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Chapter 12

Perhaps the greatest impact of quantitative methods has been in distribution, where they result in billions saved every year.

Transportation and Assignment Problems

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The Transportation Problem: Scheduling Shipments

The following capacity, demand, and unit costs apply for plants and warehouses.

The linear program involves one variable for each cell in the above:

Xij = quantity shipped from plant i to warehouse j i = J, S, T and j = F, N, P, Y

To Warehouse

From Plant Frankfurt New York Phoenix Yokohama Capacity

Juarez $19 $ 7 $ 3 $21 100

Seoul 15 21 18 6 300

Tel Aviv 11 14 15 22 200

Demand 150 100 200 150 600

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The Transportation Problem: LP Formulation

The following objective applies.

Minimize C = 19XJF + 7XJN + 3XJP +21XJY

+ 15XSF +21XSN +18XSP + 6XSY

+ 11XTF +14XTN +15XTP +22XTY

Subject to:

XJF + XJN + XJP + XJY = 100 (Juarez Capacity)

XSF +XSN + XSP + XSY = 300 (Seoul Capacity)

XTF +XTN + XTP + XTY = 200 (Tel Aviv Capacity)

XJF + XSF + XTF = 150 (Frankfurt Demand)

XJN + XSN + XTN = 100 (New York Demand)

XJP + XSP + XTP = 200 (Phoenix Demand)

XJY + XSY + XTY = 150 (Yokohama Demand)

where all Xij’s > 0

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Solving Transportation Problems

Although transportation problems may be solved using the general procedure for any linear program, there is a faster way.

The transportation method is a special-purpose algorithm utilizing the features of the shipping schedule and constraint forms: The quantities in each row sum to the row total

(capacity). The quantities in each column sum to the

column total (demand). It is user-friendly with limited arithmetic.

Small problems may be easily solved by hand. Symbols are not needed.

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Solving theTransportation Problem

Get a starting solution by filling a blank shipment schedule using a procedure like the northwest corner method. Compute total cost, C = $11,500.

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Solving theTransportation Problem

Get a set of row and column numbers (so non-empty cells have cost = row no. + col. no.). Use zero for first row.

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Solving theTransportation Problem

Find the entering cell (now empty). It will have best improvement (cost – row no. – col. no.). Then find the closed-loop path, off-setting shifts, if its quantity is raised.

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Solving theTransportation Problem

Change quantities, moving the maximum amount (i.e., the minimum losing-cell quantity, 100, around path). C change = impvt. × qty. = $19(100) = $1,900. Do next iteration.

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Solving theTransportation Problem

Previous iteration changes cost by $19 × 50 = $950. Continue until no improvements are possible. Here, the change in cost is $23 × 100 = $2,300.

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Solving theTransportation Problem

There was a tie for exiting variable. Cell TY ended up with (0) quantity. Only one cell can go blank. Here zero is the smallest losing quantity. Only (0) moves. C won’t change.

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Solving theTransportation Problem

Here, the change in cost is $1 × 100 = $100.

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Solving theTransportation Problem

Since there is no further improvement possible, the optimal solution has been reached.

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Solving Transportation Problems with QuickQuant

The ski-distribution problem is entered:

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Solving Transportation Problems with QuickQuant

After the data are entered, the run menu is pulled down and Detailed Solve selected.

That launches individual iterations to be seen on screen.

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Solving Transportation Problems with QuickQuant

The first iteration provides:

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Solving Transportation Problems with QuickQuant

After all iterations, the solution found before is provided:

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Assignment Problem

The following data apply for persons and jobs.

The linear program involves one variable for each cell in the above:

Xij = Fraction of time person i is assigned to job j

i = A, B, C and j = D, G, L

Time to Complete One Job

Individual Drilling Grinding Lathe

Ann 5 min. 10 min. 10 min.

Bud 10 5 15

Chuck 15 15 10

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Assignment Problem The following objective applies.

Minimize C = 5XAD + 10XAG + 10XAL

+ 10XBD + 5XBG + 15XBL

+ 11XCD + 14XCG + 15XCL

Subject to:

XAD + XAG + XAL = 1 (Ann’s Availability)

XBD + XBG + XBL = 1 (Bud’s Availability)

XCD + XCG + XCL = 1 (Chuck’s Availability)

XAD + XBD + XCD = 1 (drill-press requirement)

XAG + XBG + XCG = 1 (grinder requirement)

XAL + XBL + XCL = 1 (lathe requirement)

where all Xij’s > 0

Solution: XAD = 1 (Ann to Drilling) XBG = 1 (Bud to Grinding)

XCL = 1 (Chuck to Lathe) C = 20

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Solving Assignment Problems Assignment problems are solved using the transportation

method. Here’s an iteration of a larger problem.

