PART I.  Multiple Choice

1.             Which of the following represents valid constraints in linear programming?

a.   2X ≥ 7X × Y

b.   2X × 7Y ≥ 500

c.   2X + 7Y ≥ 100

d.   2X2 + 7Y ≥ 50

e.   All of the above

 

2.             In sensitivity analysis, a zero shadow price for a resource means that

a.   the resource is scarce

b.   the resource constraint was redundant

c.   the resource has not been used up

d.   something is wrong with the problem formulation

e.   none of the above

 

3.             The feasible region plotted on the diagram below is consistent with which one of the following constraints?

a.   5x1 - x2 ≤ 0

b.   5x1 - x2 ≥ 0

c.   x1 + 5x2 ≤ 0

d.   x1 + 5x2 ≥ 0

e.   x1 - x2 ≤ 0

 

4.             The point of intersection between 2x1 +  7x2 = 10 and -3 x1 + 2x2 = 10 is:

a.       x1 = 1, x2 = 3

b.      x1 = 1, x2 = -3

c.       x1 = 2, x2 = -2

d.      x1 = 2, x2 = 2

e.       x1 = -2, x2 =2

 

5.             Which point is in the feasible solution space defined by the two constraints given below:

                (1) 7x + 3y < 21                        

                (2) x - y < 3

 

          a.   x = 2, y = 1

          b.   x = 1, y = -5

          c.   x = 8, y = 1

          d.   x = 4, y = 4

          e.   x = 2, y = 8

 

6.             C&A company has four stores to be serviced by a warehouse.  The stores’ relative locations and load information are listed as follows:

Stores

X-Coordinate

Y-Coordinate

Load

A

12

15

1000

B

22

9

450

C

16

14

750

D

14

15

550

Use the information given and the center of gravity method to determine the coordinates for the warehouse.

a.  x = 13.75, y = 15.13

b.   x = 16, y = 13.25

c.   x = 13.25, y = 16

d.   x = 15.13, y = 13.75

e.   x = 4, y = 3.17

 

Questions 7 and  8.

A product is produced at three plants and shipped to four warehouses.  The transportation costs per unit, plant capacity, and warehouse demand are given as follows:

 

Warehouse 1

Warehouse 2

Warehouse 3

Warehouse 4

Capacity

Plant A

12

9

13

12

300

Plant B

8

14

7

11

300

Plant C

14

10

13

10

400

Demand

200

350

150

300

 

 

7.             C&A wants to formulate the above problem as a linear program to minimize total shipping cost.  How many decision variables should be used?

a.  4

b.  6

c.  8

d.  12

e.  15

 

8.             C&A wants to formulate the above problem as a linear program to minimize total shipping cost.  How many constraints should be used?

a.  6

b.  7

c.  8

d.  9

e.  10

 

9.             Forecasts:

a.   become more accurate with longer time horizons

b.   are rarely perfect

c.   are more accurate for individual items than for groups of items

d.   all of the above

e.   none of the above

 

10.         The primary purpose of the mean absolute deviation (MAD) in forecasting is to:

a.   estimate the trend line

b.   eliminate forecast errors

c.   measure forecast accuracy

d.   seasonally adjust the forecast

e.   all of the above

 

11.         If two variables were perfectly correlated, the correlation coefficient would equal to:

a.   0

b.   less than 1

c.   exactly 1

d.   -1 or +1

e.   greater than 1

 

12.         Which one of the following statements about forecasting is FALSE?

a.   You should use the simple moving average method to estimate the mean demand of a time series that has a pronounced trend and seasonal influences.

b.   The weighted moving average method allows forecasters to emphasize recent demand over earlier demand.  The forecast will be more responsive to change in the underlying average of the demand series.

c.   The most frequently used time series forecasting method is exponential smoothing because of its simplicity and the small amount of data needed to support it.

d.   In exponential smoothing, higher alpha values place greater weight on recent demands in computing the average.

e.  Casual methods are used when historical data are available and the relationship between the factor to be forecast and other external and internal factors can be identified.

 

Questions 13 and 14.

C&A Pizza must forecast weekly demand for special pizzas so that pizza shells can be ordered weekly. Recent demand has been:

Week

No. of Special Pizzas

1

25

2

23

3

20

4

22

5

23

6

24

 

13.         Using a three-week moving average, what is the forecast for week 7?

a.   20

b.   21

c.   22

d.   23

e.   24

 

14.         If a four-week weighted moving average were used, what would be the forecast for week 7? (The weights are 0.75, 0.25, 0.15, and 0.10 with the least weight applies to the most recent week.)

a.   21.08

b.   23.24

c.   26.35

d.   27.08

e.   29.05

 

15.         In setting the control limits to monitor forecast accuracy through tracking signal, when forecast errors are normally distributed with a mean of 0, a control limit of  three-standard deviation spread from the mean will be equivalent to:

a.       0 MAD

b.      +/- 1 MAD

c.       +/- 2 MAD

d.      +/- 3 MAD

e.       +/- 4 MAD

 

16.         A company forecasts the demand for its product using a trend adjusted exponential smoothing model.  The initial forecast is 300 units with a trend of 50 units.  If actual demand was 325 and if alpha = 0.30 and beta = 0.20, then the trend‑adjusted forecast for the next period would be:

a.       342.5

b.      383.5

c.       391.0

d.      356.0

e.       432.5

 

Questions 17 and 18.

