Question

Module 3 Individual Problems

M3_IND1. A furniture cabinet maker produces two types of cabinets, Classic and Modern, that house and hide LCD TVs. The resource requirements and profit for the two types of cabinets are shown below.

800 12 350

The firm has a budget of $185,000 to spend on materials. The firm has 2,000 labor hours are available for use. What is the best combination of furniture cabinets to be made? Solve this two decision variable problem using the LP Graphing utility.

1. a) What is the profit (value of the objective function) for the optimal solution?

2. b) How many Classic models should be produced (based on the optimal solution)?

3. c) How many Modern models should be produced (based on the optimal solution)?

4. d) Is the production of 80 Classic models and 160 Modern models feasible (not asking if it is

   optimal). Does it fall in the feasible region?

5. e) Is the production of 160 Classic models and 100 Modern models feasible (not asking if it is

   optimal). Does it fall in the feasible region?

Resource Requirements and Profitability

Model

Classic Modern

Materials ($/unit) Labor (hrs./unit) Profit ($/unit)

600 4 250

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M3_IND2. The fabrication department for an automobile component plant is scheduling its work for next month. The plant produces the following four components: A1, B2, C3, and D4. Each component must go through three departments during the fabrication process. After fabrication, each valve is inspected by a human being, who spends 15 minutes per valve. There are 500 inspection hours available for the month. The time required (in hours) for each department to work on each component is shown in the following table. Also shown are the minimum number of components that must be produced for the month and the unit profit for each component.

	COMPONENTS TO BE PRODUCED
DEPARTMENT	A1 B2 C3 D4 CAPACITY OF EACH DEPARTMENT (hours)
DRILLING	0.90 0.50 0.6 0.7 1400
MILLING	0.80 0.55 0.7 0.7 1400
ASSEMBLY	1.10 0.60 0.50 0.65 1400
MINIMUM OF EACH PRODUCT TYPE NEEDED	300 500 600 400
PROFIT ($/UNIT)	$14 $10 $11 $15

Formulate and solve the problem in Excel to determine the number of each product to manufacture that meets the requirements and maximizes profits.

1. a) What is the maximum profit based on your optimal solution (the value of the objective function)?

2. b) How many A1's should be manufactured based on your optimal solution?

3. c) How many B2's should be manufactured based on your optimal solution?

4. d) How many C3's should be manufactured based on your optimal solution?

5. e) How many D4's should be manufactured based on your optimal solution?

6. f) What is the total number of hours used in the drilling department based on your optimal

   solution?

7. g) What is the total number of hours used in the milling department based on your optimal

   solution?

8. h) What is the total number of hours used in the assembly department based on your optimal

   solution?

9. i) What is the total number of hours used in the inspection department based on your optimal

   solution?

2

M3_IND3. A snack company packages and sells three different canned party mixes that contain a total of 1 lb. of nuts. These three different products (Plain Nuts, Mixed Nuts, and Premium Mix) include a mix of four possible types of nuts (peanuts, cashews, almonds, and walnuts). The table below show the number of lbs. of each ingredient in each product type, the amount of ingredient available, and the revenue generated by selling each type of product. What should their production plan be to maximize their revenue? There is on additional piece of information that impacts their production plan and should be included in your formulation. Past demand indicates customers purchase at least three times as many cans of Plain Nuts as Mixed Nuts. Your formulation should include a constraint that states that the number of cans of Plain Nuts produced should be at least three times the number of cans of Mixed Nuts produced. Formulate and solve the problem in Excel to determine the number of each product to produce that meets the requirements and maximizes revenues. (Note: Consider this an average amount of cans produced – the number of cans does not need to be an integer).

0.8 0.25 500

0.2 0.25 0.2 300

0.25 0.4 120

0.25 0.4 100

$2.25 $5.65 $7.85

1. a) What is the maximum revenue based on your optimal solution (the value of the objective function)?

2. b) How many cans of Plain Nuts should be produced based on your optimal solution (enter two decimal places)?

3. c) How many cans of Mixed Nuts should be produced based on your optimal solution (enter two decimal places)??

4. d) How many cans of Premium Mix should be produced based on your optimal solution (enter two decimal places)??

5. e) After producing the number of cans of each product as suggested in your optimal solution, which of the ingredients has not been totally used by your production plan?

PRODUCT

INGREDIENTS

PEANUTS (lbs./can) CASHEWS (lbs./can)ALMONDS (lbs./can) WALNUTS (lbs./can) REVENUE ($/UNIT)

PLAIN MIXED PREMIUM INGREDIENT NUTS NUTS MIX AVAILABILITY (lbs.)

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M3_IND4. A gear manufacturer is planning next week’s production for four types of gears. Becausethere are limited resources in the plant for production, the manufacturer can outsource the gears by purchasing these gears from a regional supplier. The regional supplier can supply a maximum of 400 units of each type of gear. The table below shows the exact demand for the gears, the revenue per unit, and the outsource cost per unit if the gears are purchased from the supplier. The manufacturer generates the same revenue per unit for the gears regardless of whether the gear is manufactured in their plant and then sold to their customers or outsourced from their supplier and then sold to their customers.