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Solving Assignment Problems This problem has tying optimal solutions. Any

combination of the following (with job sharing) would also be optimal.

Why don’t we just enumerate all possibilities? Wouldn’t that be faster?

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Solving Assignment Problems

For the six-person and job problem, there are

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720possibilities (without job sharing). Try doing this by trial and error.

For ten persons and jobs, the number of possibilities would be

10! = 10 × 9 × 8 × 7 × 6! = 3,328,800 The transportation method is a very

efficient way to solve such problems, especially with the computer.

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Solving Transportation Problems with a Spreadsheet

Step 1: Write out the formulation table.

Step 2: Put the formulation table into a spreadsheet.

Step 3: Use Excel’s Solver to obtain a solution.

Spreadsheets can be used to solve transportation problems just like they are used to solve linear programs.

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The Formulation Table(Figure 12-16)

FromPlant Frankfurt New YorkPhoenix Yokohama Capacity

Juarez 19 7 3 21 100Seoul 15 21 18 6 300Tel Aviv 11 14 15 22 200Demand 150 100 200 150 C(min)

To Warehouse

The formulation table arranges the problem in a tabular format, as shown below for the ski distribution problem. The shipping costs are shown in the table with the plant capacities in the right-hand margin and the warehouse demands in the lower margins.

The formulation table arranges the problem in a tabular format, as shown below for the ski distribution problem. The shipping costs are shown in the table with the plant capacities in the right-hand margin and the warehouse demands in the lower margins.

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The Formulation Table in a Spreadsheet

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2345678

A B C D E F

FromPlant Frankfurt New York Phoenix Yokohama Capacity

Juarez 19 7 3 21 100Seoul 15 21 18 6 300Tel Aviv 11 14 15 22 200Demand 150 100 200 150 C(min)

To Warehouse

Ski Shipment-Scheduling Illustration

The numbers in the Excel spreadsheet come from the formulation table.

The numbers in the Excel spreadsheet come from the formulation table.

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The Expanded Spreadsheet (Figure 12-17)

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23456789

101112131415161718

A B C D E F G H

FromPlant Frankfurt New York Phoenix Yokohama Capacity

Juarez 19 7 3 21 100Seoul 15 21 18 6 300Tel Aviv 11 14 15 22 200Demand 150 100 200 150 Cost

Solution $0FromPlant Frankfurt New York Phoenix Yokohama Total

Juarez 0 0 0 0 0Seoul 0 0 0 0 0Tel Aviv 0 0 0 0 0Total 0 0 0 0

To Warehouse

To Warehouse

Ski Shipment-Scheduling Illustration

121314

F=SUM(B12:E12)=SUM(B13:E13)=SUM(B14:E14)

15B C D E

=SUM(B12:B14) =SUM(C12:C14) =SUM(D12:D14) =SUM(E12:E14)

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F=SUMPRODUCT(B5:E7,B12:E14)

1. The expanded spreadsheet contains the previous Excel spreadsheet in the upper portion (A1:F8) and a table for the solution (shipping quantities) in the lower portion (A10:F15).

1. The expanded spreadsheet contains the previous Excel spreadsheet in the upper portion (A1:F8) and a table for the solution (shipping quantities) in the lower portion (A10:F15).

2. All the formulas necessary to use Solver are in the expanded table.

2. All the formulas necessary to use Solver are in the expanded table.

3. The objective function is in cell F9. The formulas is =SUMPRODUCT(B5:E7,B12:E14)

3. The objective function is in cell F9. The formulas is =SUMPRODUCT(B5:E7,B12:E14)

4. The sum of the shipments from Juarez is in cell F12. Its formulas is =SUM(B12:E12) and it is copied down to cells F13:F14.

4. The sum of the shipments from Juarez is in cell F12. Its formulas is =SUM(B12:E12) and it is copied down to cells F13:F14.

