Linear trend, least-square, regress equation & linear programs

1. The league of American Theatres and producers, Inc, collects a variety of statistics for Broadway plays, such as gross revenue, playing time, and number of new productions. The following data show the season attendance ( in millions) for Broadway shows from 1990 to 2001 ( the world almanac 2002)

Season Attendance (in millions) Season Attendance ( in millions)
1990-1991 7.3 1996-1997 10.6
1991-1992 7.4 1997-1998 11.5
1992-1993 7.9 1998-1999 11.7
1993-1994 8.1 1999-2000 11.4
1994-1995 9.0 2000-2001 11.9
1995-1996 9.5

a) Plot the time series and comment on the appropriateness of a linear trend.
b) Develop the equation for the linear trend component for this time series
c) What is the average increase in attendance per person?
d) Use the trend equation to forecast attendance for the 2001-2002 seasons.

2. The management of a chain of fast food restaurants wants to investigate the relationship between the daily sales volume (in dollars) of a company restaurant and the number of competitor restaurants within a 1 mile radius. The following data have been collected:

Number of competitors within 1 mile Sales
1 3600
1 3300
2 3100
3 2900
3 2700
4 2500
5 2300
5 2000

a) Develop the least-square estimated regression equation that relates daily sales volume to the number of competitor's restaurants within a 1 mile radius.
b) Use the estimated regression equation developed in part (a) to forecast the daily sales volume for a particular company restaurant that has four competitors within a 1 mile radius.

Chapter 7

1. Which of the following mathematical relationships could be found in a linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why.
a) -1A + 2B ≥ 70
b) 2A - 2B = 50
c) 1A - 2B² ≥ 10
d) 3² A + 2B ≥ 15
e) 1A + 1B = 6
f) 2A + 5B + 1AB ≥ 25

2. Find the solutions that satisfy the following constraints:
a) 4A + 2B ≥ 16
b) 4A + 2B ≥ 16
c) 4A + 2B = 16

3. For the linear program
Max 2A + 3B
s.t.
1A + 3B ≥ 6
5A + 3B ≥ 15
A, B ≥0
Find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?

4. For the linear program
Max 4A + 1B
s.t.
10A + 2B ≥ 30
3A + 2B ≥ 12
2A + 2B ≥ 10
A, B ≥ 0
a) Write this problem in standard form
b) Solve the problem using the graphical solution procedure.
c) What are the values of the three slack variables at the optimal solution?

5. Consider the following linear program:
Min 2A + 2B
s.t.
1A + 3B ≥ 12
3A + 1B ≥ 13
1A - 1B = 3
A, B ≥ 0
a) Show the feasible region
b) What are the extreme points of the feasible region?
c) Find the optimal solution using the graphical solution procedure.