17.         C&A has the following data on customer demand for the past 4 weeks:

 

Mon

Tue

Wed

Thu

Fri

Sat

Week 1

210

178

250

215

160

180

Week 2

215

180

250

213

165

185

Week 3

220

176

260

220

175

190

Week 4

225

178

260

225

176

190

Averages

217.5

178

255

218.3

169

186.3

            What is the seasonal index for Wednesday?

a.       1.07

b.      1.25

c.       1.93

d.      0.91

e.       0.83

 

18.         If C&A’s projected demand for week 5 is 1320, what will be the seasonal adjusted forecast for Wednesday?

a.       275

b.      220

c.       330

d.      412.5

e.       1650

 

19.         Given the forecast demand and actual demand for C&A’s product, what is the tracking signal for year 6 if MAD is 12?

Year

Forecast Demand

Actual Demand

1

78

71

2

75

80

3

83

101

4

84

84

5

88

60

6

85

73

 

a.       +1

b.      -1

c.       +2

d.      -2

e.       cannot be determined

 

20.  The first step managers need to take when making facility location decisions is

a.     Begin negotiations with governments for several potential locations

b.     Identify specific location possibilities

c.     Gather information on location alternatives

d.     Evaluate specific sites

e.     Identify the location factors that are dominant for the business

 

PART II.  Problem Solving

Question 1.

You have been assigned a job of recommending the mix of ingredients that go into the C&A Resin.  Your recommendation has to be based on the following information:

Ingredient

Cost/Pound

Mixing Instructions or Requirements

B22 Binder

$0.30

No more than 4 pounds can be used for each pound of P55 Petroite

T90 Zinc Oxide

$0.20

Maximum of 60,000 pounds is available per year

P55 Petroite

$0.40

At least 1 pound must be used for every 4 pounds of B22 Binder

In addition, at least 400,000 pounds of C&A Resin will be needed next year.

Formulate the problem as a linear program to find the least cost mixing recommendation for C&A Resin.

 

Question 2.

C&A Company makes serveral lines of skirts, dresses, and sport coats for women.  Recently it was suggested that the company should re-evaluate its resources allocation to these products that would maximize contribution to profits and overhead.  Each product must pass through the cutting and sewing departments.  In addition, each product requires the same fabric.  The following data were collected for the study:

Product

Cutting (hours)

Sewing (hours)

Materials (yards)

Skirt

1

1

1

Dress

3

4

1

Sport Coat

4

6

4

The cutting department has 100 hours of capacity.  The sewing department has 180 hours of capacity.  60 yards of materials are available.  Each skirt contributes to $5 to profits and overhead.  Each dress $17, and each sport coat $30. 

 

C&A uses the following linear program to determine the most profitable product mix:

      Let x1 = the amount of skirts to be made

            x2 = the amount of dresses to be made

            x3 = the amount of sport coats to be made

 

       Maximize Z = 5 x1 + 17 x2 + 30 x3

        Subjet to:

            x1 + 3 x2 + 4 x3 < 100 [Cutting capacity]

            x1 + 4 x2 + 6 x3 < 180 [Sewing capacity]

               x1 + x2 + 4 x3 < 60   [Materials availability]

                      x1, x2, x3  > 0

 

Answer the following questions based on the given Excel output:

Adjustable Cells

 

 

 

Cell

Name

Original Value

Final Value

 

$X$1

Skirts

0

0

 

$X$2

Dresses

0

20

 

$X$3

Sport Coats

0

10

 

Adjustable Cells

 

 

 

 

 

 

 

 

Final

Reduced

Objective

Allowable

Allowable

 

Cell

Name

Value

Cost

Coefficient

Increase

Decrease

 

$X$1

Skirts

0

-2.5

5

2.5

1E+30

 

$X$2

Dresses

20

0

17

5.5

9.5

 

$X$3

Sport Coats

10

0

30

38

7.333333333

 

 

 

 

 

 

 

 

Constraints

 

 

 

 

 

 

 

 

Final

Shadow

Constraint

Allowable

Allowable

 

Cell

Name

Value

Price

R.H. Side

Increase

Decrease

 

$B$3

Cutting

100

4.75

100

32

40

 

$B$4

Sewing

140

0

180

1E+30

40

 

$B$5

Materials

60

2.75

60

40

26.66666667

 

(a)    What is the optimal product mix?         

(b)  What is the maximum profit?

(c) What is the effect of an additional yard of materials on C&A's profit?

(d)  How far can you go in reducing the sewing department’s capacity before this constraint becomes binding?

(e)  What is the optimal production mix if the profit margein of skirts increases by $5?

(f)   Someone offers 10 more cutting hours at $1.75 above its standard cost, what would be the maximum acheivable profit?

 

Question 3.

C&A wants to use Excel Solver to find a solution to the following transportation problem.  The tables are set up as follows:

 

 

(a)     What should be entered in cell B11 above?

(b)    What should be entered in cell F8 above?

(c)     What should be entered in cell B14 above?

(d)    What should be entered in cell F17 above?

(e)     What should be entered in the “Set Target Cell:” parameter?

(f)      What should be entered in the “Equal To:” parameter?

(g)     What should be entered in the “By Changing Cells:” parameter?

(h)     What should be entered in the “Subject to the Constraints:” parameter?

 

Question 4.

Demand for heart transplant surgery at C&A Hospital has increased steadily in the past few years, as shown in the following table:

Year

Time code (x)

No. of surgery (y)

1993

-3

45

1994

-2

50

1995

-1

52

1996

0

55

1997

1

58

1998

2

60

1999

3

55

 

(a)    C&A decides to use linear regression to predict the number of surgeries for year 2000.  What is the regression equation for C&A's surgery data?

(b)   What is the number of surgeries for year 2005 using the regression equation in (a)?

(c)    How should you determine whether or not regression is a good forecasting model for C&A?

(d)   Using exponential smoothing with a=0.2 and a 1998 forecast of 65, find the number of surgeries for year 2000.

(e)    How should you determine whether regression or exponential smoothing will be a better forecasting model for C&A?