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Network Models

Network Models. Lecture 18 The Transportation Algorithm II. The Initialized Transportation Simplex Tableau. LCSP. Recall at the end of the last lecture we had initialized the Transportation Simplex Tableau starting from the results of the Least Cost Starting Procedure.

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Network Models

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  1. Network Models Lecture 18 The Transportation Algorithm II FOMGT 353 Introduction to Management Science

  2. The Initialized TransportationSimplex Tableau LCSP • Recall at the end of the last lecture we had initialized the Transportation Simplex Tableau starting from the results of the Least Cost Starting Procedure. FOMGT 353 Introduction to Management Science

  3. Transportation Simplex cont… FOMGT 353 Introduction to Management Science

  4. The Transportation Simplex Method –Identify Entering Variable • Recall from the Generalized Simplex Method that a variable is an Entering Variable in a Minimization problem if its Cij-Zij is the most negative of the Cij-Zij values of the non-basic variables. • So we need to calculate Cij-Zij values for all the non-basic variables. • So first we need to calculate Ui and Vj values for the Basic Variables using the relationship: • Cij – Ui + Vj = 0 FOMGT 353 Introduction to Management Science

  5. Transportation Simplex cont… FOMGT 353 Introduction to Management Science

  6. The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Ui + Vj = 0 for x12 to calculate V2. • C12 – U1 + V2 = 0 => 12 – 0 + V2 = 0 • => V2 = -12 FOMGT 353 Introduction to Management Science

  7. The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3. • C13 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100 FOMGT 353 Introduction to Management Science

  8. The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Ui + Vj = 0 for x32 to calculate U3. • C32 – U3 + V2 = 0 => 10 – U3 + (-12) = 0 • => U3 = -2 FOMGT 353 Introduction to Management Science

  9. The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Ui + Vj = 0 for x31 to calculate V1. • C31 – U3 + V1 = 0 => 5 – (-2) + V1 = 0 • => V1 = -7 FOMGT 353 Introduction to Management Science

  10. The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2. • C21 – U2 + V1 = 0 => 4 – U2 + (-7) = 0 • => -3 – U2 = 0 => -3 = U2 FOMGT 353 Introduction to Management Science

  11. The Next Steps.. FOMGT 353 Introduction to Management Science

  12. The Most –ve Cij – Zij is “–1” so x11 is the Entering Variable! The Transportation Simplex Method –Identify Entering Variable LCSP • We use the relationship Cij – Zij = Cij – Ui + Vj for x11 to calculate C11 – Z11. • We use the relationship Cij – Zij = Cij – Ui + Vj for x22 to calculate C22 – Z22. • C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-3) + (-12) = 2 • We use the relationship Cij – Zij = Cij – Ui + Vj for x33 to calculate C33 – Z33. • We use the relationship Cij – Zij = Cij – Ui + Vj for x23 to calculate C23 – Z23. • C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-3) + (-100) = 3 • C33 – Z33 = C33 – U3 + V3 • C33 – Z33 = 100 – (-2) + (-100) = 2 • C11 – Z11 = C11 – U1 + V1 • C11 – Z11 = 6 – 0 + (-7) = -1 FOMGT 353 Introduction to Management Science

  13. The Pivot FOMGT 353 Introduction to Management Science

  14. - + + - The Transportation Simplex Pivot LCSP x11 is the Entering Variable! • We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable. • Label the Entering Variable with a “+” and the other variables around the Cycle “-”, “+” and “-”. FOMGT 353 Introduction to Management Science

  15. The Transportation Simplex Pivot LCSP • Then work out the max x11 can be increased by without decreasing x12 or x31 below 0. i.e. Min(x12, x31) = 10000 - + - + FOMGT 353 Introduction to Management Science

  16. The Transportation Simplex Pivot LCSP • Increase x11 and x32 by 10,000. • Decrease x12 and x31 by 10,000. FOMGT 353 Introduction to Management Science

  17. A Reminder… FOMGT 353 Introduction to Management Science

  18. The Transportation Simplex LCSP • Now, we start again calculating the revised values for Ui and Vj. FOMGT 353 Introduction to Management Science

  19. The Transportation Simplex LCSP • We use the relationship Cij – Ui + Vj = 0 for x11 to calculate V1. • C11 – U1 + V1 = 0 => 6 – 0 + V1 = 0 • => V1 = -6 FOMGT 353 Introduction to Management Science

  20. The Transportation Simplex LCSP • We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3. • C13 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100 FOMGT 353 Introduction to Management Science

  21. The Transportation Simplex LCSP • We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2. • C21 – U2 + V1 = 0 => 4 – U2 + (-6) = 0 • => -2 – U2 = 0 => -2 = U2 FOMGT 353 Introduction to Management Science

  22. The Transportation Simplex LCSP • We use the relationship Cij – Ui + Vj = 0 for x31 to calculate U3. • C31 – U3 + V1 = 0 => 5 – U3 + (-6) = 0 • => U3 = -1 FOMGT 353 Introduction to Management Science

  23. The Transportation Simplex LCSP • We use the relationship Cij – Ui + Vj = 0 for x32 to calculate V2. • C32 – U3 + V2 = 0 => 10 – (-1) + V2 = 0 • => 11 + V2 = 0 => V2 = -11 FOMGT 353 Introduction to Management Science

