MATH 251 ACTIVITY 2: Sensitivity Analysis in linear programming
WHY:
In any linear programming problem there are parameters that are estimated or that are subject to change. The linear
programming model allows us to readily asses the effects of many of the most critical changes. In particular, we can
look a the effects (on the optimal solution and/or the optimal value of the objective) of a change in 1.) a coefficient
of the objective function or 2.) the right-hand side of a constraint.
LEARNING OBJECTIVES:
1.
Work as a team, using the team roles.
2.
Know how to determine whether changes in objective function coefficients will change the optimal solution of a linear
programming problem
3.
Know how to determine the effects on the objective of small changes in a constraint.
4.
Know how determine what is a "small" change in a constraint.
CRITERIA:
1.
Success in completing the exercises.
2.
Success in working as a team
RESOURCES:
1.
Your class notes on linear programming and the notes on "Sensitivity Analysis " [attached as p. 3 to this sheet]
2.
Your text – section 2.4
3.
Microsoft Excel and the workbook TOYCOproduction.xls available in the Course Documents section of the
Blackboard site (Linear Programming Notes folder)
4
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
3.
Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade.
4.
Be prepared to discuss your results.
EXERCISES:
We will work with the problem given below [TOYCO company – production]. The LP model is given below, and the spreadsheet
model is prepared (but not solved] in a model available as TOYCOproduction.xls in the Course Documents area [Linear
Programming Documents folder] of the Blackboard site for the course.
1.
2.
Read through the situation [next page] and the LP model, open the spreadsheet model, and use the solver to find the optimal
solution. Save the Sensitivity analysis (it will produce a second sheet in the Excel workbook).
Use the solution and the sensitivity analysis to answer the following questions.
a.) What is the optimal solution? What is the value of the solution?
b.) Which constraints are binding constraints? Give the slack or surplus for each of the others.
c.) Why is the reduced cost for the number of trucks equal to 0?
d.) How low would the profit per car have to go to change the optimal profit mix? How high?
e.) If the company decided that it must make some trucks, what effect would that have on the daily proft(increase? decrease?
how much per truck?)? How can you tell?
f.) If the profit per train decreased to $4, what would be the effect on the optimal production plan? what would be the effect
on profit?
g.) What effect would there be on the profit if the time available for Operation 3 increased to 450 minutes? Why does it make
sense that the shadow price for available Operation 3 time is 0? How low would the available time on Operation 3 have to
drop to change this shadow price?
h.) Suppose the company could extend the available time for Operation 1 by 20 minutes per day at a cost of $.50 (per minute).
Use the shadow price to explain why this would be worthwhile for the company. Would this change the oprimal production
mix?
i.) If problems with machines reduce the available daily time for Operation 2 to 400 minutes, what affect will this have on
profit (be specific – a dollar amount)? Why can’t we predict [without re-writing and re-solving the problem] the effect of a
decrease to 290 minutes?
CRITICAL THINKING QUESTIONS:(answer individually in your journal)
1.
What ideas in this section are easiest to understand? Hardest?
2.
The text indicates (p.71-72) that removing a variable from a problem cannot result in a better optimal value for the objective
[If variable is 0 in optimal solution, removing it from problem won’t change optimal solution or value. If variable is not 0 in
optimal solution, removing it will change optimal solution – and will cause optimal value to get worse (usually) or stay the
same (sometimes) – that is, in Max problem new optimal value will be smaller or same, in Minimization it will be larger or
same].
Can you explain why this should be true?
3.
Since any change in the right side of a binding constraint will change the optimal solution, why would anyone bother looking
at the shadow price?
