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 changes in 1.) coefficients of
the objective function and 2.) right-hand sides of the constraints.
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 today's notes "Sensitivity Analysis "
2.
Your text – section 2.4
3.
Microsoft Excel and the workbook Lawrence2_46tmp.xls available on the Public server and in the course Documents section
of the Blackboard site
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 problem #46 from chapter 2 of the text which is on the CD that comes with the text. The situation and the LP
model are reproduced below. There is a spreadsheet model prepared (but not solved) on the Public server (P: drive) at
Public>Courses>Math>Math251>Lawrence2_46tmp.xls
1.
2.
Read through the situation 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 number of Techhelp secretaries equal to 0?
d.) How high would the number of pages per day for transferred secretaries have to go to make it desirable to use at least one
transferred secretary?
e.) If the company decided that there must be one transferred in-house secretary on the project, what effect would that have on
the number of pages produced per day? How can you tell?
f.) What effect would there be on the objective if the number of available workstations was reduced to 9? Why does it make
sense that the shadow price for the number of workstations is 0? What is the lower bound for the range of feasibility for the
number of workstations?
g.) Suppose, in monetary terms, Boeing feels that each additional potential page is worth $1 and the cost of a supervisor’s
time for technical assistance is $25 per hour. Using shadow prices, show why having up to two additional hours of
supervisory assistance is profitable to the company. Why can’t we tell whether three more hours would be worth the cost?
h.) Show (by re-solving the problem – assuming the values given in g) why hiring a second full-time employee (increasing the
hours of available technical support to 16) would not be in Boeing’s best interest.
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 deleting a variable from a problem cannot result in a better optimal value for the objective
[No change in value – because no change in optimal solution – if the value of the variable is 0 in the optimal solution.
Optimal value will (rarely) stay the same (but solution will change) or (usually) get worse (smaller in a Maximization, larger
in a Minimization) if the value of the variable is not zero in the optimal solution. 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)
Finish answering the questions for the exercises you solved for today (p. 101 #4, 5, 19, 24 add #28, 29
Read the rest of Chapter 2
Example – Exercise 46 from Chapter 2 of Lawrence & Pasternak
ADMINISTRATIVE SUPPORT. The federal government has asked Boeing Aircraft to resubmit its bid
to be the prime contractor for a modified version of the space station. Boeing’s design division is expected to need temporary
additional secretarial support to handle a substantial increase in paperwork that will be generated until the bid is completed. Boeing
can hire temporary personnel from Techhelp, a local temporary help firm specializing in technical secretarial support, or it can
temporarily transfer secretarial staff from other divisions within the company.
Given the results of qualifications tests, it appears that an in-house secretary who is transferred can produce about 40 pages of
usable work daily. Such work will require approximately two hours a day in technical assistance from supervisory personnel. Because
he or she is less familiar with the Boeing environment than Boeing employees, it is estimated that a Techhelp employee can produce
only 30 pages of work per day. Because of the Techhelp employee’s higher degree of technical secretarial training, however, each
employee only requires approximately one hour of technical assistance from supervisory personnel.
Boeing has allocated up to 10 workstations to the design division for additional secretarial support, so that
the division can utilize at most 10 workers. It has further allocated one full-time employee for technical assistance providing eight
hours of technical support time available daily.
a. Formulate a linear program to determine the optimal number of temporary Techhelp employees to hire
and in-house transfers needed to maximize the total potential number of pages produced per day. Assume
that fractional values represent workers assigned part-time to the division.
Setup:
If we let X1 = number of transferred in-house secretaries to use on the project
And X2 = number of secretaries to be hired from Techhelp for the project
Our objective is the number of pages that can be produced per day:
z = 40 X1 + 30 X2 and we want to Maximize
Our model is
Maximize 40 X1 + 30 X2
Subject to
2 X1 + X2 ≤ 8 [Technical supervision time]
X1 + X2 ≤ 10 [Available workstations]
X1, X2 ≥ 0 [nonnegativity]