AGEC 352
Midterm Exam Two
November 19, 2008
Name:__________________
Lab: 11:30 / 12:30 / Other
LP 1 below is a linear program with two decision variables (x,y) and the objective variable Z. Use
this problem to answer the first 3 questions.
min Z  2 x  3 y
s.t.
Constraint 1 : x  13
Constraint 2 : y  9
(LP 1)
1) The shadow price on constraint 1 for this problem would be:
a. 2
b. 2/3
c. -2
d. -2/3
2) True or False. Adding a  constraint to this problem will either decrease the optimal value of
the objective variable or leave it unchanged.
3) True or False. Non-negativity constraints on x and y would be redundant constraints in LP1.
LP 2 below is a linear program with three decision variables (q,r,s) and the objective variable M. Use
this problem to answer the next 3 questions.
max M  1.6q  0.4r  0.5s
s.t.
Constraint 1 : r  q  10
(LP 2)
Constraint 2 : s  10
Non - negativity : q, r , s  0
4) True or False. Removing the decision variable q from this problem will lower the optimal value
of the objective variable.
5) The shadow price on constraint 1 for this problem would be:
a. 0
b. 0.4
c. 1.6
d. 4
6) True or False. The allowable decrease on constraint 2 for this problem will be 10 (i.e. the
shadow price for this constraint is unchanged if we lower the RHS by 10 or fewer units).
A construction company quarries gravel and hauls it to its different construction sites. There are two
quarries (Quarry A and Quarry B) and five different construction sites (C1 through C5) all located at
different locations. The unit cost (per ton cost) matrix is given below as the cost of moving one ton
of gravel from a quarry to a construction site.
Unit Cost Matrix
From/To
Quarry A
Quarry B
C1
24.7
26.8
C2
19.3
22.6
C3
18.7
24.3
C4
21.2
23.7
C5
20.1
22.5
Capacity (maximum tons of rock that can be shipped) from each quarry is fixed at 350 tons. The
minimum requirements for tons of rock at each construction site are:
C1 = 100, C2 = 227, C3 = 82, C4 = 98, and C5 = 193
Use the information for the construction company to answer the following two questions:
7) If the company is trying to lower its shipping costs and can only expand capacity at one of the
two quarries, which quarry’s capacity should be expanded?
a. Quarry A
b. Quarry B
c. This can only be determined by solving a transportation cost minimization LP.
8) True or False. The transportation cost minimization problem for this construction company is
not a balanced transportation problem.
The sensitivity information for the decision variables of the construction company’s transportation
cost minimization problem is given below. Use this information to answer the next three questions.
Sensitivity Information for Variables (Routes)
Solution
Objective
Transport Cost Allowable Allowable
Name
Value
Penalty
(per ton)
Increase
Decrease
Quarry A C1
0.00
0.40
24.7
1E+30
0.40
Quarry A C2
227.00
0.00
19.3
0.80
21.80
Quarry A C3
82.00
0.00
18.7
3.10
21.20
Quarry A C4
41.00
0.00
21.2
0.10
0.80
Quarry A C5
0.00
0.10
20.1
1E+30
0.10
Quarry B C1
100.00
0.00
26.8
0.40
26.80
Quarry B C2
0.00
0.80
22.6
1E+30
0.80
Quarry B C3
0.00
3.10
24.3
1E+30
3.10
Quarry B C4
57.00
0.00
23.7
0.80
0.10
Quarry B C5
193.00
0.00
22.5
0.10
22.50
9) True or False. If a temporary shutdown occurs at Quarry B and the company is forced to ship
five tons of gravel from Quarry A to construction site C1, then the company’s costs will increase
by $2.00.
10) True or False. The worst route available to the company is from Quarry B to construction site
C1 because the per ton transport cost is highest along that route.
11) Road repairs shut down the route used from Quarry B to construction site C3. What impact will
this have on the company’s total cost of transporting gravel?
a. Total costs will increase.
b. Total costs will decrease.
c. Total costs will not change.
d. More information is needed.
The sensitivity information for the constraints of the construction company’s transportation cost
minimization problem is given below. Use this information to answer the next four questions.
Sensitivity Information for Constraints
Solution
Shadow
Name
Value
Price
Sources
Quarry A
350
-2.5
Quarry B
350
0
Destinations
C1
100
26.8
C2
227
21.8
C3
82
21.2
C4
98
23.7
C5
193
22.5
Constraint
R.H. Side
350
350
100
227
82
98
193
12) True or False. The allowable increase (not shown) for Quarry A must be zero.
13) True or False. The allowable increase (not shown) for construction site C1 must be zero.
14) Which is the worst destination available to the construction company from the perspective of
minimizing transportation costs?
a. C1
b. C2
c. C3
d. C4
e. C5
15) The construction company only ships full loads of gravel and asks you why you did not use
integer constraints in the original formulation of the model. Which of the following is a correct
response to this concern?
a. Integer constraints are not required because we can round the solution off afterwards.
b. Integer constraints are not required because the RHS of constraints are integers
in this balanced transportation problem.
c. Shut up fool, I know what I am doing!
The Ag Econ team needs to assign its three gymnasts to the three events competitive events listed in
the table below1. The preliminary scores from tryouts are given in the table below. Coach Foster
wants to assign the gymnasts to events in a way that maximizes the aggregate (total) score while
ensuring that only one person is assigned to compete in each event. Use this information to answer
the following two questions.
