The Analytic Hierarchy Process What it is and how it used R. W. Saaty, Mathematical Modelling 87 Network Topology Design using Analytic Hierarchy Process Noriaki Kamiyama, Daisuke Satoh, IEEE ICC 08 Design Data Center Networks using Analytic Hierarchy Process Noriaki Kamiyama, IEEE ICC 10 Final Project Professor : Hsueh-Wen Tseng Reporter : Bo-Han Wu 7100093012
Outline Part I : The Analytic Hierarchy Process Analytic Hierarchy Process (AHP) Step1: Constructing Hierarchies Step2: Pair-wise Comparisons Synthesis of Priorities Part II : Network Topology Design using AHP Introduction Network Topology Design Applying AHP to Network Topology Evaluation Numerical Evaluation Part III : Design Data Center Network using AHP Introduction Data center network design using AHP Numerical Evaluation Conclusion 2
Analytic Hierarchy Process (AHP) Multiple-criteria decision-making Can be used for multiple decision makers Used to prioritize alternatives Normally three kinds of elements Problem P Evaluation criteria V Alternative plan A 3
Step1: Constructing Hierarchies Structure the decision problem in a hierarchy Max 7 criteria in a layer Goal on top Decompose into sub-goals Identify criteria (attributes) to measure achievement of Alternatives added to bottom 4
Step1: Constructing Hierarchies (cont.) Example : 5
Step2: Pair-wise Comparisons Comparison of the alternatives based on the criteria 6 Ratio Scales (1~9) Intensity of Importance Definition 1 Equal Importance 3 Moderate Importance 5 Strong Importance 7 Very Strong Importance 9 Extreme Importance 2, 4, 6, 8 For compromises between the above Reciprocals of above Rationals In comparing elements i and j - if i is 3 compared to j - then j is 1/3 compared to i Force consistency Measured values available
Step2: Pair-wise Comparisons(cont.) Ratio Example: S11>S22>S33>S12>S13>S23 R(S11, S11) = 1 R(S11, S22) = 2 R(S11, S33) = 3 R(S11, S12) = 5 R(S11, S13) = 7 R(S11, S23) = 9 Intensity of Importance Definition 1 Equal Importance 3 Moderate Importance 5 Strong Importance 7 Very Strong Importance 9 Extreme Importance 2, 4, 6, 8 For compromises between the above Reciprocals of above Rationals In comparing elements i and j - if i is 3 compared to j - then j is 1/3 compared to i Force consistency Measured values available 7
Step2:Pair-wise Comparisons (cont.) Judge Matrix: r ij >0 r ii =1 r ji =1/r ij 1 1/ 1/ 1 1/ 1 1 1 1 2 1 2 12 1 12 2 1 2 21 1 12 m m m m m m m m ij r r r r r r r r r r r r r A 8
Step2:Pair-wise Comparisons (cont.) Judge Matrix Example: S11>S22>S33>S12>S13>S23 s11 s22 s33 s12 s13 s23 s11 1.000000 2.000000 3.000000 5.000000 7.000000 9.000000 s22 0.500000 1.000000 2.000000 3.000000 5.000000 7.000000 s33 0.333333 0.500000 1.000000 2.000000 3.000000 5.000000 s12 0.200000 0.333333 0.500000 1.000000 2.000000 3.000000 s13 0.142857 0.200000 0.333333 0.500000 1.000000 2.000000 s23 0.111111 0.142857 0.200000 0.333333 0.500000 1.000000 9
Step2:Pair-wise Comparisons (cont.) Calculating eigenvalue and eigenvector max Eigenvalue Eigenvector W Calculating T j and W i T a j * ij n i 1 a T ij j a ij, i, j 1, 2, n 1, 2, n ; j T * i W i n j1 * Ti n a * ij, i 1,2, n, i 1,2, n 10
Example: Judge Matrix * ij s11 s22 s33 s12 s13 s23 s11 1.0000 2.0000 3.0000 5.0000 7.0000 9.0000 s22 0.5000 1.0000 2.0000 3.0000 5.0000 7.0000 s33 0.3333 0.5000 1.0000 2.0000 3.0000 5.0000 s12 0.2000 0.3333 0.5000 1.0000 2.0000 3.0000 s13 0.1429 0.2000 0.3333 0.5000 1.0000 2.0000 s23 0.1111 0.1429 0.2000 0.3333 0.5000 1.0000 Tj( 行和 ) 2.2873 4.1762 7.0333 11.8333 18.5000 27.0000 T i a ij sij/tj s11 s22 s33 s12 s13 s23 Ti Wi s11 0.4372 0.4789 0.4265 0.4225 0.3784 0.3333 2.4769 0.4128 s22 0.2186 0.2395 0.2844 0.2535 0.2703 0.2593 1.5255 0.2542 s33 0.1457 0.1197 0.1422 0.1690 0.1622 0.1852 0.9240 0.1540 s12 0.0874 0.0798 0.0711 0.0845 0.1081 0.1111 0.5421 0.0903 s13 0.0625 0.0479 0.0474 0.0423 0.0541 0.0741 0.3281 0.0547 s23 0.0486 0.0342 0.0284 0.0282 0.0270 0.0370 0.2035 0.0339 11 Step2:Pair-wise Comparisons (cont.) * n j1 * W i * Ti n
