Answer All Problems

Part I (24 points) Answer the following questions as true (T) or false (F)

1. A feasible solution to a linear programming (LP) problem satisfies at least one of the constraints.
2. If the objective function has the same slope as a constraint in a LP problem, then the problem may have alternative optimal solutions (an infinite number of solutions).
3. The graphical solution method may be used to solve any LP problem.
4. Changing a coefficient in the objective function of a LP problem may have an impact on the feasible region.
5. If a point on one side of a constraint line is feasible, no point on the other side of the constraint line is feasible.
6. The number of constraints must be less than the number of decision variables in a LP problem.
7. In the optimal solution of a LP problem, all the available resources are fully utilized.
8. If the first derivative of a function at a point is zero while the second derivative is negative, then this point is a maximum point.
9. In a LP problem, factors such as profit per unit, production time per unit, and production capacity are known as "uncontrollable inputs" while production quantity would be the "controllable input."
10. The slope of the objective function MIN 5X1 + 2X2 is -2/5.
11.  A binding constraint has zero slack associated with it.
12.  Fixed cost depends on production volume.

Part II   Answer all the following short answer problems. For partial credit, you must show all work.
 

1. (21 points) Consider the following linear programming problem

MAX 2X1 + 3X2

subject to X1 + X2 < 10

2X1 + X2 > 4

X1 +3X2 < 24

2X1+X2< 16

where X1, X2> 0
 

a) Prepare a graph which shows each constraint and the feasible region.

b) Identify the optimal solution on your graph. What are the values of the decision variables and objective function at the optimal solution?

c) Suppose that c1 is increased from 2 to 2.5 (c2 remains as 3). What is the new optimal solution (decision variables and value of the objective function)?

d) Suppose c2 is decreased from 3 to 1 (c1 remains at 2). What is the new optimal solution?
 

2. (13 points) Northfield Bank and Trust has $100 million to allocate between two classes of earning assets, loans and securities (bonds). To achieve an acceptable level of risk, the bank has found that it is necessary to hold $30 in securities for every $100 in loans. To meet the needs of the community it serves, Northfield has decided to loan out at minimum $20 million. The bank currently earns a 5% return on securities and a 10% return on loans. Based on the above information, set up a linear programming model (objective function and constraints). The goal of the bank is to maximize return on its assets. There is no need to find the optimal solution to this problem.
 

3. (14 points) There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is $45.

a) Write an expression for total cost.

b) Write an expression for total revenue .

c) What is the "break even" point of production?

d) Graph the total cost and the total revenue functions for the above problem. Show the break even point on your graph.
 

4. (10 points) Using a graph, illustrate the following concepts:

a) infeasible solution

b) unbounded solution

Use a separate graph for each concept. Make up your own constraints and objective function.
 
 

5. (9 points) Graph the feasible region associated with the following constraints.

-6X1 + 5X2 < 60

5X1 - 2X2 < 0

where X1,X2> 0
 

6. (12 points) Based on the information below, answer the following questions:

a) What is the value of the objective function at the optimal solution?

b) What are the slack/surplus values for each of the constraints?

c) Would an increase in the right hand side of constraint 1 from 100 to 101 increase, have no impact on ,or decrease the value of the objective function?

d) Would an increase in the right hand side of constraint 2 from 80 to 81 increase, have no impact on, or decrease the value of the objective function?
 

Linear Programming Problem
MAX 5X1 + 5X2
S.T.
1) X1<100
2) X2<80
3) 2X1 + 4X2<400

Optimal Solution

Objective Function Value =
 
 

Variable	Value	Reduced Costs
X1	100.0	0.0
X2	50.0	0.0
Constraint	Slack/Surplus	Dual Prices
1		2.5
2		0.0
3		1.25

 

OBJ COEFFICIENT RANGES
Variable	Lower Limit	Current Value	Upper Limit
X1	2.5	5.0	No upper limit
X2	0.0	5.0	10.0
RIGHT HAND SIDE RANGES
Constraint	Lower Limit	Current Value	Upper Limit
1	40.0	100.0	200.0
2	50.0	80.0	No upper limit
3	200.0	400.0	520.0