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Transportation

Transportation. Chapter 7. Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137. Chapter 6 Organization. A brief history of the transportation problem An engineering modeling approach to problem-solving using mathematics

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Transportation

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  1. Transportation Chapter 7 Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137

  2. Chapter 6 Organization • A brief history of the transportation problem • An engineering modeling approach to problem-solving using mathematics • Modeling and engineering • A transportation problem from an engineering modeling perspective • 7.1 Example Transportation Problem • 7.2 Integer Solutions • 7.4 Validating Models • 7.5 Sensitivity Analysis

  3. 7.1 Example Transportation Problem • We need to transport oranges from California (CA) and Florida (FL) to Michigan (MI), Missouri (MO), and North Carolina (NC)

  4. 7.1 Example Transportation Problem • Supply of oranges • CA – no more than 50.3 tons of oranges • FL – no more than 49.7 tons of oranges • Demand of oranges • MI – at least 33.6 tons of oranges • MO – at least 36.3 tons of oranges • NC – at least 30.1 tons of oranges

  5. 7.1 Example Transportation Problem • Since the oranges are shipped by truck, costs to consider include • Shipping costs per tons of oranges

  6. MI (33.6) $10×NofTCA, MI CA (50.3) $9×NofTFL, MI $9×NofTCA, MO MO (36.3) NC (30.1) $12×NofTCA, NC $8×NofTFL, MO $6×NofTFL, NC Supply in tons FL (49.7) Demand in tons Shipping Costs NofT => Number of Tons 7.1 Example Transportation Problem

  7. MI (33.6) $10×NofTCA, MI CA (50.3) $9×NofTFL, MI $9×NofTCA, MO MO (36.3) NC (30.1) $12×NofTCA, NC $8×NofTFL, MO $6×NofTFL, NC Supply in tons FL (49.7) Demand in tons Shipping Costs NofT => Number of Tons 7.1 Example Transportation Problem

  8. Supply in tons Demand in tons Shipping Costs MI (33.6) $10×NofTCA, MI $9×NofTFL, MI CA (50.3) MO (36.3) $9×NofTCA, MO $8×NofTFL, MO FL (49.7) NC (30.1) $12×NofTCA, NC $6×NofTFL, NC NofT => Number of Tons 7.1 Example Transportation Problem

  9. 7.1 EX Transportation Problem – LP Formulation • Define • Decision Variables, Objective Function, Constraints • Create Excel Spreadsheet • Input decision variables, objective function, constraints, RHS constraint values • Use Solver to find the solution • Output final values, objective function value, LHS constraint values • Analyze results using Answer and Sensitivity Reports

  10. 7.1 EX Transportation Problem – Decision Variables • Let xij be the number of tons of oranges shipped from supply state i to demand state j • i = 1 (CA) or 2 (FL) • j = 1 (MI), 2 (MO), 3 (NC) • Much better than the notation NofTCA, MI

  11. 7.1 EX Transportation Problem – Objective Function • Total Shipping Costs per route • Minimize Shipping Cost • z = $10x11 + $9x12 + $12x13 + $9x21 + $8x22 + $6x23

  12. 7.1 EX Transportation Problem – Supply Constraints • CA and FL have a limited number of tons of oranges to distribute to demand states • CA has a maximum supply of 50.3 tons x11 + x12 + x13≤ 50.3 tons • FL has maximum supply of 49.7 tons x21 + x22 + x23≤ 49.7 tons

  13. 7.1 EX Transportation Problem – Demand Constraints • MI, MO, and NC have a minimum number of tons of oranges to receive from the supply states • MI has minimum demand of 33.6 tons x11 + x21≥ 33.6 tons • MO has minimum demand of 36.3 tons x12 + x22≥ 36.3 tons • NC has minimum demand of 30.1 tons x13 + x23≥ 30.1 tons

  14. 7.1 EX Transportation Problem – NonNegativity Constraints • Cannot have a negative amount of oranges • xij≥ 0 where i= 1, 2 and j = 1, 2, 3

  15. 7.1 EX Transportation Problem – Excel Formulas = SUMPRODUCT($B$7:$G$7,B9:G9) = SUMPRODUCT($B$7:$G$7,B12:G12) = SUMPRODUCT($B$7:$G$7,B16:G16)

  16. 7.1 EX Transportation Problem – Excel Solver

  17. 7.1 EX Transportation Problem – Solution • The minimum cost for shipping 100 tons of oranges from the supply states to the demand states is $823.70 • More specifically

  18. 7.2 Integer Solutions • What if we wanted our number in tons to be an integer value? • How would our LP formulation change?

  19. 7.2 Integer Solutions - Objective Function • Total Shipping Costs per route • Minimize Shipping Cost • z = $10x11 + $7x12 + $15x13 + $8x21 + $10x22 + $5x23

  20. 7.2 Integer Solutions - Solution • The minimum cost for shipping 100 tons of oranges from the supply states to the demand states is $790 • More specifically

  21. 7.4 Validate Our Model • Use data from government websites to validate the shipping costs per ton

  22. In Summary • Transportation problems are specific types of linear programming problems that address moving products from supply to demand • The problems in chapter 7 are all balanced transportation problems

  23. Practicing Transportation • Optional – Read pp. 1 – 5 for background information • Work on problems 7.17 – 7.23 (pp. 6-10 – 6-12), Cape Fear River Basin Water Supply Plan, individually or in pairs • In the HW solutions, the answer to 7.21 is incorrect. The optimal solution should be $32,000 - not $36,000. • When solving these problems, make sure to increase your precision (add more zeros)

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