Be specific and document your work clearly.
1. The Ace Manufacturing Company produces two lines of its product, the super and the regular. Resource requirements for production are given in the Table below. There are 1,600 hours of assembly worker hours available per week, 700 hours of paint time, and 300 hours of inspection time. Regular customers will demand at least 150 units of regular line and 90 of the super.
|
Product line
|
Profit Contribution
|
Assembly time (hr.)
|
Paint time (hr.)
|
Inspection time (hr.)
|
|
Regular
|
$50
|
1.2
|
.8
|
.2
|
|
Super
|
$75
|
1.6
|
.9
|
.2
|
The linear programming formulation for this product mix problem is:
Decision variables
x1 = units of regular produced
x2 = units of super produced
Formulation
Maximize Z = 50x1 + 75x2
s.t.
1.2x1 + 1.6x2 ≤ 1600 Assembly time
.8x1 + .9x2 ≤ 700 Paint time
.2x1 + .2x2 ≤ 300 Inspection time
x1 ≥ 150 Regular demand
x2 ≥ 90 Super demand
x1, x2 ≥ 0
Answer the following questions on this page and the next two pages referring to the above formulation and the printout on the next page
a. What is the optimal solution (be specific, how much x1 how much x2 what is the value of the objective function)?
b. If demand for regular increased by 10, what will happen to the optimal solution (Z) and the decision variables x1 and x2?
c. If demand for super increased by 10, what will happen to the optimal (Z) and the decision variables x1 and x2?
d. If the profit contribution of regular decreased to 30, what will happen to the optimal solution (Z and decision variables)?
e. If the profit contribution of super decreased to 55, what will happen to the optimal solution (Z and decision variables)?
2. Hands-on is a company that features a product line of winter gloves for the entire family- men, women, and children. They want to decide what mix of these three types of gloves to produce.
The Hands-on's: manufacturing labor force is unionized. Each full-time employee works a 40-hour week. In addition, by union contract, the number of full-time employees can never drop below 20. Nonunion, part-time workers also can be hired with the following union-imposed restrictions
(1) Each part-time worker works 20 hours per week, and;
(2) There must be at least two full-time employees for each part-time employee.
In terms of the manufacturing process, all three types of gloves are made out of the same 100 percent genuine cowhide leather. Hands-on has a long-term contract with a supplier of the leather and receives a 5,000 square-foot shipment of material each week. The material requirements and labor requirements, along with the gross profit per glove sold (Not considering labor costs), are given in the following table below:
|
Glove
|
Material Required
(Square Feet)
|
Labor Required
(Minutes)
|
Gross Profit
(per pair of gloves)
|
|
Men's
|
2
|
30
|
$8
|
|
Women's
|
1.5
|
45
|
10
|
|
Children's
|
1
|
40
|
6
|
Each full-time employee earns $13 per hour, while each part-time employee earns $10 per hour. Management wishes to know what mix of each of the three types of gloves to produce per week, as well as how many full-time and part-time workers to employ while they would like to maximize their net profit-their gross profit from sales minus their labor costs.
a. Formulate and solve a linear programming model to determine the best mix of gloves and employees to have to maxmize their net profit. Be clear and identify/describe the decision variables, constraints and objective function.
3. An energy utility company (EnCo) has to replace its old generating equipment. Two choices are under consideration: one is to invest in technology similar to what they have used but a newer model; the other is to try a newer technology that promises higher efficiency, output and profits, but, may not produce as advertised. The following table shows the anticipated profits under the two choices and two states of nature.
a. Find the recommended solution using the three game theory solution methods, Maximax, Maximin, and Minimizing Opportunity Loss.
b. EnCo estimates that the probability of the new equipment working as advertised is 0.65 (therefore not working as advertised is 0.35). Use TreePlan to draw the decision tree for EnCo. What would be the Expected Value (EV) using the probabilities of the new technology equipment producing as advertised, or not producing as advertised.
c. How sensitive is the EMO solution found in b) to changes in the probability of the new technology not performing as advertised? At what probability for the new technology performing as advertised does the choice shift to choosing the current technology equipment?
From DA chapter Ratick Q 4
4. You are deciding to invest in undeveloped property that may eventually be bought by a real estate developer to expand their current residential development. It will cost you $50,000 to purchase the property. If the developer decides to expand you will be able to sell that property to the developer for $75,000. If the real estate developer does not expand you will be able to sell the land for $35,000. You estimate the probability of the land being developed to be 0.4.
a. Develop this decision tree. Given the Expected Value (EV) calculations, should you purchase the property? Why?
b. Before making your decision you have the option of paying an expert in the real estate business who has offered to predict if the real estate developer will expand and purchase your property. If the expert is perfectly reliable (his predictions have always come true), what is the maximum amount you should pay him for his advice; that is what is the EVPI? (Hint: Create another decision tree that has the first decision that of paying for this advice or not paying for the advice. Add the decision tree for choice of purchasing the property to both of these decision paths.)