Calculating AW m m m m m m m m ij AW AW AW W W W r r r r r r W r AW 1 1 1 1 2 1 2 21 1 12 1 1 1 Step2:Pair-wise Comparisons (cont.) 12
Step2:Pair-wise Comparisons (cont.) Calculating maximum eigenvalue : max n (AW ) n w i i i consistency CI(Consistency Index) : CI max n n 1 CI <0.1 13
Step2:Pair-wise Comparisons (cont.) CR (Consistency Ratio) : CR CI RI CR<0.1 Random Index (RI) Random Index Table m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 R.I. 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.48 1.56 1.57 1.59 14
Synthesis of Priorities AWi 6 * Wi Aw/6wi s11 2.5230 2.4769 1.0186 s22 1.5505 1.5255 1.0164 s33 0.9330 0.9240 1.0098 s12 0.5458 0.5421 1.0068 s13 0.3288 0.3281 1.0022 s23 0.2044 0.2035 1.0044 lamda 6.0582 CI=(lamda-6)/5 0.011639015 CR=CI/RI 0.009386302 RI=1.24 15
Synthesis of Priorities (cont.) Alternatives: Judge Matrix * ij 16 C P R SL QS C 1.000000 3.000000 5.000000 7.000000 9.000000 P 0.333333 1.000000 3.000000 5.000000 7.000000 R 0.200000 0.333333 1.000000 3.000000 5.000000 SL 0.142857 0.200000 0.333333 1.000000 3.000000 QS 0.111111 0.142857 0.200000 0.333333 1.000000 Tj( 行和 ) 1.787302 4.676190 9.533333 16.333333 25.000000 T i a ij Alpha ij C P R SL QS Ti* Wi C 0.559503 0.641548 0.524476 0.428571 0.360000 2.514097 0.502819496 P 0.186501 0.213849 0.314685 0.306122 0.280000 1.301158 0.260231588 R 0.111901 0.071283 0.104895 0.183673 0.200000 0.671752 0.134350441 SL 0.079929 0.042770 0.034965 0.061224 0.120000 0.338888 0.067777667 QS 0.062167 0.030550 0.020979 0.020408 0.040000 0.174104 0.034820809 * n j1 * W i * Ti n
Synthesis of Priorities (cont.) AWi 5 * Wi AWi/5wi C 2.7431 2.5141 1.0911 P 1.4135 1.3012 1.0864 R 0.6991 0.6718 1.0407 SL 0.3409 0.3389 1.0059 QS 0.1773 0.1741 1.0185 lamda 5.2426 CI=(lamda-5)/4 0.0607 CR=CI/RI 0.0542 RI=1.12 17
Synthesis of Priorities (cont.) Sum of weights 18 R-C R-P R-R R-SL s11 0.4128 0.0422 0.4128 0.0556 s22 0.2542 0.1767 0.1540 0.2778 s33 0.1540 0.5200 0.0339 0.2778 s12 0.0903 0.0422 0.2542 0.0556 s13 0.0547 0.0422 0.0903 0.0556 s23 0.0339 0.1767 0.0547 0.2778 W A=WR RANK 0.557892 0.2778 1 0.263345 0.2133 3 0.121873 0.2361 2 0.056890 0.0943 4 0.0544 6 0.0892 5
Outline Part I : The Analytic Hierarchy Process Analytic Hierarchy Process (AHP) Step1: Constructing Hierarchies Step2: Pair-wise Comparisons Synthesis of Priorities Part II : Network Topology Design using AHP Introduction Network Topology Design Applying AHP to Network Topology Evaluation Numerical Evaluation Part III : Design Data Center Network using AHP Introduction Data center network design using AHP Numerical Evaluation Conclusion 19
Introduction Network topology need to simultaneously consider multiple criteria Cost Reliability Throughput etc Need to reflect the relative importance of each criterion when evaluating the network topology This paper propose to use a linear-transformed value of each criterion when constructing weights in AHP 20
Network Topology Design Evaluation Criteria Total node count : Total link length : Sum of path lengths weighted by path traffic : Amount of traffic on maximally loaded link : 21
Network Topology Design (cont.) Making Topology Candidates We can choose any physical topology from the candidate set Let z denote the number of positions where we can put a link The number of topologies obtained by putting links at any links at any possible position is Set logical paths are deployed using a greedy algorithm Eliminate from the candidate set all the topologies with links that do not accommodate any path 22
