SCIENCE MANAGEMENT (MGMT 306)
Spring Review Problems for Exam 1
Reminders:
- The exam is on February 14, Tuesday, 6:30 7:30 p.m. in EE129 or EE170. Room and seating
assignments will be announced on Katalyst.
- The exam will be closed book and notes. You may bring a single 8.5 x 11 sheet of crib notes written
on both sides to use in the exam.
- Students who are eligible for a make-up exam or need extra time must contact the instructor at least one
week prior to the exam date.
- The exam will consist of two parts:
Part 1: Multiple choice questions on basic concepts of linear and non-linear programming, the
graphical solution method, Excel model and Solvers parameters of linear programs, and sensitivity
analysis of linear programs using the Excel output.
Part 2: Linear programming modeling questions.
Question I.
Using the following constraints and their graphical representation, answer the following multiplechoice questions. The shaded region with corner points A, B, C, D is the feasible region.
1
a.
b.
c.
d.
1. If the objective function is to Minimize -x + 2y, which of points A, B, C, D is optimal?
Point A
Point B
Point C
Point D
2. If the objective function is to Maximize 3x + 5y, which of points A, B, C, D will be optimal?
a. Both C and D
b. Only A
c. Only C
d. Only D
3. At which of the points A, B, C, D does constraint 3 have a surplus (is not binding)?
a. Only A
b. Both A and B
c. Only C
d. Both C and D
4. If x y 2 is added as an additional constraint, which of the points A, B, C and D will still
be feasible?
a. C and D will be feasible, A and B will be infeasible
b. A, C and D will be feasible, B will be infeasible
c. B will be feasible, A, C and D will be infeasible
d. None of A, B, C, D will be feasible
a.
b.
c.
d.
5. Which of the following changes would make the feasible region infeasible (empty)?
Add y 6 as a new constraint
Add x + y 5 as a new constraint
Add x 7 as a new constraint
Delete constraint 3
6. Which of the following changes would make the feasible region unbounded?
a. Delete constraint 4
b. Delete constraints 1 and 2
c. Delete constraints 2 and 4
d. None of the above
7. Which of the following are components of a mathematical model for decision making?
a. Decision variables
b. An objective function and constraints
c. Input parameters.
d. All of the above.
2
Which of the following statement(s) on linear programming models is/are correct?
I. If a linear programming model has an optimal solution, then the feasible region cannot be
empty.
II. In order to obtain a feasible solution to an otherwise infeasible problem, some of the
constraints must be relaxed.
III. Changing the objective function will never make an infeasible problem feasible.
a. Only I
b. Only I and II
c. Only II and III
d. I, II and III
9.
Which of the following cannot be a reason for conducting sensitivity analysis?
a. Some of the input parameters of the model may not have been known with certainty.
b. Some of the input parameters of the model are approximations or estimates.
c. The model represents a dynamic environment in which some of the parameters are
subject to change by time.
d. None of the above.
10. Which of the following statement(s) is/are correct, given an optimal solution to a linear
program?
I. If a constraint has a shadow price of zero, then the current optimal solution satisfies
the constraint as equality.
II. There is always some amount by which the right-hand side value of a non-binding
constraint can be changed without affecting the optimal solution.
III. If the shadow price of a constraint is negative, a unit decrease in the right-hand side
value of the constraint (assuming the change is within range of feasibility) results in an
increase in the optimal objective function value.
a. Only I
b. Only I and II
c. Only II and III
d. I, II and III
11. Which of the following statement(s) is (are) true?
a) A function of one variable is concave in a region if its slope (rate of change) in that region
is always non-decreasing.
b) Excel solver can always correctly solve a maximization problem that has an objective
function that is convex and constraints that are all linear.
c) For some non-linear programs, Excel solver can get stuck at a local optimal solution and
never find the global optimal solution.
d) None of the above is true.
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4
Question 12 is based on the following linear programming model and Excel worksheet:
Maximize 3x + 2y
s.t.
x + y 10
Constraint 1
2x + y 12
x + 3y 12
Constraint 3
x, y 0
Constraint 2
12. Which of the following could be entered into cell B12 for the left-hand-side formula of
Constraint 2?
a.
b.
c.
d.
5
=2x + y
=B8*x + C8*y
sumproduct(B4:C4,$B$8:$C$8)
=sumproduct(B4:C4,B8:C8)
Question III.
The Pluto auto company makes two models of their Zippy sport car, a Regular model and a Turbo
model. The only difference between the two cars is their engines, all the rest of the parts for either
car
costs $10,000. The Zippy Regular sells for $15,000, the Zippy Turbo for sells $17,000. Pluto
currently has 30 regular engines and 25 turbo engines in stock. Pluto can purchase additional regular
engines for $2,000 and additional turbo engines for $3,400. The factory line can only assemble a
total of 100 cars this month. Pluto already has orders for 50 Turbo models which must be filled.
