Construction of designs for twophase experiments with Tabu search
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1 Construction of designs for twophase experiments with Tabu search Nha Vo-Thanh In collaboration with Professor Hans-Peter Piepho University of Hohenheim
2 Common bean experiment A breeding program by CIAT (the International Center for Tropical Agriculture) with common beans The goal is to breed new varieties - cooking time for common beans is one of the key trait 2
3 Common bean experiment Phase 1: a field experiment - 10 common beans (treatments) with 6 replications [- ] - 10 field blocks of size 6 [-0] Phase 2: a lab experiment - 60 samples from Phase 1 [Ti and Bj] - 15 days (4 samples/day, 1 machine) [D1-15] - response: cooking time for the common beans We are interested in - common beans - field blocks and days (blocking factors) 3
4 Common bean experiment 24 beans are selected from the same plot 4
5 Common bean experiment 10 Treatments 10 Field Blocks (FB) 6 Plots in FB 10 treatments 60 samples 15 Days 4 Observations in D 60 tests Factor-allocation diagram for the common bean experiment 5
6 Experimental plan Phase 1 Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 6 plots 6
7 Experimental plan Phase 1 Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 7
8 Experimental plan Phase 1 Treatment Field Block Treatment 0 Field Block Samples Treatment 0 Field Block Treatment Field Block
9 Experimental plan Phase 2 Day 1 Day 2 Day 3 Day 4 Day 5 4 samples 4 samples 4 samples 4 samples 4 samples Day 6 Day 7 Day 8 Day 9 Day 10 4 samples 4 samples 4 samples 4 samples 4 samples Day 11 Day 12 Day 13 Day 14 Day 15 4 samples 4 samples 4 samples 4 samples 4 samples 9
10 Experimental plan Phase 2 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day
11 Two-phase experiments Phase 1: Treatments are arranged in b 1 blocks of size k 1 Phase 2: The samples obtained from Phase 1 are arranged in b 2 blocks of size k 2 Responses are measured at the end of Phase 2 Nonresolvable designs 11
12 Treatment Field Block Day D1 D1 D1 D1 D1 D1 D2 D2 D2 D2 D2 D2 D15 D15 D15 D15 D15 D15?? Common bean experiment Replicate 1 Replicate 6 12
13 Possible solutions Phase 1: finding an arrangement of 10 treatments in 10 blocks of size 6 Phase 2: finding an arrangement of 10 treatments in 15 days of size 4 => combining two arrangements (CycDesigN, or SAS) However, this approach doesnot take both block effects in two phases into account at the same time! Finding all possible arrangements? > 1 billion designs 13
14 Design of experiments Computer search algorithms Factor-Allocation diagram Anticipated models Design criteria (A-, D-, C-, E-, V-optimality) Construction methods Simulated annealing Tabu search Variable neighborhood search Compare design options 14
15 Response: - cooking time: y ijk, Y Statistical models - ith treatment, jth block, kth day Explanatory variables - Phase 1: common bean effects (trt i, X) - Phase 1: field block effects (block j, Z 1 ) - Phase 2: day effects (day k, Z 2 ) - Random errors (ε ij, ε ik, ε ijk ) Statistical models (fixed effects) - model 0: µ + trt i + block j + ε ij (Phase 1) - model 1: µ + trt i + day k + ε ik (Phase 2) - model 2: y ijk = µ + trt i + block j + day k + ε ijk X 15
16 A statistical model - Y = Xµ + Z 1 β 1 +Z 2 β 2 + ε Optimality criteria - M µ = Q (a.k.a. a reduced model), where M = X T VX ; Q = X T VY; V = I Z (Z T Z) - Z T - Z = [Z 1 Z 2 ] - M is the treatment information matrix A-optimality - minimizes the average pairwise variance of the treatment effects - is computed from M - D-optimality - maximizes the determinant matrix(m) - is computed from M 16
17 Optimality criteria Variance-covariance matrix [ M - ] t 1 t 2 t 3 t n t 1 V 11 Cov 12 Cov 13 Cov 1n t 2 Cov 12 V 22 Cov 23 Cov 2n t 3 Cov 13 Cov 23 V 33 t n Cov 1n Cov 2n Cov 3n V nn Pairwise variances - v 12 = var( t1 t2 ) = (V 11 + V 22-2Cov 12 ) σ 2 - n(n-1)/2 = 45 pairwise comparisions - n = 10 17
18 Optimality criteria For a block design, the average variance of all estimated pairwise comparisons - v = 2δ2 v(v 1) treatments i<j v ij, where v is the number of For a complete block design, the average variance of all estimated pairwise comparisons - 2δ 2 /r, where r is the number of replications The overall average efficiency factor (John and Williams 1995) is defined as: 2δ2 /r 1 v 18
