12-Introduction to Discrete Optimization

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  • DSC1007 Lecture 10 Discrete Optimization

  • Sensitivity Analysis: How the optimal solution and objective value change if one of aij, bi, or cj changes Changing cj rotates the isoquant and direction to optimize The optimal solution will not change if within the sensitivity range Changing bi shifts (parallel) the corresponding constraint The binding constraints and shadow price will not change if within the sensitivity range Changing aij rotates the corresponding constraint Shadow price: The marginal change of the objective value because of an additional unit of resource/requirement Excel Solvers Answer Report and Sensitivity Report

    2

    Recap: Last Lecture .

  • The Lego Game Revisited We have seen fractional solutions for certain numbers of large and small bricks. What if we insist that the numbers of chairs and tables to produce must be whole numbers?

    Scenario 1. Some decision variables have to be integers.

  • Transportation Problem Revisited What if there is a requirement that each retail outlet can only be supplied by one plant? What if Outlet A and Outlet B cannot be supplied by the same plant?

    Scenario 2. There is a need to make choice or consider some logical constraints.

  • Computer Production Problem Revisited

    Scenario 3. There is a need to convert some nonlinear constraints into linear constraints.

  • Discrete Optimization is an optimization problem in which some (or all) of the decision variables must be integer-valued Examples: Integer variables: Number of workers, cars, machines, etc. Binary variables: YES/NO decisions Mixed Integer Program (MIP): Some variables are integer-valued Why do we need discrete optimization? A very powerful modeling method! CPLEX reports that > 90% of their clients problems are MIPs Various applications: Sudoku

    Discrete Optimization

  • Taxonomy Integer Optimization/Programming (IP): Linear optimization with all the decision variables being integer valued Binary Optimization: Linear optimization with all the decision variables being binary valued Mixed Integer Optimization/Programming (MIP): Linear optimization with some of the decision variables being integer or binary valued

    Discrete Optimization

  • One of the most powerful modeling frameworks for modeling Yes/No type of decisions Usually 1 denotes YES and 0 denotes NO Various use of binary variables

    Using Binary Constraints

  • $15,000 available to invest 7 investments to choose from Goal: Choose investments to maximize total NPV Investment Cash required NPV 1 $5,000 $16,000 2 $2,500 $8,000 3 $3,500 $10,000 4 $6,000 $19,500 5 $7000 $22,000 6 $4,500 $12,000 7 $3,000 $7,500

    Example: Capital Budgeting Models

    *No partial investment is allowed for each choice

  • Decision variables: x1, , x7 Binary: xj = 1 if investment j is chosen, and 0 otherwise Objective: Total return Constraints: Cash constraint, binary constraint Example: Capital Budgeting Models

    Investment Cash required NPV 1 $5,000 $16,000 2 $2,500 $8,000 3 $3,500 $10,000 4 $6,000 $19,500 5 $7000 $22,000 6 $4,500 $12,000 7 $3,000 $7,500

  • Formulation: Example: Capital Budgeting Models Investment 1 $5,000 $16,000 2 $2,500 $8,000 3 $3,500 $10,000 4 $6,000 $19,500 5 $7000 $22,000 6 $4,500 $12,000 7 $3,000 $7,500

  • Solution: Select investments 1, 2 and 5. Total NPV = $46,000 Example: Capital Budgeting Models

    Investment Cash required NPV 1 $5,000 $16,000 1 2 $2,500 $8,000 1 3 $3,500 $10,000 0 4 $6,000 $19,500 0 5 $7000 $22,000 1 6 $4,500 $12,000 0 7 $3,000 $7,500 0

  • Knapsack problem: n: number of treasures to pick from b: the total weight (or volume) that the knapsack can hold aj: value of each treasure j cj: weight (or volume) of each treasure j xj: whether to choose treasure j

    Formulation of the Knapsack Problem

  • It is not so easy!

    In reality, there may be additional conditions to be considered: At most two investments can be selected Investments 1 and 2 cannot be selected

    together Investment 5 can be selected only if

    Investment 7 is selected.

  • At most one event (among 3) can occur For two candidates, it is the same as an either-or (not both) relationship

    Common Uses of Binary Variables

  • All or Nothing can occur How about all-or-nothing constraint for three events?

