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AGEC 352
Midterm I
October 6, 2008
NAME:_________________
LAB (11:30 or 12:30 or V)
Each question is worth three points. Your name is worth five points. Filling in the answer
sheet is worth five points. See the answer sheet for additional instructions.
The first 5 questions make use of the following linear programming problem written in
standard algebraic form. This is a farm production problem with options for the farmer to
plant corn (C), soybeans (S), or oats (A) each of which contribute to the farm’s gross margin
(GM) defined as revenue minus variable costs. The available resources of the farm are land
(75 acres), labor (500 hours), and capital (2,500 dollars).
max GM  75C  60S  90 A
subject to :
Land :
C  S  A  75
Labor : 10C  5S  5 A  500
Capital : 50C  30S  20 A  2500
Non  neg : C  0; S  0; A  0
1)
True or False. Any feasible plan will include some positive amount of oats (A)
production since it has the highest per acre return of any crop.
2)
The most limiting resource in soybean (S) production is which of the following?
a)
Land
b)
Labor
c)
Capital
d)
None of the above
3)
True or False. The most limiting resource for corn (C) is land.
4)
The farmer currently splits her land equally between the three crops (i.e. 25 acres per
crop). Which of the following statements is true regarding her current plan?
a)
The farmer’s current plan (25 acres per crop) is optimal because it makes use
of all available resources.
b)
The farmer’s current plan (25 acres per crop) is not optimal because a
reallocation of an acre of land from corn to oats is feasible and generates a
higher return.
c)
Both of the above statements are true.
d)
Neither a) nor b) is true.
5)
True or False. Given the information from the farmer’s problem on returns per acre
and resource use, oats (A) production is always a better choice than soybeans (S)
production.
Equations (1)-(4) below depict a cost and returns model for a profit maximizing pie
producer in algebraic form. Let  be profits, R be revenue, C be costs, Q pies be the quantity
of pies demanded, and P pies be the price charged for pies. Use this information for questions
6-10.
  R C
R Q P
C  20
Q pies  20  2P pies
pies
pies
(1) (Profits)
(2) (Revenue)
(3) (Costs)
(4) (Demand)
6)
Equation (4) above gives the relationship of pies sold Q pies to the price P pies charged
for pies in dollars (i.e. the demand equation). Which statement below is correct given
this relationship between price and quantity?
a) Increasing the pie price by $ 0.50 decreases the number of pies sold by 1.
b) Decreasing the pie price by $ 1.00 decreases the number of pies sold by 2.
c) At a price of $ 9.00, the company facing this demand equation will sell 20
pies.
d) All of the above are correct statements
e) None of the above are correct statements
7)
True or False. Given the demand equation in (4) and the cost equation in (3),
increasing the pie price will reduce the quantity sold and thus will reduce costs of the
company facing this demand.
8)
True or False. Substitution of equation (4) into (2) shows that revenue is a non-linear
function of the pie price.
9)
What revenue is earned when the pie producer charges a price of $5.00?
a)
$ 35.00
b)
$ 40.00
c)
$ 45.00
d)
$ 50.00
10)
True or False. If the producer wants to earn maximum profits then she should
charge $4.00 per pie.
11)
Which of the following is not part of an optimal LP solution?
a)
Objective variable value
b)
Shadow prices
c)
Optimal activity levels
d)
All of the above are part of an optimal LP solution.
12)
True or False. The feasible space of a linear program is determined by the
intersection of the objective equation and the constraints.
The next four questions make use of the following linear programming problem written in
standard algebraic form. This is a furniture production problem with options for the
business to manufacture chairs (H) and tables (T) each of which contribute to the business’
gross margin (GM) defined as revenue minus variable costs. The available resources of the
factory are labor (300 hours) and lumber (300 board feet).
max GM  75 H  100T
subject to :
Labor : 20 H  30T  300
Lumber : 10 H  40T  300
Non  neg. : H  0; T  0
13)
True or False. Lumber is the most limiting resource for Table (T) production while
labor is the most limiting resource for Chair (H) production.
14)
Which of the following statements about the tables and chair problem is not correct?
a)
The lumber constraint gives all combinations of chairs and tables that can be
produced with the 300 available board feet of lumber.
b)
Tables generate higher earnings and are less costly in to produce in terms of
available resources (i.e. labor and lumber use).
c)
The non-negativity constraints indicate that production levels of zero are
feasible.
d)
None of the above. (i.e. all the above statements are correct).
15)
True or False. If we wanted to explicitly consider the possibility that the company
can hire labor in addition to the 300 available hours, we would need to make which
of the following additions to the model?
a)
Add a decision variable for hiring labor.
b)
Add a term to the objective equation reflecting the cost of hiring labor.
c)
Add a term to the constraint reflecting the addition of available labor
through hiring.
d)
All of the above.