19 Quality of solution Computer search algorithms Metaheuristics search Optimal solution Nearly optimal solution Time 19
20 Search strategy Stage 1 Assign treatments to blocks in Phase 1 Stage 2 Assign samples to blocks in Phase 2, taking treatments and blocks in Phase 1 into account Tabu search 20
21 Search strategy Stage 2 Assign samples to blocks in Phase 2, taking treatments and blocks in Phase 1 into account Tabu search 21
22 Neighborhood designs Phase 1 Swap 2 treatments Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 22
23 Neighborhood designs Phase 1 Swap 2 treatments Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 23
24 Neighborhood designs Phase 1 Swap 2 treatments Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 24
25 Neighborhood designs Phase 1 Swap 2 treatments Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 25
26 Neighborhood designs Phase 2 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day
27 Neighborhood designs Phase 2 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day
28 Neighborhood designs Our example: - The number of treatments = 10 - Phase 1: The number of blocks = 10 The block size = 6 - Phase 2: The number of blocks = 15 The block size = 4 The total number of all neighborhood designs - Phase 1: = Phase 2: =
29 Local search SD = A starting design Step 1 Step 2 Step 3 SD All neighborhood designs (SD) BND = Select the best neighborhood design Compare with SD SD = BND n 0 iterations 29
30 Quality of solution Local search Iter k? Iter 2 Iter 1 Solution 30
31 Quality of solution Local search Tabu list Solution 31
32 Quality of solution Tabu search Tabu list move to a worse solution add a solution to a Tabu list Solution 32
33 Tabu search SD = A starting design; T = empty Step 1 Step 2 Step 3 SD All neighborhood designs (SD) BND = Select the best neighborhood design not in T SD = BND n 0 iterations 33
34 Stage 1: Steps 0 and 1 Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 Design efficiency =
35 Stage 1: Steps 0 and 1 Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 Design efficiency =
36 Stage 1: Steps 0 and 1 Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 Design efficiency =
37 Stage 1: Step 2 Iteration Pos 1 Pos 2 Design efficiency swap
38 Stage 1: Step 2 Iteration Pos 1 Pos 2 Design efficiency swap 1 0 swap
39 Stage 1: Step 3 Block Block 2 Block 3 Block 4 Block 5 Block 6 Block 7 Block 8 Block 9 Block 10 Design efficiency =
40 Stage 2: Steps 0 and 1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Day 11 Day 12 Day 13 Day 14 Day 15 40
41 Stage 2: Steps 0 and 1 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day Design efficiency =
42 Stage 2: Step 2 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day Design efficiency =
43 Stage 2: Step 2 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day Design efficiency =
44 Stage 2: Step 2 Iteration Pos 1 Pos 2 Design efficiency D4 D12 swap 1 [, ] [, ]
45 Stage 2: Step 2 Iteration Pos 1 Pos 2 Design efficiency D4 D12 swap 1 [, ] [, ] D7 D13 swap 2 [, ] [, ]
46 Stage 2: Step 3 Day 1 Day 2 Day 3 Day 4 Day Day 6 Day 7 Day 8 Day 9 Day Day 11 Day 12 Day 13 Day 14 Day Design efficiency =
47 Results Design efficiency starting design optimal design One blocking factor (model 0) Two blocking factors (model 2) A 1 = A 2 = A 1 -A 2 =
48 Common bean experiment Phase 1: a field experiment - 68 common beans (treatments) with 3 replications - 17 blocks of size 12 Phase 2: a lab experiment samples from the first experiment - 4 machines (51 samples) - response: cooking time for the common beans We are interested in - common beans - field blocks and machines (blocking factors) 48
49 Results Design efficiency One blocking factor (model 0) Two blocking factors (model 2) optimal design A 1 = A 2 = A 1 -A 2 =
50 Conclusion We introduced two-phase experiments We proposed a sequential approach to obtain designs for two-phase experiments We proposed a general algorithm based on Tabu search to obtain designs for two-phase experiments 50
51 Thank you for your attention! Questions? 51
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