MATH 251 ACTIVITY 4: Spreadsheet solution and Extensions of Transportation problems
WHY:
Although transportation models can be modeled and solved as linear programming models, the particular structure
and the large number of variables (relative to the constraints) makes use of a special structure for the spreadsheet
model preferable
Many optimization problems that are not directly about moving commodities nevertheless fit the transportation
problem setup and solution methods. In particular, division of work among several work centers (with
profitability/cost for each type of work, each type of center known) and scheduling of production for different time
periods often fit transportation-type models – though these tend to be maximization (rather than minimization).
LEARNING OBJECTIVES:
1.
Work as a team, using the team roles.
2.
Learn how to set up & solve transportation models on a spreadhseet.
3.
See how the model can be extended to deal with special requirements or different problems.
CRITERIA:
1.
Success in completing the exercises.
2.
Success in working as a team
RESOURCES:
1.
Your class notes on transportation models – and the attached sheet “Spreadsheet Model of a transportation problem”
2.
The spreadsheet example “Transportationexample.xls” available on Blackboard [setup for P&T Trucking example]
3.
Your text section 4.2
4.
Microsoft Excel, running on the campus computer network
5.
45 minutes
PLAN:
1.
Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3
2.
Work through the exercises given here - be sure everyone understands all results. Hand in your solutions and email me the
Excel workbook (as an attachment).[send a copy to each member of the team, also to use for added work].
3.
Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade.
EXERCISES:
1.
Model and solve the problem (Using Excel) and answer questions a and b (not c and d) for Exercise #8 on p.241 in your text.
Be sure you actually write out your answer (using words). Some things to note:
a. Demands are greater than the supply. You must either set up a dummy “supply” or use “less than or equal” constraints
(rather than “equal”) on the demands.
b. Figuring out the per unit cost for each route involves handling costs at both ends as well as the mileage cost.
2.
Model and solve the problem in the production example in the notes (“Transportation problems”). Note that product 3 cannot
be produced at plant 2 – set the cost at least ten times as large as any other cost. [If that doesn’t force the cell to have value 0,
re-run the model with an even higher cost]
CRITICAL THINKING QUESTIONS:(answer individually in your journal)
1.
If the amounts in Exercise 2 represented per unit profits, how would we handle the problem? How would we represent the fact
that Plant 2 cannot manufacture product 3?
2.
If the supply limits and demand levels are integers, our hand solution method will always give integer solutions (the amounts
in each cell will be integers) - why?
SKILL EXERCISES:(This is the assignment due next class) Be sure you give the solutions in words, and answer all questions. Text
p. 239 (Ch 4) #8c,d (need to add additional constraints – can add new rows below the “solutions” table and new constraints in
Solver), 12, 21, 30, 32
Spreadsheet model of a transportation problem:
[extracted & slightly amplified from typed notes]
The spreadsheet model closely parallels the setup for the hand solution. It is not convenient to try to put a cost and an amount into the
same cell, but it is convenient to make a table of costs with supply and demand . Thus we have two tables, the parameter table
(containing the costs, with demand and supply amounts along the outside] and the solution table containing the changing cells –
repreesenting the variables]. We do not write out the variables and constraints in algebraic form.
Each table has a row for each source, a column for each destination. We do not need to balance the table with dummy rows or
columns, because the Solver [unlike the hand solution] will allow use of inequality constraints.
Each table also has a column & a row (usually shown as “marginal” values) for "totals" - to set up constraints formula in total is
=(sum(row)) or =sum(column)
The solution table also has:
1. A cell for total cost/profit: formula is “= sumproduct (parameter table cells, solution table cells”)
2. An additional row and an additional column [along the outside] for calculating totals – to compare to constraints
[for row “=sum(row range)” for column “=sum(column range)”
The constraints are of the form rowtotal <= supply , columntotal >= demand
3. A cell (shown at corner) for “total cost’ or “total profit” calculated by “=sumproduct(solution table range,
parameter table range)”
If some entry in solution must be 0, or must be "at least" some value, this also gives a constraint which can be entered in the solver.
Solver setup involves minimizing (if it's cost) or maximizing (if it's profit) the "total cost/profit cell" subject to the
constraints - also need the "Assume Linear Model" and "Assume Non-Negative" options. We do not need to specify that the solution
values must be integer - that will take care of itself.
See the Excel file “Transportationmodel.xls” available via Blackboard [with formulas and solve setup filled in] – cell entries (but not
formulas) shown here:
P&T Trucking Company model - transporation problem
Amount shipped
Destination
Sacra
Salt Lake
Rapid
Source
Bellinhgam
Eugene
Albert Lea
Total
0
>=
80
Demand
Source
Albuq
Bellingham
Eugene
Albert Lea
Demand
Sacra
464
352
995
80
0
>=
65
Costs
Destination
SaltLake
513
416
682
65
0
>=
70
Rapid c
654
690
388
70
0
>=
85
Albu
totals
0
<=
0
<=
0
<=
152535 Total Cost
Supply
867
791
685
85
Supply
75
125
100
75
125
100