  24. Another Reminder… FOMGT 353 Introduction to Management Science

  25. The Transportation Simplex LCSP • We use the relationship Cij – Zij = Cij – Ui + Vj for x12 to calculate C12 – Z12. • C33 – Z33 = C33 – U3 + V3 • C33 – Z33 = 100 – (-1) + (-100) = 1 • We use the relationship Cij – Zij = Cij – Ui + Vj for x23 to calculate C23 – Z23. • C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-2) + (-11) = 2 • We use the relationship Cij – Zij = Cij – Ui + Vj for x22 to calculate C22 – Z22. • C12 – Z12 = C12 – U1 + V2 • C12 – Z12 = 12 – 0 + (-11) = 1 • We use the relationship Cij – Zij = Cij – Ui + Vj for x33 to calculate C33 – Z33. • C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-2) + (-100) = 2 FOMGT 353 Introduction to Management Science

  26. The Transportation Simplex LCSP • There are no negative Cij – Zij values so stop. This is an optimal solution! • We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal. FOMGT 353 Introduction to Management Science

  27. Another Example Starting from the Initial Feasible Solution generated using the Vogel’s Approximation Starting Procedure FOMGT 353 Introduction to Management Science

  28. The Transportation Simplex Method –Identify Entering Variable VAM • We use the relationship Cij – Ui + Vj = 0 for x11 to calculate V1. • C11 – U1 + V1 = 0 => 6 – 0 + V1 = 0 • => V1 = -6 FOMGT 353 Introduction to Management Science

  29. The Transportation Simplex Method –Identify Entering Variable VAM • We use the relationship Cij – Ui + Vj = 0 for x21 to calculate U2. • C21 – U2 + V1 = 0 => 4 – U2 + (-6) = 0 • => U2 = -2 FOMGT 353 Introduction to Management Science

  30. The Transportation Simplex Method –Identify Entering Variable VAM • We use the relationship Cij – Ui + Vj = 0 for x13 to calculate V3. • C12 – U1 + V3 = 0 => 100 – 0 + V3 = 0 • => V3 = -100 FOMGT 353 Introduction to Management Science

  31. The Transportation Simplex Method –Identify Entering Variable VAM • We use the relationship Cij – Ui + Vj = 0 for x33 to calculate U3. • C33 – U3 + V3 = 0 => 100 – U3 + (-100) = 0 • => U3 = 0 FOMGT 353 Introduction to Management Science

  32. The Transportation Simplex Method –Identify Entering Variable VAM • We use the relationship Cij – Ui + Vj = 0 for x32 to calculate V2. • C32 – U3 + V2 = 0 => 10 – 0 + V2 = 0 • => 10 – 0 + V2 = 0 => -10 = V2 FOMGT 353 Introduction to Management Science

  33. The Transportation Simplex Method –Identify Entering Variable VAM x31 is the Entering Variable! • We use the relationship Cij – Zij = Cij – Ui + Vj for x12 to calculate C12 – Z12. • We use the relationship Cij – Zij = Cij – Ui + Vj for x31 to calculate C31 – Z31. • We use the relationship Cij – Zij = Cij – Ui + Vj for x23 to calculate C23 – Z23. • We use the relationship Cij – Zij = Cij – Ui + Vj for x22 to calculate C22 – Z22. • C22 – Z22 = C22 – U2 + V2 • C22 – Z22 = 11 – (-2) + (-10) = 3 • C12 – Z12 = C12 – U1 + V2 • C12 – Z12 = 12 – 0 + (-10) = 2 • C23 – Z23 = C23 – U2 + V3 • C23 – Z23 = 100 – (-2) + (-100) = 2 • C31 – Z31 = C31 – U3 + V1 • C31 – Z31 = 5 – 0 + (-6) = -1 FOMGT 353 Introduction to Management Science

  34. - + - + The Transportation Simplex Pivot VAM • We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable. x31 is the Entering Variable! • Label the Entering Variable with a “+” and the other variables around the Cycle “-”, “+” and “-”. FOMGT 353 Introduction to Management Science

  35. The Transportation Simplex Pivot VAM • Then work out the max x31 can be increased by without decreasing x11 or x33 below 0. i.e. Min(x11, x33) = 5000 - + - + FOMGT 353 Introduction to Management Science

  36. The Transportation Simplex Pivot VAM • Increase x31 and x13 by 5,000. • Decrease x11 and x33 by 5,000. FOMGT 353 Introduction to Management Science

  37. The Transportation Simplex VAM • We have seen this before! • It is the optimal solution!! • We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal. FOMGT 353 Introduction to Management Science

  38. A More Complex Pivot • If we look at the result of applying the North West Corner method to derive a Basic Feasible Solution. • We will see in the following slides, that we need to undertake a more complex pivot involving a cycle which has more than four elements… FOMGT 353 Introduction to Management Science

  39. A More Complex Pivot NWC x13 is the Entering Variable! • We look for a “Cycle” of at least 3 Basic Variables and the Entering Variable. • A cycle which includes the four corners looks good, but if we use this we get 6 Basic Variables! FOMGT 353 Introduction to Management Science

  40. A More Complex Pivot NWC • So we need to find a “Cycle” of at least 5 Basic Variables and the Entering Variable. - + Remember x13 is the Entering Variable! - + 5000 - + • We then need to find the minimum leaving variable… • So, add 5000 to x13, x32, x21 and deduct 5000 from x33, x22 and x11 to complete the pivot. FOMGT 353 Introduction to Management Science

  41. A More Complex Pivot NWC • This time we can find a simpler pivot involving 3 Basic Variables and the Entering Variable. - + 5000 - + FOMGT 353 Introduction to Management Science

  42. The Transportation Simplex NWC • We have seen this before! • It is the optimal solution!! • We ship 10,000 from Boston to New York, with 10,000 slack capacity in Boston. We ship 10,000 from Hartford to New York, 5,000 from Worcester to New York and 20,000 from Worcester to Montreal. FOMGT 353 Introduction to Management Science

  43. Reading and Homework. • Read Network Model Algorithms Supplement 5 Section III • Read the Assignment Problem Handout. FOMGT 353 Introduction to Management Science

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