5. The sum of the shipments to Frankfurt is in cell B15. Its formula is =SUM(B12:B14) and it is copied to cells C15:E15.

5. The sum of the shipments to Frankfurt is in cell B15. Its formula is =SUM(B12:B14) and it is copied to cells C15:E15.

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Using Excel’s Solver to Solve Transportation Problems

Click on Tools on the menu bar, select the Solver option, and the Solver Parameters dialog box shown next appears.

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The Solver Parameters Dialog Box(Figure 12-18)

1. Enter the value of the objective function, F9, in the Target Cell line, either with or without the $ sign.

1. Enter the value of the objective function, F9, in the Target Cell line, either with or without the $ sign.

2. The Target Cell is to be maximized so click on Min in the Equal To line.

2. The Target Cell is to be maximized so click on Min in the Equal To line.

3. Enter the decision variables in the By Changing Cells line, B12:E14.

3. Enter the decision variables in the By Changing Cells line, B12:E14.

4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.

4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.

NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

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The Add Constraint Dialog Box1. Enter B15:E15 (or $B$15:$E$15) in the Cell Reference line. These cells give the total amount shipped to each warehouse.

1. Enter B15:E15 (or $B$15:$E$15) in the Cell Reference line. These cells give the total amount shipped to each warehouse.

2. Enter = as the sign because the shipments must be equal to the requirements, given next in Step 3.

2. Enter = as the sign because the shipments must be equal to the requirements, given next in Step 3.

3. Enter the requirements for each warehouse B8:E8 in the Constraint line (or =$B$8:$E$8).

3. Enter the requirements for each warehouse B8:E8 in the Constraint line (or =$B$8:$E$8).

4. Click Add and repeat Steps 1 - 3 for the shipments from each plant to make sure they are equal to the plant capacities. After this, click OK.

4. Click Add and repeat Steps 1 - 3 for the shipments from each plant to make sure they are equal to the plant capacities. After this, click OK.

Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.

Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.

If you need to change a constraint, the Change Constraint dialog box functions just like this one.

If you need to change a constraint, the Change Constraint dialog box functions just like this one.

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Solver’s Answer Report

To get Solver’s Answer Report, highlight Answer Report in the Report box of the Solver Results dialog box before clicking the OK button.

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Spreadsheet withOptimal Solution and Answer Report (Figure 12-19 )

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23456789

101112131415

A B C D E F G H I J K

FromPlant Frankfurt New York Phoenix Yokohama Capacity

Juarez 19 7 3 21 100Seoul 15 21 18 6 300Tel Aviv 11 14 15 22 200Demand 150 100 200 150 C(min)

Solution $6,250FromPlant Frankfurt New York Phoenix Yokohama Total

Juarez 0 0 100 0 100Seoul 50 0 100 150 300Tel Aviv 100 100 0 0 200Total 150 100 200 150

To Warehouse

To Warehouse

Ski Shipment-Scheduling Illustration

Cell Name Original Value Final ValueB12 Juarez Frankfurt 0 0C12 Juarez New York 0 0D12 Juarez Phoenix 0 100E12 Juarez Yokohama 0 0B13 Seoul Frankfurt 0 50C13 Seoul New York 0 0D13 Seoul Phoenix 0 100E13 Seoul Yokohama 0 150B14 Tel Aviv Frankfurt 0 100C14 Tel Aviv New York 0 100D14 Tel Aviv Phoenix 0 0E14 Tel Aviv Yokohama 0 0

1. To solve other problems:1. To solve other problems:

4. For bigger problems or if dummies are needed, insert additional rows or columns. Insert them in the middle of the table and not at the beginning or the end. Copy the formulas in column F and row 15 to any new cells created by the insertions. Check to make sure the ranges of the formulas in the Solver Parameters dialog box are correct.

4. For bigger problems or if dummies are needed, insert additional rows or columns. Insert them in the middle of the table and not at the beginning or the end. Copy the formulas in column F and row 15 to any new cells created by the insertions. Check to make sure the ranges of the formulas in the Solver Parameters dialog box are correct.

2. Enter the data: the capacities in cells F5:F7 and their source names in cells A5:A7, the demands in cells B8:E8 and the destination names in cells B4:E4, and the costs in cells B5:E7.

2. Enter the data: the capacities in cells F5:F7 and their source names in cells A5:A7, the demands in cells B8:E8 and the destination names in cells B4:E4, and the costs in cells B5:E7.

3. To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button.

3. To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button.

5. Only a portion of the Answer Report is shown here because of the lack of space.

5. Only a portion of the Answer Report is shown here because of the lack of space.