GEAR TYPE

Demand

RevenueOutsource Cost

GEAR A GEAR B GEAR C GEAR D

650

$13.75

$9.20

500

$12.50

$9.75

450

$16.90

$11.00

550

$18.50

$11.75

PRODUCT

When the gears are manufactured in the own plant, the gears must be processed through three different departments: forming, hardening, and deburring. The table below shows the processing time (in hours) for each type of gear in the departments as well as the capacity for each department and the cost per hour for processing the gears in those departments. The cost per hour for processing the gears is provided so that you can calculate the manufacturing cost.

0.30 0.25 0.31 0.40 400 $8.75

Formulate and solve this problem in Excel to determine the production and/or outsource plan which will meet the requirements and maximize the profit. (Hint: processing costs in the second table effect only the profit for the gears that are manufactured and not the gears that are outsourced)

1. a) How much profit does the company for all gears they make and buy in your solution (the value of the objective function)? (enter to the nearest integer)

2. b) If you could add one hour of capacity to any department to increase profit - adding one hour of capacity to which department would generate the biggest increase in profit: Forming, Hardening, or Deburring?

3. c) Which of the following constraints have slack? (Choose all constraints with slack): Forming, Hardening, or Deburring

4. d) In your solution, how many Gear C's should the company make?

5. e) If the cost per hour of the hardening process increases to $12/hr. - how many Gear D's should

   the company make with this new process cost?

PROCESS

GEAR A GEAR B (hrs./unit) (hrs./unit)

GEAR C GEAR D (hrs./unit) (hrs./unit)

DEPARTMENT CAPACITY (hours)

Forming Hardening Deburring

0.37 0.43

0.45 0.52 500

4

COST ($/hr.)

$9.50

0.40 0.37

0.42 0.32 400

$7.90

M3_IND5. An investor wishes to invest all of her $6.5 million in a diversified portfolio through a commercial lender. The types of investments, the expected annual interest rate for the investment, and the maximum allowed percentage of the total portfolio that the investment can represent are shown in the table below:

6.20% 25%

8.00% 25%

4.45% 30%

7.50% 15%

8.90% 10%

She wants at least 40% of her total investment in non-mortgage instruments. Furthermore, she wants no more than 35% of her total investment to be in high-yield and high-risk instruments (i.e. expected interest rate of investment is 8% or greater). Formulate and solve this problem in Excel to determine how her money should be diversified in a manner which will meet the requirements and maximize the amount of interest income. (Hint: Make sure that the LHS and RHS of constraints are the same units)

1. a) What is the expected total interest income generated from the investment strategy (the value of the objective function)?

2. b) Based on your solution, how much should be invested in government sponsored mortgage loans?

3. c) Based on your solution, how much should be invested in stock investments?

4. d) If you could increase the maximum allowed for the investments (in order to increase overall

   return) - which would you choose: conventional mortgage loans, bond investments, or

   governmental sponsored mortgage loans.

5. e) If the return on low-income mortgage loans was reduced to 4%, how much should be invested

   in these low-income mortgage loans based on your new solution?

INVESTMENT

EXPECTED INTEREST

MAXIMUM ALLOWED (% of total portfolio)

Low-income mortgage loans Conventional mortgage loans

7.40% 20%

Government sponsored

mortgage loans Bond investments Stock investments Futures trading

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M3_IND6. A student project at WCU was initiated to try to determine the impact of implementation of new technologies. The students want to survey both distance and residential undergraduate students in the four different years at Western (first year, sophomore, junior, and senior). They have estimated that it will cost them $5.50 to survey first year and sophomore residential students and $8.00 to survey junior and senior residential students. The cost to interview distance students is slightly higher. It will cost $6.75 for first year and sophomores and $9.50 for junior and seniors. For statistical validity they want to interview at least 900 students. They feel that there are certain criteria that they must adhere to:

• At least 25% of first year students surveyed should be distance students

• At least 20% of sophomore students surveyed should be distance students

• At least 35% of junior students surveyed should be distance students

• At least 40% of senior students surveyed should be distance students

• No more than 35% of all the students surveyed should be first year students

• Juniors and seniors should be at least 45% of the students surveyed

• Each of the eight types of students must be represented in the survey by at least 10% of the

  total interviews

  Formulate and solve this problem in Excel to determine the number of each type of student that should be surveyed that meets the requirements and minimizes the cost to carry out the interviews.

1. a) What is the minimum cost in your optimal solution (the value of the objective function)?

2. b) If the cost of surveying first year and sophomore residential students increases from $5.50 to

   $7.00 – what is the new minimum cost in your optimal solution?

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Answer 1

(1)

Decision Variables:

x = Number of Classic models

y = Number of Modern models

P = Profit

Objective Function:

Maximize P = 250x + 350y

Constraints:

600x + 800y ≤ 185000 [C1]

4x + 12y ≤ 2000 [C2]

x,y ≥ 0

Solution:

Optimal Solution: x = 155, y = 115, P = $79000

(a) $79000

(b) 155

(c) 115

(d) No

(e) Yes