16)
True or False. Producing six tables and six chairs is a corner point of the feasible
space for this problem.
The next six questions deal with the sensitivity report from the furniture production problem given
in algebraic form on the previous page. Note that I have changed the heading on what Excel calls
‘Reduced Cost’ to ‘Objective Penalty.’ The column ‘Solution Value’ for decision variables represents
the choices indicated by the optimal production plan and the column ‘Usage Amount’ in the
constraints reflects the amount of each resource used in implementing the optimal production plan.
Decision Variables
Cell
Name
$B$5 Chairs
$C$5 Tables
Solution
Value
15.00
0.00
Objective
Penalty
0.00
-12.50
Objective
Coefficient
75.00
100.00
Allowable
Increase
?
12.50
Allowable
Decrease
8.33
Infinite
Usage
Amount
300.00
150.00
Shadow
Price
3.75
?
Constraint
R.H. Side
300.00
300.00
Allowable
Increase
300.00
Infinite
Allowable
Decrease
300.00
150.00
Constraints
Cell
Name
$D$7 Labor LHS
$D$8 Lumber LHS
17)
True or False. The objective penalty for tables indicates that deviating from the optimal plan
to produce one table will reduce earnings by the company.
18)
Which of the following is a correct statement about the missing (see the ?) ‘Allowable
Increase’ for the Chairs decision variable in the sensitive information above?
a)
The allowable increase should be zero because the optimal plan indicates only chairs
should be produced.
b)
The allowable increase should be ‘Infinite’ because the optimal plan indicates only
chairs should be produced.
c)
Neither a) nor b) is a correct statement.
19)
True or False. The missing (see the ?) shadow price on lumber will be zero.
20)
True or False. If the GM per table increases to 125 dollars the optimal number of chairs will
change.
21)
Which of the following are true statements given the sensitivity report for the furniture
factory?
a)
The company has surplus labor.
b)
The lumber constraint is binding.
c)
The solution occurs at a corner point.
d)
All of the above are true statements.
22)
What is the maximum gross margin earned by the furniture company?
a)
925 dollars
b)
1,025 dollars
c)
1,125 dollars
d)
1,225 dollars
e)
None of the above
A profit maximizing bakery makes and sells two types of bread, sourdough (S) and raisin (R). The
per loaf ingredient requirements and per loaf profits for each type of bread are given in the table
below. The bakery has the following resources available to make bread: 800 ounces of flour, 150
ounces of butter, 300 ounces of raisins, and 200 ounces of yeast. Use this information and the
information in the table below to answer the next four questions.
Flour per loaf
Butter per loaf
Raisins per loaf
Yeast per loaf
Sourdough
(S)
8
1
0
1
Raisin
(R)
8
1
4
1
Profit per loaf
$1.50
$1.95
23)
True or False. The objective equation for this problem is:
Max Profit = 1.50*S + 1.95*R.
24)
True or False. The yeast constraint for the LP is written as:
Yeast: S + R ≤ 200 (ounces of yeast)
25)
True or False. Raisins are the most limiting resource for raisin bread production.
26)
True or False. An optimal plan for the bakery is to only make raisin bread since the return
per loaf is higher and both types of loaves use the same amount of yeast, butter, and flour.
500
450
400
350
300
250
200
Dollars
150
100
50
0
-50
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-100
-150
-200
-250
-300
-350
Quantity Supplied
Revenue
Cost
Profits
The above figure plots the total revenue, costs, and profits measured in dollars (y-axis) for a
company as a function of the quantity supplied by the company (x-axis). The company’s objective is
to maximize profits by choosing the quantity of its product to supply. Use the above graph to
answer questions 27-29.
27)
Which quantity given below will earn the greatest profit for the company?
a)
6.00
b)
8.00
c)
10.00
d)
None of the above
28)
True or False. The cost graph has a positive slope.
29)
At what quantity below does this company break-even (i.e. have costs that are equal to
revenues)?
a)
2
b)
4
c)
6
d)
8
Soybeans
Extreme Points
Feasible Space
Corn
30)
Which of the following statements is correct given the above figure of a linear programming
feasible space. Note that ‘extreme point’ is synonymous with the term ‘corner point’.
a)
To identify an optimal crop plan we would need only to check the three marked
‘extreme points’ to find the highest profit level.
b)
Changing the returns to corn production will change the shape of the feasible space.
c)
Adding a constraint for an additional resource must shrink the feasible space.
d)
The feasible space is defined by four resource constraints and two non-negativity
conditions.
BONUS: Review your exam answers and identify the one question you are most unsure of
correctly answering. On the exam answer sheet write FREE as your answer. Do this
for only one question and you must write FREE to receive credit.