Spring 2015Mathematics in

Management Science

Mixture Problems

What are these?

Examples

Algebra Review

A Mixture Problem Example

A candy manufacturer has: 1000 lbs of chocolate,200 lbs of nuts,100 lbs of fruit.

They make 3 mixes:special – 3 lbs choc, 1 lb nuts, 1 lb fruit;regular – 4 lbs choc, .5 lbs nuts, no fruit;purist – 5 lbs only chocolate.

These sell for $10, $6, $4 per pound.

How much of each should they make?

Mixture Problems

To combine limited resources into products so that the profit from selling these products is maximized.

A Production Policy tells us how many units of each product to make (without violating resource constraints).

An Optimal Production Policy is a PP which yields maximum profit.

Common Features of MPs

Resources – limited, known quantities

Products – combine/mix resources to get

Recipes – describe how many units of each resource need to make each product

Profits – products earn known profit

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

Find an OPP.

Another MP Example

Your company makes orange juice. It takes 10 oranges to make 1 carton of juice. You have 200 oranges.

What are possible production policies?

Answer

You can produce anywhere between 0 and 20 cartons of juice.

Feasible Region

All possible production policies; given via

resource constraints.

Have algebraic representation of FR:

Let x be number of cartons of juice.

Then 0 ≤ x ≤ 20 .

Have geometric rep (i.e. a picture) of FR:

0 5 10 2015

Feasible region for the orange juice problem

Number of cartons of juice

x

Recipes, Constraints, Production Variables

Recipe 10 oranges to make 1 carton of juice

Constraints200 oranges available

Production Variable x = number of cartons to make

Profit??

Making Profit

Suppose 50 cents profit on each carton.

Question

How many cartons of juice should be produced to maximize profit?

Answer 20, of course.

This will use up all the oranges, and

yield a profit of 20 x 50 cents = $10.

One Resource & Two Products

Suppose make either juice or frozen concentrate.It takes 5 oranges to make a can of concentrate.

Now what are the production possibilities?

Examples of Production Options:• 20 cartons of juice, no concentrate• No juice and 40 cans of concentrate• 10 cartons of juice and 20 cans of concentrate• 15 cartons of juice and 10 cans of concentrate

Production Variables & Feasible Region

Make x cartons of juice: use 10x oranges.

Make y cans concentrate: use 5y oranges.

Make x cartons and y cans: use

10x + 5y oranges.

Resource constraint: have 200 oranges.

Feasible Region:

10x + 5y ≤ 200 , x ≥ 0, y ≥ 0

What does picture of FR look like?

Algebra Review

• Number line • Inequalities• Coordinates• Cartesian plane• Lines ax + by = c• Linear Inequalities ax + by ≤ c

0 1 2 3 4 5x2 ≤ x ≤ 5

Coordinates

Coordinates used to identify a location (aka, a pt) with a set of numbers (e.g., latitude & longitude) so that we can refer to it and everyone knows where we are talking about.Can do this in one dimension (a number line), two dimensions (a coordinate plane), and all higher dimensions.Construct coordinates by introducing a base point (the origin) and intersecting lines through the base point (the axes) and then measuring distances parallel to the axes to other points.

y

2

1

0

One Dimension (1-D):

x

2

0 1 2 3

Two Dimensions (2-D):

x

2

1 2 3

3 (3, 2)

Numbers on axes are for convenience—save us from getting a ruler to measure distances.For 2-D get a pair of numbers for each point (thecoordinates) and order is important!

(horizontal distance, vertical distance).

To “plot a point” means to locate and mark the point with the given coordinates.Get a “picture” for data that consists of pairs of numbers by treating the pairs of numbers as coords and then plotting the points.

Example

Plot the points: (−1, 2), (2, −1), (2, 1)

y

x 0 1 2 3

2

1(2, 1)

(2, )

( 2)

The axis labels give the variables we use to refer to point coords. Usually, the axes are labeled x and y , so a generic point is (x, y ).

If we say y = 2, we mean that the vertical (second) coordinate is 2, so the point has a vertical distance from the origin of 2.

There are many (infinitely many) points that satisfy this condition.We specify one of these by giving its x coord.

Coords & the xy-Plane

x

y

(2,5)

(0,0) (2,0)

(0,5)(6,3)

(3,8)

y = 5x = 3

Equations (e.g., 2x + 3y = 7) are constraints that only some points (i.e., their coords) will satisfy.For example, (2, 1) and (−1, 3) both satisfy

2x + 3y = 7 , but neither (1, 1) nor (1, 2) satisfy this equation.

All pts that satisfy an equation give the solution set for the equation; we can graph an equation by plotting all pts in its solution set. This gives a picture of the solution set.

Can also do this for inequalities;e.g., 2x + 3y ≤ 7.

Picturing Inequalities

x

y

(0,0)

x ≥ 0

y ≥ 0 x ≥ 0 & y ≥ 0

Since equations are associated with a graphs (pictures of the solution set), we can use pictures to help us solve the problems we will look at.

All equations of the typeax + by = c (e.g. x+5y=10)

have graphs which are straight lines; this is the general equation of a line.

Drawing LinesEvery equation of the form ax + by = c can

be pictured as a line in the xy-plane.

First determine the x-intercept & y-intercept.

These are the points where the line crosses the x-axis & x-axis respectively. You find these by setting y=0 or x=0 respectively.

Then sketch the line joining these points.

An Example 10x+5y=200

(20,0)

(0,40) 10x+5y=200

10x+5y≤200

x intercept

y intercept

An Example 5x+2y=60

(12,0)

(0,30) 5x+2y=60

5x+2y≤60

x intercept

y intercept