Network Topology Design (cont.) Network Model Using the Japanese archipelago model 23
Applying AHP to Network Topology Evaluation We apply AHP to network topology evaluation Layer 0 : the target problem P, which is choosing optimum network topology Layer 1 : the evaluation criteria Vi are located in the middle layer Layer 2 : the candidate topologies are located in the bottom layer 24
Applying AHP to Network Topology Evaluation (cont.), a linear- Use the normalized value of transformed value of Define as a and b are arbitrary real numbers Weights : 25
Applying AHP to Network Topology Evaluation (cont.) Weights in descending order for each criterion 26
Numerical Evaluation 27
Numerical Evaluation (cont.) Example of scenarios for criteria comparison 28
Numerical Evaluation (cont.) 29
Outline Part I : The Analytic Hierarchy Process Analytic Hierarchy Process (AHP) Step1: Constructing Hierarchies Step2: Pair-wise Comparisons Synthesis of Priorities Part II : Network Topology Design using AHP Introduction Network Topology Design Applying AHP to Network Topology Evaluation Numerical Evaluation Part III : Design Data Center Network using AHP Introduction Data center network design using AHP Numerical Evaluation Conclusion 30
Introduction Network topology and data center location strongly affect various evaluation criteria, such as cost, path length, and reliability Design data center networks by evaluating both network topology and data locations simultaneously using AHP Investigate the results of applying the proposed design method to three areas: Japan, USA, and Europe 31
Data center network design using AHP Constructing candidate set of data center network The constrains that all candidates need to satisfy are To maintain the connectivity between all pairs of N nodes at any single link failure(slf) Have no links unused by traffic during normal operation as well as any SLF Generate candidate data center network satisfying constraints The total number of candidates : N : nodes ; S : data center 32
Data center network design using AHP (cont.) Applying AHP to data center network evaluation Use the normalized value of, a lineartransformed value of Define as a and b are arbitrary real numbers Weights : 33
Data center network design using AHP (cont.) Evaluation Criteria Criterion Related to Cost : CM1 : CM2 : CM3 : ; Criterion Related to Quality : the average path length of data center services ; where V is the node set 34
Numerical Evaluation Node location and population 35
Numerical Evaluation (cont.) Distribution of node population ratio and relative link length 36
Numerical Evaluation (cont.) Results using CM1 37 Best three data center networks in CM1 when S = 1
Numerical Evaluation (cont.) Results using CM2 38 Best data center network in CM2 when S = 1
Numerical Evaluation (cont.) Results using CM3 39 Best three data center networks in CM3 when S = 1
Conclusion This paper presented a design method of data center networks using AHP Generate candidates for data center networks satisfying the requirements that connectivity between all pairs of nodes be maintained at single link failures (SLFs) and that no links are unused by traffic during normal operation and at any SLFs. Evaluate the generated candidates by AHP using two criteria, i.e., the total link cost and the average path length 40
Question Why use the AHP to design data center network? Ans : Network topology and data center location strongly affect various evaluation criteria, such as cost, path length, and reliability; therefore, these criteria with different respective units need to be considered simultaneously when designing a data center network. The analytic hierarchy process (AHP) is a way to make a rational decision considering multiple criteria. 41