Keeney
Gramig
Balagtas
Beam
10
8
7
Vault
10
5
6
Floor
10
9
7
16) True or False. In Coach Foster’s optimal assignment of gymnasts, Keeney will compete in the
vault.
17) True or False. In Coach Foster’s optimal assignment of gymnasts, Gramig will compete on the
beam.
LP 3 below depicts the profit maximization problem for a pizza maker who has to decide how many
of each type of pizza to make. The two pizza types are regular (R) and deluxe (D) which have per
pizza profits of $2.25 and $2.65 respectively. Use LP 3 to answer the following two questions.
max P  2.25 R  2.65 D
subject to :
Sauce :
8 R  8 D  440
Dough :
16 R  16 D  1000
Sausage :
3R  9 D  275
Cheese :
8 R  12 D  500
Mushrooms : 4 D  100
Non  neg . : R  0; D  0
(LP 3)
18) Which of the following statements is correct regarding the pizza maker’s optimization problem?
a. The non-negativity constraints indicate that some positive quantity of each pizza must be
produced.
b. In the optimal solution, 220 units of sauce must be used on each type of pizza.
c. The dough constraint in this problem is irrelevant.
d. None of the above statements are correct.
19) True or False. Mushrooms are the most limiting resource for deluxe pizzas.
I of course realize that the beam is a women’s event and that the names I used are male faculty members in the
department. Do not let this confuse you as you answer the question as it is irrelevant to solving the quantitative model.
Take a one point bonus for reading the whole footnote. Let me know you read it by drawing a circle around your answer
for number 17.
1
The sensitivity information for the pizza maker’s problem is given below. Use this information to
answer the next five questions.
Decision Variables
Name
Quantity Regular
Quantity Deluxe
Solution
Value
40.00
15.00
Objective Profit per Allowable
Penalty
Pizza
Increase
0.00
2.25
0.40
0.00
2.65
0.72
Allowable
Decrease
0.48
0.40
Solution
Value
440
880
255
500
60
Shadow Constraint Allowable
Price R.H. Side Increase
0.18
440
60.00
0.00
1000
1E+30
0.00
275
1E+30
0.10
500
13.33
0.00
100
1E+30
Allowable
Decrease
17.78
120.00
20.00
60.00
40.00
Constraints
Name
Sauce LHS
Dough LHS
Sausage LHS
Cheese LHS
Mushrooms LHS
20) True or False. The optimal solution earns the pizza maker $129.75 in profits.
21) Which of the lists below correctly identifies the binding constraints from the pizza maker’s
optimal solution?
a. Dough, Sausage, and Mushrooms
b. Sauce and Cheese
c. Sauce and Sausage
d. None of the lists correctly identify all of the binding constraints
22) Papa John had some pizza dough go bad and wants to buy 100 units of dough from the pizza
maker. Which value below indicates a price for the 100 units of dough the pizza maker could
accept and not be made worse off (i.e. have lower profits)?
a. $100.00
b. $50.00
c. $5.00
d. Free
e. All of the above
23) The pizza maker wants to increase profits by adjusting the prices he charges. He estimates that
he can raise the price of a deluxe pizza enough to increase the profit margin by twenty percent
without lowering demand. Which of the following is a correct statement regarding this potential
price increase?
a. It is not possible to raise the price of deluxe pizzas without lowering profits.
b. If the pizza maker raises the price of deluxe pizzas as indicated it is still optimal
to make 40 regular and 15 deluxe pizzas.
c. If the pizza maker raises the price of deluxe pizzas as indicated a new optimal plan will
be required.
24) True or False. The shadow price information indicates that the pizza maker should expand his
ingredient purchases by buying more cheese.
The pizza maker decides to raise the price of deluxe pizzas enough so that the profit per deluxe
pizza increases to $4.00. We re-solve our model and tell him that the optimal plan is to now produce
33.33 regular pizzas, and 14.44 deluxe pizzas. This earns a profit of $167.36 for the pizza maker with
the binding constraints now being sausage and cheese. Use this information to answer the following
two questions.
25) True or False. If we round the activities (pizza numbers) down to 33 regular and 14 deluxe
pizzas, profits for the pizza maker will be lower than $167.36.
26) True or False. If we solve the model using integer constraints, the profit level for the pizza
maker will be lower than $167.36, but at least as high as the profits we would get if we used
rounding to get integer values for pizzas.
27) True or False. George Stigler (JFE, 1945) solved a linear program in his seminal article on the
“Cost of Subsistence” to determine the minimum cost diet.
28) True or False. A linear programming solution can have no more non-zero shadow prices than
there are decision variables (i.e. activities) available to the decision maker.
29) Which of the following is not true about a balanced transportation problem?
a. One of the binding constraints will have a zero shadow price.
b. One of the routes will have a zero activity level and zero objective penalty.
c. All source constraints will bind.
d. All destination constraints will bind.
30) True or False. Dynamic models are distinguished from static models by the fact that they cover
multiple periods with decisions in earlier periods affecting available choices in latter periods.
Bonus: Please choose the one question in which you are least confident of your answer and
write FREE in the appropriate spot on the answer sheet.