    Common Uses of Binary Variables

  • Conditional: Event 1 must occur if Event 2 occurs Note that Event 1 can occur without Event 2

    Common Uses of Binary Variables

  • Assigning 70 workers to 70 jobs If solved with brutal force, it would take forever More details: i: index for workers j: index for jobs pij: performance of Worker i doing Job j IP Formulation Decision variables: xij whether to assign Worker i to Job j (binary) Objective: total performance = Constraints: Exactly one job for each Worker i Exactly one worker for each Job j Binary

    Dantzigs Assignment Problem

  • Math formulation:

    70x70 = 4,900 (binary) decision variables 70 + 70 + 4,900 = 5,040 constraints binary

    1

    1

    subject to

    maximize

    70

    1

    70

    1

    70

    1

    70

    1

    ij

    iij

    jij

    i jijij

    x

    x

    x

    xp

    =

    =

    = =

    =

    =

    Dantzigs Assignment Problem for each worker (row) i = 1, 2, , 70 for each job (column) j = 1, 2, , 70

  • Solution (after 0.09s on Excel Premium Solver with Gurobi engine) Dantzigs Assignment Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

    1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

    10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 53 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 54 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 56 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 58 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 59 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 61 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 62 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 65 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 66 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 68 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 69 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  • Conditional Range: If Event x does not occur, then y=0. If Event x occurs, then 0 y M. Variable x is an ON/OFF switch The range of variable y depends on the switch If OFF, y must be zero If ON, y can be anything in the given range 0 y M Example: The production rlow will be zero if one decides to Shut down a production plant

    Common Uses of Binary Variables

  • Cost of production consists of Gixed and variable costs Variable cost is per unit cost Fixed cost is a lump sum cost incurred as long as anything is produced Example of a typical cost function:

    A Fixed Cost Revisited

  • Five products 4000 labour hours and 4500 square yards of cloth available in a given week Machines are rented on weekly basis

    Goal: Find the optimal production plan that maximizes weekly prorit

    Product

    Shirts $1500 2 3.0 $35 $20 $15

    Shorts $1200 1 2.5 $40 $10 $30

    Pants $1600 6 4.0 $65 $25 $40

    Skirts $1500 4 4.5 $70 $30 $40

    Jackets $1600 8 5.5 $110 $35 $75

    Textile Manufacturing at Great Threads

  • Decision variables: y1, , y5: On/Off switches x1, , x5: Quantities to produce Formulation 1:

    Not quite correct: Inter-dependence between xs and ys!

    Textile Manufacturing at Great Threads

  • Formulation 2:

    Not good: Non-linear! Why is linearity important?

    Textile Manufacturing at Great Threads

  • Formulation 3:

    How to choose M? Any sufriciently large number

    Textile Manufacturing at Great Threads

  • Set Covering Problem

  • Western Airlines runs rlights among 12 cities. Western wants to set up a hub system. Some of the 12 cities will be chosen as hubs. Each hub is used for connecting rlights to and from cities within 1000 miles of the hub.

    Goal: To determine the smallest number of hubs that Western will need.

    Hub Location At Western Airlines A sample airlines hub system. Source: http://www.airlinepilotforums.com

  • Set 1: the 12 cities to be covered by the selected hubs. Set 2 consists of sets of cities that each of the 12 cities can cover Hub Location At Western Airlines

    City Cities within 1000 miles

    Atlanta (AT) AT, CH, HO, NO, NY, PI

    Boston (BO) BO, NY, PI

    Chicago (CH) AT, CH, NY, NO, PI

    Denver (DE) DE, SL

    Houston (HO) AT, HO, NO

    Los Angeles (LA) LA, SL, SF

    New Orleans (NO) AT, CH, HO, NO

    New Year (NY) AT, BO, CH, NY, PI

    Pittsburgh (PI) AT, BO, CH, NY, PI

    Salt Lake City (SL) DE, LA, SL, SF, SE

    San Francisco (SF) LA, SL, SF, SE

    Seattle (SE) SL, SF, SE

    One element of Set 2

    Set Covering concept: to find the least number of elements from Set 2 that are able to cover all elements of Set 1

  • Hub Location At Western Airlines Input data: which cities are covered by which potential hubs