5. KMR industries is considering plant expansion to enable the production of a new product. The company's CEO must decide whether to do a large-scale or medium scale expansion. The demand for the new product is uncertain, which the company will view in three categories, low demand, medium demand, or high demand, the probability estimates for the three categories are 0.20 low demand, 0.50 medium demand, and 0.30 high demand. Letting the random variable x represent annual profit in $1000s, the firm's analysts developed the following profit forecasts for the low, medium and high demand forecasts:
|
|
|
Medium Scale
|
Large Scale
|
|
|
Expansion
|
Profits
|
Expansion
|
Profits
|
|
|
x
|
f(x)
|
y
|
f(y)
|
|
Demand
|
Low
|
50
|
0.20
|
0
|
0.20
|
|
Medium
|
150
|
0.50
|
100
|
0.50
|
|
High
|
200
|
0.30
|
300
|
0.30
|
a. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing profits?
b. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?
6. Consider a binomial experiment with 2 trials and p=0.04.
a. Compute the probability of 1 success in the two trials f(1).
b. Compute f(0).
c. Compute f(2).
d. Find the probability of at least one success.
e. Find the expected value, variance and standard deviation for this experiment.
7. The quarterly sales data in number of copies sold for a college textbook over the past three years is as follows:
|
Quarter
|
Year 1
|
Year 2
|
Year 3
|
|
1
|
1690
|
1800
|
1850
|
|
2
|
940
|
900
|
1100
|
|
3
|
2625
|
2900
|
2390
|
|
4
|
2500
|
2360
|
2615
|
a. Construct a time series plot. What type of patterns exists in the data?
b. Use a regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr 1 = 1 if Quarter 1; 0 otherwise, Qtr 2 = 1 if Quarter 2, 0 otherwise; Qtr 3 = 1 if Quarter 3, 0 otherwise; Qtr 4 = 1 if Quarter 4, 0 otherwise.
c. Compute the quarterly forecasts for the next year.
d. Let t=1 to refer to the observations in quarter 1 of year 1; t=2 to refer to the observation in quarter 2 of year 1; ...; and t=12 to refer to the observations in quarter 4 or year 3 (see table below). Using the dummy variables defined in part (b) above, and also using t, develop a regression equation to account for seasonal effects and any linear trend in the time series. Based upon the seasonal effects in the data and the linear trend, compute the quarterly forecasts for the next year (Year 4).
|
Year
|
Quarter
|
t
|
Textbook Sales
|
|
1
|
Quarter 1
|
1
|
1690
|
|
1
|
Quarter 2
|
2
|
940
|
|
1
|
Quarter 3
|
3
|
2625
|
|
1
|
Quarter 4
|
4
|
2500
|
|
2
|
Quarter 1
|
5
|
1800
|
|
2
|
Quarter 2
|
6
|
900
|
|
2
|
Quarter 3
|
7
|
2900
|
|
2
|
Quarter 4
|
8
|
2360
|
|
3
|
Quarter 1
|
9
|
1850
|
|
3
|
Quarter 2
|
10
|
1100
|
|
3
|
Quarter 3
|
11
|
2930
|
|
3
|
Quarter 4
|
12
|
2615
|
8. Building a new plant warehouse consists of 9 major activities (A through I). The activities and their immediate predecessors are shown in the following table.
|
Activity
|
A
|
B
|
C
|
D
|
E
|
F
|
G
|
H
|
I
|
|
Immediate Predecessor
|
None
|
None
|
A,B
|
A,B
|
B
|
C
|
D
|
D,F
|
E,G,H
|
Assume that activity time estimates (in days) for the warehouse construction project are as follows:
|
Activity
|
Optimistic
|
Most Probable
|
Pessimistic
|
|
A
|
3
|
5
|
6
|
|
B
|
2
|
4
|
6
|
|
C
|
5
|
6
|
7
|
|
D
|
7
|
9
|
10
|
|
E
|
2
|
4
|
6
|
|
F
|
1
|
2
|
3
|
|
G
|
5
|
8
|
10
|
|
H
|
6
|
8
|
10
|
|
I
|
3
|
4
|
5
|
a. Develop the project network.
b. What is the critical path (what are the critical activities)?
c. What is the expected time to complete the project?
d. What is the probability the project can be completed in 25 or fewer days