BUS 190 Homework 2

BUS 190        Homework 2

For this week’s homework, you will just be asked to set up the linear program as we did in class, and then to rewrite it in the ‘algebraic form’.  Homework 3 will involve graphing the constraints and solving the linear program using that graph, so if you want to get a head start on Homework 3 and are sure of your algebraic setup, you can start the graphs. DO NOT INCLUDE graphs here on HW 2.

1. Suppose a small company is in the business of making golf bags to be sold at a local golf course where state championships are often held. They make 2 types of golf bags, a standard bag and a deluxe bag.  The table below gives the information for limitations of materials and labor hours, and also for the profit for each type of bag.  The materials and labor hours are the number of hours available per week, and the profit is the profit in $ per week as well.

Units are per bag	Standard Leather (square feet)	Premium Leather (square feet)	Cutting and Sewing Labor hours	Finishing Labor Hours	Profit ($)
Standard	7	1	3	1	25
Deluxe	0	9	4	2	65
Total available	210	180	120	40

1. What are the decision variables (use NUN!)

The decision variables are S for the number of standard bags and D for the number of deluxe bags.

1. What is the objective function? (make sure to use dimensional analysis to set it up)

The objective function is to maximize profit. The formula is max (z) = 25S + 65D.

$25                            S of standard bags                     $65           D of deluxe bags

--------                  *                                             +            -------     *                         = $25S + $65D

1 standard bag                                                                 1 deluxe bag

1. What are the constraints? Make sure to use dimensional analysis to set them up.

The constraints are:

7           * S standard leather sq ft          0 * D standard leather sq ft             

-----                                                       +        -------                                         <= 210

1 standard leather sq ft                        standard leather sq ft                           

1           * S premium leather sq ft          9 *D premium leather sq ft             

-----                                                       +        -------                                         <= 180

1 premium leather sq ft                        premium leather sq ft                          

3           * S cslabor hours          4 * D cslabor hours             

-----                                                       +        -------                                         <= 120

1 cslabor hours                        cslabor hours                           

1          * S flabor hours          1 * D flabor hours   

-----                                                       +        -------                                         <= 140

1 flabor hours                           flabor hours                              

7S+0D <= 210

1S + 9D <= 180

3S + 4D <= 120

1S + 1D <= 40

1. Summarize the linear program in algebraic form. You do not need to include units here, but you should label your constraints.

Z = 25S + 65D

s.t.

standard leather constraint: 7S+0D <= 210

premium leather constraint: 1S + 9D <= 180

cutting and sewing labor hours constraint: 3S + 4D <= 120

finished labor hours constraint: 1S + 1D <= 40

And

S >= 0, D >= 0

2. You have just inherited $100,000 from a relative, and you want to invest the money so that you get the maximum annual return you can.  You are thinking of putting your money into two types of stock mutual funds. One fund S concentrates in social media companies, and the other fund I concentrates in companies that specialize in infrastructure construction and maintenance.  You also are sensitive to risk in investment, so you visited a financial consultant to determine your risk tolerance.   You are also worried that since social media isn’t a tangible product, you want to limit the number of shares you own in that mutual fund to be less than 1000. The table below summarizes the information you have at hand, per share.

Cost ($)  per share	Risk (unitless)	Max shares	Expected annual return ($1,000s)
Social media	$50	0.5	1	5
Infrastructure	$30	0.25	7
Total	$100,000	700	1000

1. What are the decision variables (use NUN!)

The decision variables are S as the number of units purchased in fund S and I as the number of units purchased in fund I.

1. What is the objective function? (make sure to use dimensional analysis to set it up)

The objective function is maximizing expected annual return (z) = 5S + 7I

5S          * 1 shares            7I *  shares

--------                  +             -------                   = 5S + 7I

1 share                              1 share

1. What are the constraints? Make sure to use dimensional analysis to set them up.

5           * S shares             $30 * I shares

--------                  +             -------                   <= $100,000

1 share                              1 share

1S shares <= 1000 shares

$50S + $30I <= $100000

S <= 1000

.5S + 0.25I <= 700

1. Summarize the linear program in algebraic form. You do not need to include units here, but you should label your constraints.

z = maximized expected annual return

z = 5S + 7I

s.t.

cost constraint: 50S + 30I <= 100000

max shares constraint: S <= 1000

risk restraint: .5S + 0.25I <= 700

S >= 0, I >= 0

3. Suppose you are in the business of manufacturing white board markers. You make two types: Standard and Heavy Duty. You sell these to wholesalers for $10/box of 10 for Standard Markers and $15/box of 10 for Heavy Duty markers. Each standard marker is made up of 3 oz. of plastic and 1 fl. Oz. of marker fluid. Each Heavy Duty Marker is made up of 5 oz. of plastic and 1.5 fl. oz of marker fluid.   Plastic costs $1/lb and marker fluid costs $5/gal.  Each marker takes 30 seconds to manufacture by machine, but the machine can manufacture up to 10 at one time. Machine time costs $60/hr.

Set up the profit function for each type of marker, USING DIMENSIONAL ANALYSIS as you would have to do for a linear program objective function where you wanted to maximize profit.  It is probably easiest to figure out profit per marker rather than profit per box, but you may use either unit, as long as you are consistent and use the same unit throughout the entire problem.  SHOW ALL YOUR WORK TO GET CREDIT. Warning: do NOT skip using units at any stage. Numbers multiplied together without units in intermediate calculations, but  with units attached to them at the end will earn you ZERO points, even if your ‘number’ is correct.

Profit function ($) = (1-(.2665625+.05))x + ((10/15)-(.37190375+.05))y

4. Consider the following linear program

Max 1A + 1B

1. t.

5A + 3B ≤ 15

3A + 5B ≤ 15

A, B ≥ 0

Graph the two constraints and find the overall feasible region for the problem.  You should use the original large block graph paper I handed out first day of class and use 2 blocks per unit.

Constraint 1

5A + 3B ≤ 15

0A + 3B ≤ 15 and 5A +0B ≤ 15

B = 5

A = 3

Constraint 2

3A + 5B ≤ 15

0A + 5B ≤ 15 and 3A +0B ≤ 15

A = 5

B = 3

Test point for both (0,0)

0*5 + 0*3 ≤ 15, Correct

0*3 + 0*5 ≤ 15, Correct

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