = ( w 1 ; w 2 ; w 3 ; w 4 ) for four criteria that are relevant in a decision needs to be determined. The opinions of experts in the field on the relative importance of the criteria are elicited and combined into a matrix of observations of pairwise comparisons of the four criteria. The resultant system of log least squares (LLS) equations D b = d is b 1 − b 2 = + ln 4 ...... (1) − b 1 + 2 b 2 − b 3 = − ln 4 + ln 2 ...... (2) − b 2 + 2 b 3 − b 4 = − ln 2 + ln 8 ...... (3) − b 3 + b 4 = − ln 8 ...... (4) , where the preference vector is u = ( u 1 ; u 2 ; u 3 ; u 4 ) and b = ( b 1 ; b 2 ; b 3 ; b 4 ), with b i = ln u i . 4.1 How many observations were processed to obtain this system of LLS equations? (1) 4.2 Solve for b and determine a preference vector u = ( u 1 ; u 2 ; u 3 ; u 4 ) and corresponding weights vector w = ( w 1 ; w 2 ; w 3 ; w 4 ) for the four criteria. (7) 4.3 Complete the A matrix of pairwise comparisons of the four criteria from the definition by using the weights vector w = ( w 1 ; w 2 ; w 3 ; w 4 ) obtained from solving the LLS equations above. (6) 4.4 Is the complete A matrix of pairwise comparisons of the four criteria obtained from solving the LLS equations above errorless (perfectly consistent)? Why or why not? (3) 4.5 Describe one advantage of using the LLS method above other methods for determining the preference vector of a matrix of pairwise comparisons. (1) [18] TURN OVER

CONFIDENTIAL Page 6 of 7 DSC3704 October/November 2020 Question 6 Joan needs to buy a new car as her old car was written off by the insurance company after an accident. She decides to compare four vehicle models and designs a SMART scoring function for each of the four criteria (vehicle specifications). The specifications of the four vehicle models are as follows: Specification Car1 Car2 Car3 Car4 Price (R1 000) 220 280 320 360 Power (kW) 65 73 73 88 Capacity (cm 3 ) 1 198 1 339 1 339 1 496 Airbag quantity 2 4 6 6 The SMART scoring function for the retail price of the vehicle model is as follows: f Price ( x ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 100 if 0 ≤ x ≤ 100 , 110 − 0 , 1 x if 100 < x ≤ 200 , 150 − 0 , 3 x if 200 < x ≤ 300 , 240 − 0 , 6 x if 300 < x ≤ 400 , 0 if x > 400 , where x is the retail price of the vehicle model in thousands of rand. The SMART scoring function for the engine power of the vehicle model is as follows: f Power ( z ) = ⎧ ⎪ ⎨ ⎪ ⎩ z if 0 ≤ z ≤ 100 , 100 if z > 100 , where z is the engine power of the vehicle model in kW. The SMART scores for the engine capacity of the vehicle model are as follows: Engine CC in cm 3 Score 0 + to 1 200 20 1 200 + to 1 400 40 1 400 + to 1 600 60 1 600 + to 1 800 80 1 800 + 100 Safety and security are important features. Her SMART scores for the number of airbags are as follows: Airbag quantity Score 2 20 4 80 6 100 TURN OVER

CONFIDENTIAL Page 7 of 7 DSC3704 October/November 2020 6.1 By definition, what is the interval for SMART score values? (2) 6.2 Determine each car’s SMART score for each specification. (8) 6.3 If she bases her decision on the weighted total SMART score for each car, determine which car is her best choice if 6.3.1 only capacity and safety matter and are equally important to her. (3) 6.3.2 only power, capacity and safety matter and are equally important to her. (3) 6.3.3 all four criteria, price, power, capacity and safety, are equally important to her. (3) 6.4 Determine which car is her best choice if price is twice as important to her than each of the other three criteria. Again, she bases her decision on the weighted total SMART score for each car. Show relevant calculations. (5) [24] c UNISA 2020