Answer All Problems

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

1. Suppose you collected data for a class project such as the assets (as of the end of 1997) of the 100 largest banks in the U.S. These data are an example of "time series" data.
2. Forecast errors are all nonnegative.
3. A zero dual price associated with a particular resource constraint suggests none of that resource is being used in the optimal solution of the problem.
4. The reduced cost for a positive decision variable is 0.
5. A negative dual price associated with a minimization problem indicates that the value of the objective function will get smaller if the value of the right hand side constraint is increased by one unit.
6. The addition of a constraint such as X11=0 to an assignment problem may lead to a higher value of the objective function.
7. A change in an objective function coefficient to a value outside the "range of optimality" will lead to a change in the optimal solution.
8. The transportation problem cannot be solved if supply exceeds demand.
9. The number of decision variables (xij's) in a transportation problem is equal to the number of origins times the number of destinations.
10. Decisions variables in an assignment problem may take on the value of 0 or 1.
11. The current dual price remains valid as long as the change in the right hand side value of the related constraint is within the lower and upper limits.
12. Increasing the supply available at an origin in a transportation problem will increase transportation costs.

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

1. (14 points) A gravel company has received a contract to supply gravel to three new road projects.

a) Using the information below, write out the linear programming version of the transportation problem. The goal is to minimize transportation costs. There is no need to solve the problem.
 
 

Origin	Project 1	Project 2	Project 3
Plant 1	4	8	8
Plant 2	16	24	16
Plant 3	8	16	24

Weekly Requirements (truckloads)     Amount Available (per week)

Project 1     72                                         Plant 1         56

Project 2     102                                       Plant 2         82

Project 3     41                                          Plant 3         77
 

b) What are the possible impacts on the solution if an additional constraint was added which indicated that it is infeasible to ship from Plant 1 to Project 1 (X11=0)? There is no need to provide specific numbers in your response.
 

2. (18 points) Using the information below, answer the following questions:

a) What is the value of the objective function?

b) How much of Resource A and B will be used?  Is the marketing constraint binding?

c) What would be the impact on the objective function if the right hand side of constraint 1 were increased from 50 to 55?

d) What would be the impact on the solution if the right hand side of constraint 2 were increased from 120 to 200?

e) What would be the impact on the solution if the right hand side of constraint 2 were increased from 120 to 320?

f) How many units of Product 2 will be made if profit per unit rises from 5 to 5.5? Any impact on the value of the objective function?

In the Management Scientist output below, the decision variables (X1,X2,X3) stand for the number of units of Products 1, 2 and 3. The first constraint is a marketing restriction, the second measures Resource A and the third measures Resource B. The objective function measures profit.

MAX 6X1+5X2+8X3

S.T.

1) X1+X2+X3>50
2)2X1+2X2+3X3<120
3) 4X1+3X2+2X3<210

Objective Function Value=
 

Variable	Value	Reduced Costs
X1	48.75	0
X2	0	.75
X3	7.5	0
Constraint	Slack/Surplus	Dual Prices
1	6.25	0
2	0	2.5
3	0	.25

Objective Coefficient Ranges
 

Variable	Lower Limit	Current Value	Upper Limit
X1	5.33	6.0	16.0
X2	No lower limit	5.0	5.75
X3	5.0	8.0	9.0

Right Hand Side Ranges
 

Constraint	Lower Limit	Current Value	Upper Limit
1	No lower limit	50.0	56.25
2	105.0	120.0	315.0
3	160.0	210.0	240.0

3. (15 points) Using the information below, answer the following questions:

a) Set up the linear programming version of the assignment problem (include the objective function and constraints). Note: each project must be assigned exactly one worker.

b) Based on the computer printout, summarize the assignments. What is the value of the objective function at the optimal assignments?

c) Interpret the slack/surplus values of 0 and 1 in the computer output.

Optimal Solution
 

Variable	Value
X11	1
X12	0
X13	0
X21	0
X22	0
X23	0
X31	0
X32	0
X33	1
X41	0
X42	1
X43	0
Constraint	Slack/Surplus
1	0
2	1
3	0
4	0
5	0
6	0
7	0

4. (16 points) Using the information below, answer the following:

a) Determine the MA(2) forecast for period 6.

b) Determine the MA(3) forecast for period 6.

c) Determine the mean square error (MSE) for the MA(2) and the MA(3) forecasts.

d) Which is the better forecast (MA(2) or MA(3))?

Forecasting with Moving Averages

The Moving Average uses 2 time periods
 

Time Period	Time Series Value	Forecast	Forecast Error
1	14
2	17
3	15
4	21
5	20

5. (16 points)

Briefly describe the four components of time series data. In describing each component, provide an example of data which might exhibit this component.