# OR/MA 504 Homework #1 Solutions Fall 05

OR/MA 504 Fall ’17Homework #1page 1

OR/MA 504Homework #1 Due Wednesday Sept. 6Reiland

Course text: p. 55: #5; p. 68: #1 - #7; additional problems

p. 55, #5

Why don’t we allow an LP to have < or > constraints?

p. 68, #1 - #7

Case 1: The LP has a unique optimal solution.

Case 2: The LP has alternate optimal solutions: two or more extreme points are optimal, and the LP will have an infinite number of optimal solutions.

Case 3: the LP is infeasible: the feasible region contains no points.

Case 4: The LP is unbounded: There are points in the feasible region with arbitrarily large z-values (max problem) or arbitrarily small z-values (min problem).

In the LP in problems 1-4 identify which of Cases 1-4 above apply to the LP.

#1max z = x1 + x2

s.t.x1 + x2 ≤ 4

x1 − x2 ≥ 5

x1 , x2 ≥ 0

#2max z = 4x1 + x2

s.t.8x1 + 2x2 ≤ 16

5x1 + 2x2 ≤ 12

x1 , x2 ≥ 0

#3max z = −x1 + 3x2

s.t.x1 − x2 ≤ 4

x1 + 2x2 ≥ 4

x1 , x2 ≥ 0

#4max z = 3x1 + x2

s.t.2x1 + x2 ≤ 6

x1 + 3x2 ≤ 9

x1 , x2 ≥ 0

#5 True or False: For an LP to be unbounded, the LP’s feasible region must be unbounded.

#6. True or False: Every LP with an unbounded feasible region has an unbounded optimal solution.

#7. If an LP’s feasible region is not unbounded, we say that the LP’s feasible region is bounded. Suppose an LP has a bounded feasible region. Explain why you can find the optimal solution to the LP (without using objective function contour lines) by simply checking the objective function values at each of the feasible region’s extreme points. Why might this method fail if the LP’s feasible region is unbounded?

8.The Electrotech Corporation manufactures two industrial-sized electrical devices: generators and alternators. Both of these products require wiring and testing during the assembly process. Each generator requires 2 hours of wiring and 1 hour of testing and can be sold for a $250 profit. Each Alternator requires 3 hours of wiring and 2 hour of testing and can be sold for a $150 profit. There are 260 hours of wiring time and 140 hours of testing time available in the next production period and Electrotech wants to maximize profit.

Create a spreadsheet model for this problem and solve it using Solver.

9.Valu-Com Electronics manufactures five different models of telecommunication interface cards for personal and laptop computers. As summarized in the following table, each of these devices requires differing amounts of printed circuit (PC) board, resistors, memory chips, and assembly.

**Per Unit Requirements**

HyperLink / FastLink / SpeedLink / MicroLink / EtherLink

PC Board / 20 / 15 / 10 / 8 / 5

(square inches)

Resistors / 28 / 24 / 18 / 12 / 16

Memory Chips / 8 / 8 / 4 / 4 / 6

Assembly Labor / 0.75 / 0.6 / 0.5 / 0.65 / 1

(in hours)

The unit wholesale price and manufacturing cost for each model are as follows:

**Per Unit Revenues and Costs**

HyperLink / FastLink / SpeedLink / MicroLink / EtherLink

Wholesale Price / $189 / $149 / $129 / $169 / $139

Manufacturing Cost / $136 / $101 / $96 / $137 / $101

In their next production period, Valu-com has 80,000 square inches of PC board, 100,000 resistors, 30,000 memory chips, and 5,000 hours of assembly time available. They can sell all the product they can manufacture, but the marketing department wants to be sure they produce at least 500 units of each product and at least as twice as many FastLink cards as HyperLink cards while maximizing profit.

- Formulate an LP model for this problem

b. Create a spreadsheet model for this problem and solve it using Solver.

c.What is the optimal solution?

d. Could Valu-Com make more money if they schedule their assembly workers to work overtime?

OR/MA 504 Fall ‘16Homework #1page 1

10. A trust officer at the Blacksburg National Bank needs to determine how to invest $100,000 in the following collection of bonds to maximize the annual return.

AnnualBond / Return / Maturity / Risk / Tax Free

A / 9.5% / Long / High / Yes

B / 8.0% / Short / Low / Yes

C / 9.0% / Long / Low / No

D / 9.0% / Long / High / Yes

E / 9.0% / Short / High / No

The officer wants to invest at least 50% of the money in short-term issues and no more than 50% in high-risk issues. At least 30% of the funds should go in tax-free investments and at least 40% of the total annual return should be tax free.

- Formulate an LP model for this problem

b. Create a spreadsheet model for this problem and solve it using Solver.

c. What is the optimal solution?

- Acme Manufacturing makes a variety of household appliances at a single manufacturing facility. The expected demand for one of these appliances during the next four months is shown in the following table along with the expected production costs and the expected capacity for producing these items.

Month

1 / 2 / 3 / 4

Demand / 420 / 580 / 310 / 540

Production Cost / $49 / $45 / $46 / $47

Production Capacity / 500 / 520 / 450 / 550

Acme estimates that it costs $1.50 per month for each unit of this appliance carried in inventory (estimated by averaging the beginning and ending inventory levels each month). Currently Acme has 120 units in inventory on hand for this product. To maintain a level workforce, the company wants to produce at least 400 units per month. They also want to maintain a safety stock of at least 50 units per month. Acme wants to determine how many of this appliance to manufacture during each of the next four months to meet the expected demand at the lowest possible total cost.

a.Formulate an LP model for this problem

b.Create a spreadsheet model for this problem and solve it using Solver.

c.What is the optimal solution?

d.How much money could Acme save if they were willing to drop the restriction about producing at least 400 units per month?

- Jack Potts recently won $1,000,000 in Las Vegas and is trying to determine how to invest his winnings. He has narrowed his decision to five investments, which are summarized in the following table.

**Summary of Cash Inflows and Outflows**

**(at beginning of years)**

1 / 2 / 3 / 4

A / -1 / 0.50 / 0.80

B / -1 / <------> / 1.25

C / -1 / <-----> / <------> / 1.35

D / -1 / 1.13

E / -1 / <------> / 1.27

If Jack invests $1 in investment A at the beginning of year 1, he will receive $0.50 at the beginning of year 2 and another $0.80 at the beginning of year 3. Alternatively, he can invest $1 in investment B at the beginning of year 2 and receive $1.25 at the beginning of year 4. Entries of “<---->” in the table indicate times when no cash inflows or outflows can occur. At the beginning of any year, Jack can place money in a money market account that is expected to yield 8% per year. He wants to keep at least $50,000 in the money market account at all times and doesn’t want to place any more than $500,000 in any single investment. How would you advise Jack to invest his winnings if he wants to maximize the amount of money he’ll have at the beginning of year 4?

- Formulate an LP model for this problem.
- Create a spreadsheet model for this problem and solve it using Solver.
- What is the optimal solution?