ENGM 631 Optimization Transportation Problems

Prototype Example K-Log Lumber Mill Warehouse

Prototype Example 10 8 7 K-Log Lumber Mill Warehouse

Prototype Example 6 10 12 8 11 7 K-Log Lumber Mill Warehouse

Prototype Example 5 6 10 12 13 8 11 7 7 K-Log Lumber Mill Warehouse

Prototype RC DO OC SF AL SP 10 7 8 13 5 6 11 12

Prototype Demand Supply RC DO OC SF AL SP 10 7 8 13 5 6 11 12 150 80 120 130 100 120

Prototype Demand Supply 1 6 5 4 2 3 10 7 8 13 11 12 150 80 120 130 100

Prototype Min Z = Transportation Costs s.t. Total amount shipped from plant i = Capacity at i Demand at each Warehouse is satisfied

Prototype Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36

Prototype Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36 s.t. X14 + X15 + X16 = 130 X24 + X25 + X26 = 100 X34 + X35 + X36 = 120

Prototype Min Z = 10X14 + 7X15 + 8X16 + 13X24 + 7X25 + 5X26 + 6X34 + 11X35 + 12X36 s.t. X14 + X15 + X16 = 130 X24 + X25 + X26 = 100 X34 + X35 + X36 = 120 X14 + X24 + X34 = 150 X15 + X25 + X35 = 80 X16 + X26 + X36 = 120

Prototype (re-index warehouse) Min Z = 10X11 + 7X12 + 8X13 + 13X21 + 7X22 + 5X23 + 6X31 + 11X32 + 12X33 s.t. X11 + X12 + X13 = 130 X21 + X22 + X23 = 100 X31 + X32 + X32 = 120 X11 + X21 + X31 = 150 X12 + X22 + X32 = 80 X13 + X23 + X33 = 120

General Formulation Transportation Problem Min Z c X s t i m d j n ij = å 1 2 . , Also, requires that supply matches demand.

General Format Transportation Problem Also, requires that supply matches demand.

Excel Solver Setup

Excel Solver Setup

Excel Solver Setup Note Excel Solver does not use a special transportation problem method. It just solves the problem with the usual LP software. For larger problems Excel Solver will be considerably slower than software designed to for transportation problems

Transportation Tableau

Transportation Tableau Total Demand = Total Supply

Initial Feasible Solution Northwest Corner requires m+n-1 basic variables Vogel’s Approximation Russel’s Approximation (Not done for class)

Initial Feasible Solution Northwest Corner

Initial Feasible Solution Northwest Corner

Initial Feasible Solution Total Cost = 10(130) + 13(20) + 7(80) + 11(0) + 12(120) = $3,560

Clever Idea Suppose we can find a loop to move units around.

Clever Idea Suppose we can find a loop to move units around.

Clever Idea Suppose we can find a loop to move units around.

Clever Idea Suppose we can find a loop to move units around.

Clever Idea Suppose we can find a loop to move units around.

Clever Idea For each unit I can move around the loop, I can save -5 + 12 - 11 + 7 = 3 per unit of flow

Clever Idea I can move at most 80 units around this loop

Clever Idea I can move at most 80 units around this loop

Clever Idea Total Cost = 10(130) + 13(20) + 11(80) + 5(80) + 12(40) = $3,320 = $3,560 - 3(80)

Finding the Best Loop Basic Cell cij = ui + vj Nonbasic Cell dij = cij - ui – vj Note: book doesn’t use d’s page 321

Transportation Algorithm Arbitrarily select u2 = 0

Transportation Algorithm 13 = 0 + v1 v1 = 13 7 = 0 + v2 v2 = 7

Transportation Algorithm 10 = u1 + 13 u1 = -3 11 = u3 + 7 u3 = 4

Transportation Algorithm 12 = 4 + v3 v3 = 8

Transportation Algorithm 3 d12 = 7 -(-3) - 7 = +3

Transportation Algorithm 3 3 d13 = 8 -(-3) - 8 = +3

Transportation Algorithm 3 3 3 d23 = 5 -0 - 8 = -3

Transportation Algorithm 3 3 3 11 d31 = 6 -4 - 13 = -11

Transportation Algorithm 3 3 3 11 Note: -3 is the same thing we got earlier by finding a loop.

Transportation Algorithm 3 3 3 11 Let non-basic cell with largest -dij enter basis.

Transportation Algorithm Find a feasible loop.

Transportation Algorithm Move the maximim unit flow around the loop.

Transportation Algorithm Move the maximim unit flow around the loop. Total Cost = 10(130) + 13(20) + 7(80) + 12(120) = $3,560

Transportation Algorithm Note that ui and vj must now be recomputed from new basis. Arbitrarily select v1 = 0

Class Problem Find u1, u2, u3, v2, v3 dij for non-basic cells

Class Problem 8 14 Find u1, u2, u3, v2, v3 and dij for non-basic cells

Class Problem 14 Find most -dij. Find feasible loop for transfer.

Class Problem Find most -dij. Find feasible loop for transfer.

Class Problem Total Cost = 10(130) + 7(80) + 5(20) + 6(20) + 12(120) = $3,280 = 3,560 - 20(14)

Class Problem Arbitrarily select u2 = 0. Find other multiplier values.

Class Problem Arbitrarily select u2 = 0. Find other multiplier values.

Class Problem Arbitrarily select u2 = 0. Find other multiplier values.

Class Problem Arbitrarily select u2 = 0. Find other multiplier values.

Class Problem Find all dij values. Select largest –dij to leave basis. 11 8 3 Find all dij values. Select largest –dij to leave basis.

Class Problem Find largest -dij. Find feasible loop for transfer.

Class Problem Total Cost = 10(50) + 7(80) + 5(100) + 6(100) + 12(20) = $2,400 = 3,280 - 11(80)

Class Problem Arbitrarily select u1 = 0. Find other multiplier values.

Class Problem Arbitrarily select u1 = 0. Find other multiplier values.

Class Problem Arbitrarily select u1 = 0. Find other multiplier values.

Class Problem Arbitrarily select u1 = 0. Find other multiplier values.

Class Problem Arbitrarily select u1 = 0. Find other multiplier values.

Class Problem Find all dij values. Select largest –dij to leave basis. 8 Find all dij values. Select largest –dij to leave basis.

Class Problem 8 Find largest -dij. Find feasible loop.

Class Problem Find largest -dij. Find feasible loop.

Class Problem Total Cost = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = $2,240 = 2,400 - 8(20)

Class Problem Arbitrarily select u1 = 0.

Class Problem Arbitrarily select u1 = 0. Find other multipliers.

Class Problem Arbitrarily select u1 = 0. Find other multipliers.

Class Problem 6 3 8 8 All dij > 0 Solution is optimal.

Class Problem Z = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = 2,240 6 3

Initialization (Vogel’s)

Initialization (Vogel’s) Table 8.17 H&L

Initialization (Vogel’s) Table 8.17 H&L

Initialization (Vogel’s) Table 8.17 H&L

Initialization (Vogel’s) Table 8.17 H&L

Initialization (Vogel’s) Table 8.17 H&L

Initialization (Vogel’s) Table 8.17 H&L

Dummy Warehouse Suppose total supply exceeds total demand.

Dummy Warehouse Add dummy warehouse with 0 cost.

Dummy Supplier Suppose total demand exceeds total supply.

Dummy Supplier

Final slide Transportation Problem Northwest corner Method Transportation Tableau Method Vogler’s approximation (Initialization)