Given the formulation and Excel output below, answer the following independent questions.
R: number of regular models to assemble and sell this month
T: number of turbo models to assemble and sell this month
RE: number of additional regular engines to buy this month
TE: number of additional turbo engines to buy this month
Maximize 5000 R + 7000T 2000RE 3400TE
Subject to:
R RE 30
T TE 25
R + T 100
T 50
R, T, RE, TE 0
Micros
oft
Excel
9.0
Answer
Report
Target
Cell
(Max)
Cell Name
$G$3 Obj.
(max)
Adjusta
ble
Cells
Cell Name
6
Original
Value
0
Final
Value
487000
Original
Value
Final
Value
$B$2
$C$2
$D$2
$E$2
R
T
RE
TE
0
0
0
0
30
70
0
45
Constrai
nts
Cell Name
$G$4 RE Stock
Cell Value
30
$G$5 TE Stock
25
$G$6 Capacity
100
$G$7 T Orders
70
Formula
$G$4<=$I
$4
$G$5<=$I
$5
$G$6<=$I
$6
$G$7>=$I
$7
Status Slack
Binding
0
Binding
0
Binding
0
Not
Binding
20
Micros
oft
Excel
9.0
Sensiti
vity
Report
Adjusta
ble
Cells
Final Reduce Objective Allowabl Allowabl
d
e
e
Cell Name Value Cost Coefficien Increase Decreas
t
e
$B$2 R
30
0
5000
600
1400
$C$2 T
70
0
7000
1400
600
$D$2 RE
0
-600
-2000
600
1E+30
$E$2 TE
45
0
-3400
1400
600
Constr
aints
Final Shadow Constraint Allowabl Allowabl
e
e
Cell Name Value Price R.H. Side Increase Decreas
e
$G$4 RE
30
1400
30
20
30
Stock
7
$G$5 TE Stock
$G$6 Capacity
$G$7 T Orders
25
100
70
3400
3600
0
25
100
50
45
1E+30
20
1E+30
20
1E+30
a. How much will total profit change if the factory line becomes more efficient and can
assemble 10 more cars this month?
b. If only 40 previous orders for turbo models had to be filled, how much would total profit
change?
c. A new lower cost supplier has emerged for regular engines. At what price will Pluto consider
buying regular engines from this supplier?
d. Pluto is considering increasing the selling price for regular cars. At what minimum increased
selling price would Pluto consider changing its production plan?
e. A competitor has offered 25 extra regular engines that are compatible with Plutos cars for a
total price of $25,000. Should Pluto take the offer?
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f. Plutos quality control inspector has just found flaws in 5 of the 25 turbo engines on stock,
which means that these 5 engines cannot be used. What will the total profit be under these
conditions?
9
Question IV.
Cori Ander's Spice Company has a limited supply of two herbal ingredients that are used in the
production of seasonings. Cori uses the two ingredients, HBO1 and HBO2, to produce either
turmeric or paprika. The marketing department reports that although the firm can sell all of the
paprika it can produce, it can only sell up to a maximum of 1500 bottles of turmeric each month.
Unused herbal ingredients can be sold at $0.75 per ounce of HBO1 and $0.15 per ounce of HBO2.
15 bottles of turmeric or 20 bottles of paprika can be produced in one hour due to the capacity
of the machine that mixes the ingredients. The machine is operated for 160 hours during regular
hours each month. An additional capacity of up to 40 hours per month can be obtained using
overtime but each overtime hour costs the company $20.
The goal of the company is to maximize its monthly income (total profit less total cost).
Additional data are shown in the below table.
Seasoning
Turmeric
Paprika
Availability each
month(ounces)
Ingredients (oz/bottle)
HBO1
HBO2
5
3
2
3
10,000
Demand(bottles)
1500
Unlimited
Profit
per bottle ($)
3.50
2.50
8,500
a. Clearly define the decision variables that are needed to formulate a linear programming model for
this problem.
b. Using only the variables defined above, formulate a linear programming model.
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c. Assume additional amounts of HBO1 and HBO2 can be purchased at $1.00 and $0.30 per ounce,
respectively. Reformulate the original LP model to reflect these changes. (You do not need to
rewrite the rows of the original formulation that do not change but explicitly define new variables
if you need them).
d. Assume that the Spice Company purchases bottles for turmeric and paprika from a bottle
manufacturer (bottle costs have been accounted for in the profit values). The bottle manufacturer
can supply as many bottles as needed so long as the bottles for paprika do not exceed 70% of the
total number of bottles each month. Reformulate the original LP model to reflect these changes.
(You do not need to rewrite the rows of the original formulation that do not change but explicitly
define new variables if you need them).
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Question V.
Solve the following LP model using Excel Solver. Report the optimal solution and optimal objective
value, and output the answer report and sensitivity report.
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