1. Transportation and Assignment Problems
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

2. 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 22
200
Demand
150
600

3. 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 = (Juarez Capacity)
XSF +XSN + XSP + XSY = (Seoul Capacity)
XTF +XTN + XTP + XTY = (Tel Aviv Capacity)
XJF + XSF + XTF = (Frankfurt Demand)
XJN + XSN + XTN = (New York Demand)
XJP + XSP + XTP = (Phoenix Demand)
XJY + XSY + XTY = (Yokohama Demand)
where all Xij’s > 0

4. 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.

5. Solving the Transportation 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.

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

7. Solving the Transportation 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.

8. Solving the Transportation 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.

9. Solving the Transportation 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.

10. Solving the Transportation 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.

11. Solving the Transportation Problem
Here, the change in cost is - $1 × 100 = - $100.

12. Solving the Transportation Problem
Since there is no further improvement possible, the optimal solution has been reached.

13. Solving Transportation Problems with QuickQuant
The ski-distribution problem is entered:

14. 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.

15. Solving Transportation Problems with QuickQuant
The first iteration provides:

16. Solving Transportation Problems with QuickQuant
After all iterations, the solution found before is provided:

17. Time to Complete One Job
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.
Bud 10 5 15
Chuck

18. 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

19. Solving Assignment Problems
Assignment problems are solved using the transportation method. Here’s an iteration of a larger problem.

20. 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?

21. Solving Assignment Problems
For the six-person and job problem, there are
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
possibilities (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.

22. Solving Transportation Problems with a Spreadsheet
Spreadsheets can be used to solve transportation problems just like they are used to solve linear programs.
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.

23. The Formulation Table (Figure 12-16)
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.

24. The Formulation Table in a Spreadsheet
The numbers in the Excel spreadsheet come from the formulation table.

25. The Expanded Spreadsheet (Figure 12-17)
2. All the formulas necessary to use Solver are in the expanded table.
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).
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.
3. The objective function is in cell F9. The formulas is =SUMPRODUCT(B5:E7,B12:E14)
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.

26. 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.

27. 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.
The Solver Parameters Dialog Box (Figure 12-18)
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.
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.
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.

28. The Add Constraint Dialog Box
Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.
1. Enter B15:E15 (or $B$15:$E$15) in the Cell Reference line. These cells give the total amount shipped to each warehouse.
3. Enter the requirements for each warehouse B8:E8 in the Constraint line (or =$B$8:$E$8).
4. Click Add and repeat Steps for the shipments from each plant to make sure they are equal to the plant capacities. After this, click OK.
2. Enter = as the sign because the shipments must be equal to the requirements, given next in Step 3.
If you need to change a constraint, the Change Constraint dialog box functions just like this one.

29. 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.

30. Spreadsheet withOptimal Solution and Answer Report (Figure 12-19 )
3. To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button.
Spreadsheet withOptimal Solution and Answer Report (Figure )
1. To solve other problems:
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.
5. Only a portion of the Answer Report is shown here because of the lack of space.
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.