Wine in Your Knapsack? Jon M. Conrad, Miguel I. Gomez and Alberto J. Lamadrid 5th Annual Conference American Association of Wine Economists Free University of Bozen-Bolzano, Bolzano, June 23 rd 2011 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 1 / 28
Outline 1 Motivation Problem Formulation 2 Background Objective Multiple Solutions Different Cultivars Multiple cultivars, multiple solutions Data 3 Descriptive Detailed information 4 Results No multiple Purchases Multiple purchases Conclusions 5 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 2 / 28
Motivation Outline Motivation 1 Problem Formulation 2 Background Objective Multiple Solutions Different Cultivars Multiple cultivars, multiple solutions Data 3 Descriptive Detailed information Results 4 No multiple Purchases Multiple purchases Conclusions 5 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 3 / 28
Motivation Motivation How do you optimally build a wine cellar? Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 4 / 28
Problem Formulation Outline Motivation 1 Problem Formulation 2 Background Objective Multiple Solutions Different Cultivars Multiple cultivars, multiple solutions Data 3 Descriptive Detailed information Results 4 No multiple Purchases Multiple purchases Conclusions 5 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 5 / 28
Problem Formulation Background Background Prototype OR problem n � max X j u = u j × X j j = 1 n � Subject to w j × X j ≤ c j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n Operation, and changes, in electrical systems Prototype binary problem [Martello and Toth.(1990)]. NP-hard problem [Kellerer, Pferschy, and Pisinger(2004)]. Computing solution using Mixed Integer Programming (MIP) [Srisuwannapa and Charnsethikul(2007)]. Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 6 / 28
Problem Formulation Objective The Wine Selection Problems Open selection problem n � max X j r = r j × X j j = 1 n (1) � Subject to p j × X j ≤ B j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n = 64 Adding quantity constraints, e.g. � n j = 1 X j = 30 n � max X j r = r j × X j j = 1 n � Subject to p j × X j ≤ B (2) j = 1 n � X j = 30 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 7 / 28 j = 1
Problem Formulation Objective The Wine Selection Problems Open selection problem n � max X j r = r j × X j j = 1 n (1) � Subject to p j × X j ≤ B j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n = 64 Adding quantity constraints, e.g. � n j = 1 X j = 30 n � max X j r = r j × X j j = 1 n � Subject to p j × X j ≤ B (2) j = 1 n � X j = 30 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 7 / 28 j = 1
Problem Formulation Objective The Wine Selection Problems Open selection problem n � max X j r = r j × X j j = 1 n (1) � Subject to p j × X j ≤ B j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n = 64 Adding quantity constraints, e.g. � n j = 1 X j = 30 n � max X j r = r j × X j j = 1 n � Subject to p j × X j ≤ B (2) j = 1 n � X j = 30 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 7 / 28 j = 1
Problem Formulation Objective The Wine Selection Problems Open selection problem n � max X j r = r j × X j j = 1 n (1) � Subject to p j × X j ≤ B j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n = 64 Adding quantity constraints, e.g. � n j = 1 X j = 30 n � max X j r = r j × X j j = 1 n � Subject to p j × X j ≤ B (2) j = 1 n � X j = 30 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 7 / 28 j = 1
Problem Formulation Multiple Solutions Second Stage Solution If multiple solutions, run a second stage. n � min X j E = p j × X j j = 1 n � r j × X j ≥ r ∗ Subject to j = 1 (3) n � X j = 30 j = 1 X j ∈ { 0 , 1 } , j = 1 , 2 , . . . , n = 64 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 8 / 28
Problem Formulation Different Cultivars Extending to different cultivars Choosing more than one wine cultivar (e.g. Cabernet Sauvignon, Pinot, Zinfandel) m n k � � max X j r = r j , k × X j , k k = 1 j = 1 n k m � � Subject to p j , k × X j , k ≤ B k = 1 j = 1 (4) n k � X j , k ≥ C k = 10 j = 1 X j , k ∈ { 0 , 1 } , j = 1 , 2 , . . . , n k , k = 1 , 2 , 3 = m Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 9 / 28
