Module 3 Individual Problems

M3_IND1. A furniture cabinet maker produces two types ofcabinets, Classic and Modern, that house and hide LCD TVs. Theresource requirements and profit for the two types of cabinets areshown below.

800 12 350

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

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

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

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

4. d) Is the production of 80 Classic models and 160 Modern modelsfeasible (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 modelsfeasible (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 componentplant is scheduling its work for next month. The plant produces thefollowing four components: A1, B2, C3, and D4. Each component mustgo through three departments during the fabrication process. Afterfabrication, each valve is inspected by a human being, who spends15 minutes per valve. There are 500 inspection hours available forthe month. The time required (in hours) for each department to workon each component is shown in the following table. Also shown arethe minimum number of components that must be produced for themonth 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 numberof each product to manufacture that meets the requirements andmaximizes 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 optimalsolution?

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

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

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

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

   solution?

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

   solution?

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

   solution?

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

   solution?

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M3_IND3. A snack company packages and sells three differentcanned party mixes that contain a total of 1 lb. of nuts. Thesethree 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. ofeach ingredient in each product type, the amount of ingredientavailable, and the revenue generated by selling each type ofproduct. What should their production plan be to maximize theirrevenue? There is on additional piece of information that impactstheir production plan and should be included in your formulation.Past demand indicates customers purchase at least three times asmany cans of Plain Nuts as Mixed Nuts. Your formulation shouldinclude a constraint that states that the number of cans of PlainNuts produced should be at least three times the number of cans ofMixed Nuts produced. Formulate and solve the problem in Excel todetermine the number of each product to produce that meets therequirements and maximizes revenues. (Note: Consider this anaverage amount of cans produced – the number of cans does not needto 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 youroptimal solution (enter two decimal places)?

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

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

5. e) After producing the number of cans of each product assuggested in your optimal solution, which of the ingredients hasnot 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 productionfor four types of gears. Becausethere are limited resources in theplant for production, the manufacturer can outsource the gears bypurchasing these gears from a regional supplier. The regionalsupplier can supply a maximum of 400 units of each type of gear.The table below shows the exact demand for the gears, the revenueper unit, and the outsource cost per unit if the gears arepurchased from the supplier. The manufacturer generates the samerevenue per unit for the gears regardless of whether the gear ismanufactured in their plant and then sold to their customers oroutsourced from their supplier and then sold to theircustomers.

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 mustbe 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 thecapacity for each department and the cost per hour for processingthe gears in those departments. The cost per hour for processingthe gears is provided so that you can calculate the manufacturingcost.

0.30 0.25 0.31 0.40 400 $8.75

Formulate and solve this problem in Excel to determine theproduction and/or outsource plan which will meet the requirementsand maximize the profit. (Hint: processing costs in the secondtable effect only the profit for the gears that are manufacturedand not the gears that are outsourced)

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

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

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

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

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

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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 ina diversified portfolio through a commercial lender. The types ofinvestments, the expected annual interest rate for the investment,and the maximum allowed percentage of the total portfolio that theinvestment 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-mortgageinstruments. Furthermore, she wants no more than 35% of her totalinvestment to be in high-yield and high-risk instruments (i.e.expected interest rate of investment is 8% or greater). Formulateand solve this problem in Excel to determine how her money shouldbe diversified in a manner which will meet the requirements andmaximize the amount of interest income. (Hint: Make sure that theLHS and RHS of constraints are the same units)

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

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

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

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 newsolution?

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 Futurestrading

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

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

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

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

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

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

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

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

  total interviews

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

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

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

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

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