SKILL EXERCISES:(This is the assignment due next class)
Read sections 2.7(what can be weird), 2.8 (an extended example) and 2.9 (integer linear proramming)
Write exercises (p.101) 9, 28, 29
Situation: TOYCO Company – production [modified from Taha, Operations Research 7th ed, Prentice-Hall, 2003 p.133]
TOYCO assembles three types of toys: trucks, cars and trains, using three operations. Operation 1 can be carried out for a maximum of
430 minutes per day, Operation 2 for 460 minutes, Operation 3 for 420 minutes. Assembling a toy truck requires 1 minute in
Operation 1, 3 minutes in operation 2, and 1 minute in Operation 3. Assembling a toy car uses 2 minutes in operation 1, does not use
Operation 2, and requires 3 minutes in Operation 3. Assembling a toy trrain uses one minute in Operation 1, 2 minutes in Operration 2,
and no tme in Operation 3. The company makes $3 per toy truck, $2 per toy car and $5 per toy train, and can sell all the toys made.
Because of an existing contract, TOYCO must make at least 80 toy cars per day. The company wishes to decide on a daily production
mix to maximize profit.
Setup:
If we let X1 = number of toy trucks made per day
And X2 = number of toy cars made per day
And X3 = number of toy trains made per day
Our objective is to maximize the daily profit
z = 3 X1 + 2 X2 + 5 X3
Our model is
Maximize z = 3 X1 + 2 X2 + 5 X3
Subject to
X1 + 2X2 + X3 ≤ 430
3X1 +
+ 2X3 ≤ 460
X1 + 3X2
≤ 420
X1, X2, X3 ≥ 0 [nonnegativity]
Sensitivity Analysis Summary
What happens if we change one part of the problem (leaving everything else unchanged).
Objective Function Coefficients
Range of Optimality: As long as the coefficient remains within its range of optimality, a change in the coefficient will not
change the optimal solution [but will change the value of the solution, unless the variable has value 0] Note that this
range is often infinite in one direction (above or below the actual value)
Allowable increase: The difference between the actual value of the coefficient and the upper limit of its range of optimality
Allowable decrease: The difference between the actual value of the coefficient and the lower limit of its range of optimality
IF allowable increase or allowable decrease = 0 for some coefficient, this is a signal there is more than one optimal solution
Variables:
Reduced Cost: The reduced cost of a variable whose value is 0 in the optimal solution is the amount by which the value of
the objective will change for each unit increase in the value of the variable. It also gives the negative of the change in its
coefficient that would (change the problem to) make the variable positive in the (new optimal) solution.
If a variable has positive (non-zero) value in the optimal solution, its reduced cost is 0.
Constraints: [Changes in right-hand side]
Shadow Price: The shadow price of a constraint is the amount by which the value of the (optimal value of) the objective will
change for each unit increase in the right-hand side. [The value(for maximization) or cost(for minimization)of increasing
the right side by one unit]. The shadow price is only valid while the (changed) right-hand side stays within its range of
feasibility.
A nonbinding constraint will always have shadow price 0 . A binding constraint will always have a non-zero shadow
price.
Range of feasibility: As long as the right-hand side of a constraint remains within its range of feasibility, the shadow price is
a valid (to predict resulting changes in the value of the objective function) - the value of the objective function will
change by an amount equal to the change in the right-hand side [of this constraint] times the shadow price. A change in a
binding constraint (shadow price not 0) will always produce a change in the optimal solution (and its value)
Allowable increase: The difference between the actual value of the right-hand side and the upper limit of its range of
feasibility.
Allowable decrease: The difference between the actual value of the right-hand side and the lower limit of its range of
feasibility.
The slack in a “≤” constraint is the amount by which the actual value of the left-hand side is below the (upper) limit which is
the right-hand side.
The surplus in a “≥” constraint is the amount by which the actual value of the left-hand side is above the (lower) limit which
is the right-hand side.
A constraint is a binding constraint if the slack or surplus has value 0 – the value of the left side exactly matches the limit
which is the right-hand side. A binding constraint is one whose line makes up part of the corner which is the optimal
solution.
Other changes: A new constraint will have no effect if the optimal solution already satisfies it; otherwise, we have to re-solve the
problem to determine the effect. Dropping a non-binding constraint will not affect the optimal solution (may affect range
of feasibility for some constraints).