1. EMGT 501 HW #1 Solutions Chapter 2 - SELF TEST 18

2. Ch. 2 – 18
(a)

3. Ch. 2 – 18
(b)
(c)
s1 = 0, s2 = 0, s3 = 4/7

4. Ch. 2 – 20
(a)
Let E = number of units of the EZ-Rider produced
L = number of units of the Lady-Sport produced

5. Ch. 2 – 20
(b)

6. Ch. 2 – 20
(c)
The binding constraints are the manufacturing time and the
assembly and testing time.

7. Ch. 3 – 28
(a)
Let A = number of shares of stock A
B = number of shares of stock B
C = number of shares of stock C
D = number of shares of stock D
To get data on a per share basis multiply price by rate of
return or risk measure value.

8. Solution: A = 333.3, B = 0, C = 833.3, D = 2500
Risk: 14,666.7
Return: 18,000 (9%) from constraint 2

9. Ch. 3 – 28
(b)
Solution: A = 1000, B = 0, C = 0, D = 2500
Risk: 10A + 3.5B + 4C + 3.2D = 18,000
Return: 22,000 (11%)

10. Ch. 3 – 28
(c)
The return in part (b) is $4,000 or 2% greater, but the risk
index has increased by 3,333.
Obtaining a reasonable return with a lower risk is a preferred
strategy in many financial firms. The more speculative,
higher return investments are not always preferred because
of their associated higher risk.

11. Ch. 4 – 3 x1 = $ automobile loans x2 = $ furniture loans
x3 = $ other secured loans
x4 = $ signature loans
x5 = $ "risk free" securities

12. Solution
Annual Return $188,800 (9.44%)

13. Ch. 5 – 6
(a)

14. Ch. 5 – 6
(b)

15. Ch. 5 – 6
(c)
The original basis consists of s1, s2, and s3. It is the
origin since the nonbasic variables are x1, x2, and x3
and are all zero.
(d) 0
x3 enters because it has the largest negative zj - cj
and s2 will leave because row 2 has the only positive
coefficient.
(e)
(f)
30; objective function value is 30 times 25 or 750.
(g)
Optimal Solution: x1 = 10 s1 = 20
x2 = 0 s2 = 0
x3 = 30 s3 = 0
z = 800.

16. EMGT 501 HW #2 Due Day: Sep 27 Chapter 6 - SELF TEST 21

17. Ch. 6 – 21
Consider the following linear programming problem:
Write the dual problem.
Solve the dual.
Use the dual solution to identify the optimal solution to the original primal problem.
Verify that the optimal values for the primal and dual problems are equal.

18. Ch. 6 – 22
A sales representative who sells two products is trying to determine the number of sales calls that should be made during the next month to promote each product. Based on past experience, representatives earn an average $10 commission for every call on product 1 and a $5 commission for every call on product 2. The company requires at least 20 calls per month for each product and not more than 100 calls per month on any one product. In addition, the sales representative spends about 3 hours on each call for product 1 and 1 hour on each call for product 2. If 175 selling hours are available next month, how many calls should be made for each of the two products to maximize the commission?

19. Formulate a linear program for this problem.
Formulate and solve the dual problem.
Use the final simplex tableau for the dual problem to determine the optimal number of calls for the products. What is the maximum commission?
Interpret the values of the dual variables.

20. Duality Theory

21. One of the most important discoveries in the early development of linear programming was the concept of duality.
Every linear programming problem is associated with another linear programming problem called the dual.
The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways.

22. Primal and Dual Problems
Primal Problem
Dual Problem
Max
s.t.
Min
s.t.
for
for
for
for
The dual problem uses exactly the same parameters as the primal problem, but in different location.

23. In matrix notation
Primal Problem
Dual Problem
Maximize
subject to
Minimize
subject to
Where and are row vectors but and are column vectors.

24. Example
Primal Problem
in Algebraic Form
Dual Problem
in Algebraic Form
Max
s.t.
Min
s.t.

25. Primal Problem
in Matrix Form
Dual Problem
in Matrix Form
Max
s.t.
Min
s.t.

26. Primal-dual table for linear programming
Primal Problem
Coefficient of:
Right
Side
Coefficient
of:
Objective Function
Coefficients for
(Minimize)
Dual Problem VI VI VI
Right
Side
Coefficients for
Objective Function
(Maximize)

27. Relationships between Primal and Dual Problems
One Problem Other Problem
Constraint Variable
Objective function Right sides
Minimization
Maximization
Variables
Constraints
Unrestricted
Constraints
Variables
Unrestricted

28. The feasible solutions for a dual problem are those that satisfy the condition of optimality for its primal problem.
A maximum value of Z in a primal problem equals the minimum value of W in the dual problem.

29. Rationale: Primal to Dual Reformulation
Lagrangian Function
Max cx
s.t. Ax b
x 0
L(X,Y) = cx - y(Ax - b)
= yb + (c - yA) x
= c-yA
Min yb
s.t. yA c
y 0

30. The following relation is always maintained
yAx yb (from Primal: Ax b)
yAx cx (from Dual : yA c)
From (1) and (2), we have
cx yAx yb
At optimality
cx* = y*Ax* = y*b
is always maintained.
(1)
(2)
(3)
(4)

31. “Complementary slackness Conditions” are obtained from (4)
( c - y*A ) x* = 0
y*( b - Ax* ) = 0
xj* > y*aj = cj , y*aj > cj xj* = 0
yi* > aix* = bi , ai x* < bi yi* = 0
(5)
(6)

32. Any pair of primal and dual problems can be converted to each other.
The dual of a dual problem always is the primal problem.

33. Converted to Standard Form
Dual Problem
Min W = yb,
s.t yA c y
Max (-W) = -yb,
s.t yA -c y
Converted to
Standard Form
Its Dual Problem
Max Z = cx,
s.t Ax b x
Min (-Z) = -cx,
s.t Ax -b x

34. Min
s.t.
Min
s.t.

35. Max
s.t.
Max
s.t.

36. Question 1: Consider the following problem.
(b) Work through the simplex method step by step in tabular form.
(c) Use a software package based on the simplex method to solve
the problem.

37. Question 2:
For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead. Explain.
(a) Maximize
subject to
(b) Maximize
subject to
and
and

38. Question 3:
Consider the following problem.
Maximize
subject to
and
(a) Construct the dual problem.
(b) Use duality theory to show that the optimal solution
for the primal problem has