1. Applied Mathematics 1 Applications of the Multi-Weighted Scoring Model and the Analytical Hierarchy Process for the Appraisal and Evaluation of Suppliers / Sub- contractors

2. Applied Mathematics Multi-Weighted Scoring Model 1 st Step: Determination of appraisal / evaluation criteria for the selection of supplier 2 nd Step: Creation of Evaluation Tables + Selection of appropriate weights for each criterion The weights rank the criteria according to their importance / value In some cases, weights can also be used as the maximum credits/points one can get for each criterion 3 rd Step: Evaluation of Suppliers for each criterion

3. Applied Mathematics Multi-Weighted Scoring Model The weights typically depend on our goals (e.g. a strategic goal of our organization) or the particular characteristics of the item / service we want to procure For example, if we want to buy a common item widely available in the market, we will put high weights on criteria related to fast deliveries or criteria related to the reduction of processing costs For example, if we want to buy an item with complex specifications, we will put high weights on criteria related to quality The weight can be expressed as a percentage or simply as a liner distribution from 1 to 10

4. Applied Mathematics Rating of Suppliers CriteriaMax Value Supplier ΑΒΓ Technical - Understanding of the problem - Technical approach - Production capacity - Functional requirements - Quality requirements Sub-total 10 20 5 3 2 40 10 18 4 2 1 35 8 16 5 3 2 34 7 15 4 2 30 - Responsiveness - Price - Financial stability - Application of well known standards 20 10 18 16 10 9 15 20 8 12 2 8 7 Total100888559 Multi-Weighted Scoring Model Design, development and production of a specialized package for the storage of small sized and highly sensitive archeological founds

5. Applied Mathematics Multi-Weighted Scoring Model 5 Combination of the multi-weighted scoring model with other considerations, such as motivation, building of long terms relationships etc…. In class example

6. Applied Mathematics Analytical Hierarchy Process When determining weights for different criteria, subjective and other factors may increase the inconsistency of the overall evaluation process The Analytical Hierarchy Process (AHP) is a common multi-criteria decision making method that can be used to determine weights 6

7. Applied Mathematics Analytical Hierarchy Process Let a set of criteria and or suppliers First, we construct a matrix where all criteria / suppliers are assessed in pairs, using for example range from 1 to 9. aij = 1 if both criteria Oi and Oj are equally important aij = 3 if Oi is slightly more important compare to Oj aij = 5 if Oi is more important compared to Oj, aij = 7 if Oi is much more important compared to Oj aij = 9 if Oi is highly important compared to Oj.  Second, we normalize the matrix by dividing each element with the total sum of the corresponding column  Third, we calculate the average values of each row  We repeat for all criteria / suppliers

8. Applied Mathematics Analytical Hierarchy Process Example: Let 4 suppliers, i.e., S1, S2, S3 and S4 We will evaluate them according to 4 criteria, quality, cost, service and response 8 Evaluation of suppliers Quality S1 S2 S3 S4 Cost S1 S2 S3 S4 Service S1 S2 S3 S4 Response S1 S2 S3 S4

9. Applied Mathematics Analytical Hierarchy Process Our EstimationRate Very High preferred9 Highly preferred7 Strongly preferred5 Slightly preferred3 Equally preferred1 9 Intermediate values 2, 4, 6 and 8 can be used as additional ranking levels In addition, the rule of reversed ranking also applies. If a criterion or a supplier i has a particular rate with respect to j, then the reversed rate is assumed if j is compared against i.

10. Applied Mathematics Analytical Hierarchy Process Before the evaluation of supplier, we must first determine the weights for each criterion. The first step is to construct an assessment matrix for each pair of criteria. For example, we first consider that Quality is equally and perhaps very slightly preferred compared to Cost. In this case, we put 2. Then, we consider that Cost is slightly preferred compared to Service. In this case we put 3. We do the same for all pairs of criteria. However, be careful and put meaningful values 10

11. Applied Mathematics Analytical Hierarchy Process Α. Initial Matrix QualityCostServiceResponse Quality1243 Cost1/2133 Service1/41/312 Response1/3 ½1 Total25/1211/317/29 Β. Normalized matrix QualityCostServiceResponse Weight Quality12/256/118/17 1/3 = 0,457 Cost6/253/116/17 1/3 = 0,300 Service3/251/112/17 2/9 = 0,138 Response4/251/111/17 1/9 = 0,105 The weights are the average values of each row and their sum equals 1 11

12. Applied Mathematics Analytical Hierarchy Process In this particular example, the weights for each criterion are 0,457, 0,3, 0,138 and 0,105, respectively. On the basis of the above, we can conclude that Quality is one and half times (0,457/0,300) more important compared to Cost, something more that 3 times (0,457/0,138) more important compared to Service, and approximately 4 times (0,457/0,105) more important compared to Response. And so on….. 12

13. Applied Mathematics Analytical Hierarchy Process Next, we compare each supplier (in pairs) for each criterion. The same procedure applies also here; however, we repeat the AHP method for all criteria. 13

14. Applied Mathematics Α. Quality S1S2S3S4 S11561/3 S21/5121/6 S31/61/211/9 S43691 Weights0,2970,0870,0530,563 Β. Cost S111/358 S23179 S31/51/712 S41/81/91/21 Weights0,3030,5730,0780,046 C. Service S11548 S21/511/21/3 S31/4215 S41/81/41/51 Weights0,5970,1400,2140,050 D. Response S1131/51 S21/311/81/3 S35815 S4131/51 Weights0,1510,0600,6380,151 14

15. Applied Mathematics Analytical Hierarchy Process The final step is to calculate the total weighted multi-score for each supplier for all criteria: The final rank of each supplier is the result of the weighted contribution w.r.t all criteria In this particular example, suppler S1 (0,325) is the best, and must be selected. 15 QualityCostServiceResponse S1(0,457)(0,297) +(0,300)(0,303) +(0,138)(0,597) +(0,105)(0,151) =0,325 S2(0,457)(0,087) +(0,300)(0,573) +(0,138)(0,140) +(0,105)(0,060) =0,237 S3(0,457)(0,053) +(0,300)(0,076) +(0,138)(0,214) +(0,105)(0,638) =0,144 S4(0,457)(0,563) +(0,300)(0,046) +(0,138)(0,050) +(0,105)(0,151) =0,294 Total1,000