LINEAR

Linear programming techniques are used

to solve a wide variety of problems, such as optimising airline scheduling and establishing telephone lines.

•First we will write an objective function which we desire to either maximise (for example with profit) or minimise (for example with cost)

•Next we will come up with inequalities that model the conditions or constraints for the situation.

•We'll then graph the constraints to come up with a region that satisfies all constraints.

•Find the vertices or corners of the region because these will be the possible values of x and y that maximize or minimize the objective function.

•The x and y value that correspond to the maximum value (or minimum value) will be our solution.

•We'll substitute these values in the objective function to obtain the maximum or minimum value depending on our situation.

•First we will write an objective function. We desire to maximise the profit.

Let's let c be the number of consoles built per month and w be the number of widescreens built per month.

A television manufacturer makes console and widescreen televisions. The profit per unit is $71 for console televisions and $127 for widescreen televisions. Equipment in the factory allows for the manufacturer to make at most 430 console televisions and 362 widescreen televisions per month. Also, the cost to the manufacturer per unit is $493 for console televisions and $633 for the widescreen televisions. Total cost per month may not exceed $206,000. Determine the maximum profit and when it occurs.

Profit would be: 71c+127w

•Next we will come up with inequalities that model the conditions or constraints for the situation.

c = number of consoles w = number of widescreens

c 430

w 362

493c + 633w 206000

A television manufacturer makes console and widescreen televisions. The profit per unit is $71 for console televisions and $127 for widescreen televisions. Equipment in the factory allows for the manufacturer to make at most 430 console televisions and 362 widescreen televisions per month. Also, the cost to the manufacturer per unit is $493 for console televisions and $633 for the widescreen televisions. Total cost per month may not exceed $206,000.

Constraint due to limit on number of consoles that can be built

Constraint due to limit on number of widescreens that can be built

Constraint due to limit on costs

c ≥ 0 and w ≥ 0

•We'll then graph the constraints to come up with a region that satisfies all constraints.

c 430

w 362 493c + 633w 206000Easiest to plot this line by finding c and w intercepts

493(0) + 633w = 206000

w 325

493c + 633(0) 206000

c 418

(0, 325)

(418,0)

So this is the area where we shaded for all three.

c 430

w 362 493c + 633w 206000

x

y

c ≥ 0 and w ≥ 0

c 430

w 362 493c + 633w 206000

x

y

•Find the vertices of the region because these will be the possible values of c and w that maximise or minimise the objective function.

Vertice(0, 0)

Vertice(0, 325)

Vertice(430, 0)

These vertices were easy because they were along the axes. If a vertice is not on an axis, the vertice is the intersection of two lines so solve the two equations together to find the point of intersection and hence the corner.

•We'll substitute these values in the objective function to obtain the maximum or minimum value depending on our situation.

P = 71c+127w

This was our objective function. Our objective is to maximize profit.

Vertice(0, 0)

Vertice(0, 325)

Vertice(430, 0)

P = 71(0)+127(0) P = 71(0)+127(325) P = 71(430)+127(0)

P = 0 P = 41,275 P = 30,530

Since this is the largest value, to maximize profit would be at the point (0, 325). This means do not make any console TV's and make 325 widescreens each month.

•The c and w value that correspond to the maximum value (or minimum value) will be our solution.

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au