Problem Formulation Multiple cultivars, multiple solutions Multiple solutions, multiple cultivars Run a second stage for the case of multiple solutions. n k m � � min X j E = p j , k × X j , k k = 1 j = 1 m n k r j , k × X j , k ≥ r ∗ � � Subject to j = 1 k = 1 (5) n k � X j , k ≥ C k = 10 , j = 1 X j , k ∈ { 0 , 1 } , j = 1 , 2 , . . . , n k , k = 1 , 2 , 3 = m Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 10 / 28
Problem Formulation Multiple cultivars, multiple solutions Finding all solutions The pseudo-algorithm to identify multiple solutions is as follows 1: i ← 1; 2: Do first run without constraints on past solutions. 3: i ← i + 1; 4: repeat 5: Create dynamic set to store solutions ( cont ). This is implemented as a matrix with increasing number of columns in each iteration (dimensions P . c × cont , where P . c is the number of solutions for all classes, cont is the number of solutions found -initially 1). 6: Store first solution in the first column of the matrix. The implementation places each integer solution per class as a stacked column vector. 7: for all consecutive runs, i > 1 do 8: add a constraint such that the squared difference between each element of the new solution and each element of all past solutions is greater than or equal to one. This is equivalent to state that there must be at least one position of the integer solution vector (1’s and 0’s) that differs. 9: Store the new solution found in the set of past solutions ( cont ) 10: i ← i + 1; 11: end for 12: until i = N iter , run-time specified by user. Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 11 / 28
Data Outline Motivation 1 Problem Formulation 2 Background Objective Multiple Solutions Different Cultivars Multiple cultivars, multiple solutions Data 3 Descriptive Detailed information Results 4 No multiple Purchases Multiple purchases Conclusions 5 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 12 / 28
Data Descriptive Descriptive prices and ranks for the wines Example Wide range of variances for prices... Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 13 / 28
Data Detailed information Some detail on information General descriptives Cabernet Pinot Zinfandel Overall Average of rank 90.71 88.60 90.34 89.99 Average of price 106.41 73.23 44.11 84.14 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 14 / 28
Data Detailed information ... And the further detail Type Rank Cabernet Pinot Zinfandel Grand Total 84.5 Average Price ($) 58.86 47.11 32.11 49.23 Max Price ($) 220.00 95.00 55.00 220.00 Min Price ($) 10.00 19.00 18.00 10.00 92.5 Average Price ($) 102.83 86.34 48.22 86.90 Max Price ($) 484.50 370.00 140.00 484.50 Min Price ($) 27.00 25.00 24.00 24.00 97.5 Average Price ($) 350.70 411.25 - 360.79 Max Price ($) 830.00 490.00 - 830.00 Min Price ($) 75.00 332.50 - 75.00 98 Average Price ($) - - 75.00 75.00 Max Price ($) - - 75.00 75.00 Min Price ($) - - 75.00 75.00 Average Price across all ranks ($) 106.41 73.23 44.11 84.14 Max Price across all ranks ($) 830.00 490.00 140.00 830.00 Min Price across all ranks ($) 10.00 19.00 18.00 10.00 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 15 / 28
Results Outline Motivation 1 Problem Formulation 2 Background Objective Multiple Solutions Different Cultivars Multiple cultivars, multiple solutions Data 3 Descriptive Detailed information Results 4 No multiple Purchases Multiple purchases Conclusions 5 Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 16 / 28
Results No multiple Purchases Solution, Single cultivar (2003 Zinfandel) Five solutions found. Sum of rankings r ∗ = 2 , 896 32 bottles selected. Solutions differ from one another by one bottle of equivalent price and rating. 31 wines in common for all solutions. Changes in: Hartford Vineyard Zinfandel, Hartford’s Fanucchi Wood Road Vineyard, Hartford’s Dina’s Vineyard, Robert Biale’s Monte Rosso and Robert Biale’s Aldo’ Vineyard. Conrad et al. (Cornell University) Wine in your Knapsack? June 2011 17 / 28
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