Spring 2015Mathematics in

Management Science

Linear Programming

Mixture Problems

Dolls & Skateboards

Linear Programming

Mathematics for analyzing optimal allocation of resources.

All businesses have limited resources (capital, raw materials, etc).

A fundamental problem is to make production decisions to optimize profits.

Mixture Problems

Combine limited resources into products so that the profit from selling products is a maximum.

Focus on case where there are two limited resources and two possible products that can be created using these resources; but more ok too.

All Mixture Problems Include

1. list of resources to be used,2. amount of each resource available,3. list of products to be produced,4. recipes for each product,5. profit information for each product.Goal is to find Optimal Prod Policy:

determine how much of each product to make to maximize profit (w/o exceeding any resource limitations).

Dolls and Skateboards

A company makes dolls & skateboards. It has 60 buckets of plastic.

Each board uses 5 buckets of plastic and each doll 2 buckets.

Make $1 profit on each board, and 55c on each doll.

How many dolls and/or skateboards should they produce to maxmz profit?

Setting up the variables

We want to find

# of dolls and # of boards

to be produced.

Use production variables

x = number of skateboards

y = number of dolls

Mixture Charts

A mixture table summarizes given information.

Resource column: amount of resource needed to make 1 unit of product.

Profit column: amount of profit per unit of product.

Resource Constraint

Total available plastic is 60 buckets.• x skateboards requires 5x buckets• y dolls requires 2y buckets of plastic

So, resource constraint is5x + 2y ≤ 60

Also have prod constraints: x,y ≥ 0.

Feasible region

All production possibilities are given by the inequalities:

5x + 2y ≤ 60x ≥ 0y ≥ 0

Above inequalities describe the so-called feasible region.

Want (x,y) in feasible region and giving maximum profit.

Drawing Feasible Region

First, draw the line 5x + 2y = 60.

Find x- and y-intercepts:

y-intercept: point with x = 0, so y = ?

x-intercept: point with y = 0, so x = ?

y-intercept is pt (0,30)

x-intercept is pt (12,0)

Graphical representation of feasible region:

5x + 2y ≤ 60 , x ≥ 0 , y ≥ 0 .

Corner Point Principle

Production options are points of feasible region. Which of these options yields maximal profit?

The corner point principle states that we get maximum profit at a corner point of the feasible region.

Evaluate profit at all corner points, and choose point where profit is greatest.

Mixture Charts

A mixture chart summarizes given information.

Resource column: amount of resource needed to make 1 unit of product.

Profit column: amount of profit per unit of product.

Profit Formula

Profit from x skateboards is

x $1 = $x

Profit from y dolls is

y $0.55 = $(.55)y

Total profit is

P = $ x + $ (.55)y

Evaluate the profit function P= $ x + $ (.55)y at the three corner points (0,0), (0,30) and (12,0)

Conclusion Get max profit at (0,30).

OPP produce 0 skateboards and 30 dolls.

Mixture Problem Algorithm

Display all info in mixture chart.

Write down

resource constraints (RC),

profit formula (PF).

Draw feasible region & mark corner pts.

Evaluate PF at each corner point.

State OPP.

Sam’s Ski Shop

Sam makes skis and snow boards. He uses

3 buckets of plastic for a pair of skis

4 buckets of plastic for a board.

He has 24 buckets of plastic.

His profit is$100 for each pair of skis

$120 for each snow board.

Mixture Chart

RC 3x + 4y ≤ 24 , x ≥ 0, y ≥ 0

PF P = 100 x + 120 y

Products Resourceplastic

24

Profit

skis (x) 3 100

boards (y) 4 120

Feasible Region

(8,0)

3x + 4y ≤ 24

(0,6)

Corner Point Principle

Have corner points (0,0), (8,0), (0,6).

At each of these, profit P=100x+120y is

P=0 at (0,0)

P=800 at (8,0)

P=720 at (0,6)

So, optimal production policy is for Sam to make 8 pairs of skis.

Changing Profit Formula

By aggressive advertising, Sam is able to sell his snowboards for $150. So?

Get new profit formula,

P = 100 x + 150 y .

Same feasible region & corner points.

Evaluate new PF. GetP=0 at (0,0), P=800 at (8,0), P=900 at (0,6).

New OPP: produce 6 boards.

Positive Minimums

Sam lands a contract to sell 4 pairs of skis each week. Now what?

This changes one of the minimum constraints

from x ≥ 0 to x ≥ 4.

This changes the feasible region and

introduces a new corner point.

New Feasible Region

(8,0)

3x + 4y ≤ 24

(0,6)

x=4

(4,3)

(4,0)

Corner Point Principle

Have corner points (4,0), (8,0), (3,4).

At each of these, profit P=100x+150y is

P=400 at (4,0)

P=800 at (8,0)

P=850 at (4,3)

Sam’s new optimal production policy is to make 4 pairs of skis & 3 snowboards

Bikes & Wagons

Bill’s Toy Shop manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.

Each bike requires 2 hours of machine time and 4 hours of painting time.

Each wagon takes 3 hours of machine time and 2 hours of painting time.

Bikes & Wagons

Bill’s Toy factory manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.

Each bike requires 2 hours of machine time and 4 hours of painting time.

Each wagon takes 3 hours of machine time and 2 hours of painting time.

There are 12 hours of machine time and 16 hours of painting time available per day.

How many bikes/wagons to make?

Bikes & Wagons

Bill’s Toy factory manufactures bikes and wagons for profits of $12 per bike and $10 per wagon.

Each bike requires 2 hours of machine time and 4 hours of painting time.

Each wagon takes 3 hours of machine time and 2 hours of painting time.

There are 12 hours of machine time and 16 hours of painting time available per day.

How many bikes/wagons to make?

What if must make at least 2 of each?

Mixture Chart

RC 2x + 3y ≤ 12 ,

4x + 2y ≤ 16 ,

x ≥ 0, y ≥ 0

PF P = 12 x + 10 y

Products Resourcesmachine time painting time

12 16

Profit

bikes (x) 2 4 12

wagons (y) 3 2 10

4x+2y=16

2x+3y=12

(0,8)

(4,0) (6,0)

(0,4)

(3,2)

Corner Pts are

(0,0),(0,4),(3,2),(6,0)

Feasible Region

Corner Point Principle

Have corner points

(0,0), (0,4), (3,2),(4,0).

Evaluate profit P=12x+10y at each

P=0 at (0,0) P=40 at (0,4)

P=56 at (3,2) P=48 at (4,0)

Optimal production policy is to make

3 bikes & 2 wagons.

Same OPP when make 2 of each