    Potential hub City AT BO CH DE HO LA NO NY PI SL SF SE AT 1 0 1 0 1 0 1 1 1 0 0 0 BO 0 1 0 0 0 0 0 1 1 0 0 0 CH 1 0 1 0 0 0 1 1 1 0 0 0 DE 0 0 0 1 0 0 0 0 0 1 0 0 HO 1 0 0 0 1 0 1 0 0 0 0 0 LA 0 0 0 0 0 1 0 0 0 1 1 0 NO 1 0 1 0 1 0 1 0 0 0 0 0 NY 1 1 1 0 0 0 0 1 1 0 0 0 PI 1 1 1 0 0 0 0 1 1 0 0 0 SL 0 0 0 1 0 1 0 0 0 1 1 1 SF 0 0 0 0 0 1 0 0 0 1 1 1 SE 0 0 0 0 0 0 0 0 0 1 1 1

    Set 1

    Set 2

  • Decision variables: x1, , x12: binary variable to indicate if city j is selected as a hub Formulation:

    Hub Location At Western Airlines

  • General Formulation of MIP

  • The Lego problem as an LP:

    C 0 5 10 7.5

    5

    10

    (6, 2)

    Geometry of MIP T MAX 40C + 50T

    s.t. 1C + 2T 10 (Big Bricks)

    4C + 3T 30 (Small Bricks)

    C, T 0

  • What if (only) C must be an integer? Feasible set: The set of parallel line segments inside the LP feasible region

    0

    Geometry of MIP MAX 40C + 50T

    s.t. 1C + 2T 10 (Big Bricks)

    4C + 3T 30 (Small Bricks)

    C, T 0

    C 5 10 7.5

    5

    10

    (6, 2)

    T

  • What if both C and T must be integers? Feasible set: The set of dots with integer valued coordinates inside the LP feasible region

    0

    Geometry of MIP

    C 5 10 7.5

    5

    10

    (6, 2)

    T MAX 40C + 50T s.t. 1C + 2T 10 (Big Bricks)

    4C + 3T 30 (Small Bricks)

    C, T 0

  • Methods: Branch & Bound, Cutting plane, Bad news: Discrete optimization problems are usually hard to solve How hard is hard? Very, very, hard! (Clay Millennium Prize Problem: Is P = NP? ) Good news: State-of-the-art solvers (such as CPLEX) are able to solve many reasonably sized problem of practical importance When to use Linear Optimization and when to use Discrete Optimization? Trade-off between computational efriciency and optimality Do you need sensitivity analysis/shadow prices?

    Solving Discrete Optimization Problems

  • Gain A readily implementable integer solution Loss Speed in solving the problem Sensitivity report and shadow prices! Guaranteed solution (for the relaxed LP) So, ignore integer constraints for large quantities Round off the fractional solution generated by LP Round up or down? Depends on constraint feasibility Not necessary optimal, only approximately optimal

    When to use integer constraints?

  • Excel Solver (Free but not good for IP and NLP) 200 decision variables, 200 constraints Frontline Premium Solver (15-day trial version at www.solver.com) A lot more decision variables & constraints (depend on the engine) Multiple engines to choose from (E.g., Gurobi MIP, LP/ Quadratic) Excel-based IBM/ILOG CPLEX Most famous and widely used Very fast for large-scale optimization problems Need to learn modeling/programming language More expensive Many other solvers: E.g., Gurobi, Mosek, etc.

    Software for Solving Optimization Problems

  • Often there are > 1 correct formulations. Find the easiest one Difriculty: NLP > MIP > LP (generally) Sometimes we can convert nonlinear constraints to linear ones Guidelines for Formulating Optimization Problems

    X1 + 2X2 + X3 2X2 + 3X3 + 1

    - X1 - X2 + X3 1

    X1 + 2X2 + X3 2X2 + 3X3 - 1 (Because 2X1 + 3X2 - 1 < 0) Because 2X1 + 3X2 may be > 0 or < 0 No way to convert! Too bad!

  • Write your problem in the standard format By convention, write all decision variables on the left-hand side and constants on the right Standard math formulation and standard Excel formulation Decisions:

    Le& side Right side

    Constraints: a11 a12 a13 a1n =sumproduct(...) = b1 a41 a42 a43 a4n =sumproduct(...) b4 a81 a82 a83 a8n =sumproduct(...) b8

    c1 c2 c3 cn

    Objec>ve: =sumproduct(F8:L8, F21:L21)

    Guidelines for Formulating Optimization Problems

  • Whenever in doubt, use inequality instead of equality constraint Do not try to calculate or optimize. Leave the problem to Solver Avoid removing seemingly redundant constraints Do not attempt to simplify the problem for Solver

    Always double-check the Solver option Assume Linear Model Assume Non-negative

    Guidelines for Formulating Optimization Problems