D1, C6 - Linear Programming

Question 1

With linear programming, what is another term for variables (x, y, z, …)

Answer 1

Decision variables

Question 2

With linear programming, what is another term for inequalities (5x + 2y >= 10)

Answer 2

Constraints

Question 3

With linear programming, what is the area inside all the constraints called

Answer 3

The feasible region

Question 4

With linear programming, what is another term for minimising cost / maximising profit (P = 2x - 3y, C - 7x + 3y)

Answer 4

The objective function

Question 5

What is the solution to a linear programming problem called

Answer 5

The optimal solution

Question 6

From the statements given in linear programming questions, what do you first write down

Answer 6

Constraints
Objective function

Question 7

What must you specify as an additional constraint for each of the variables

Answer 7

Non-negativity
If they can be below 0

Question 8

You will need twice as many pocket diaries as desk top diaries. How do you write this as an inequality

Answer 8

Assign variables:
pocket = y
desk top = x
y needs to be at least twice as big as x
So if 2x = y
ans –> 2x <= y

Question 9

For linear programming how would you write percentages

Answer 9

As decimals

Question 10

Syrup A contains 50% fruit and syrup B contains 35% fruit. When mixed the syrup C must contain at least 40% fruit, how do you write this as an inequality

Answer 10

0.5x + 0.35y >= 0.4(x + y)
0.5x + 0.35y >= 0.4x + 0.4y
0.1x >= 0.05y
2x >= y

Question 11

What part of the liner programming graph do you shade

Answer 11

Unwanted areas

Question 12

What letter do you denote as the feasible region on a graph

Answer 12

R

Question 13

Are less than (<) and more than (>) dotted or solid lines

Answer 13

Dotted

Question 14

Are less than or equal to (<=) and more than or equal to (>=) dotted or solid lines

Answer 14

Solid

Question 15

Where are minimising or maximising solutions found on linear programming graph

Answer 15

At once of the vertices of the feasible region polygon (R)

Question 16

What are the 2 methods for finding the minimising or maximising solutions on a linear programming graph

Answer 16

Ruler-line method
Vertex-testing method

Question 17

How would you perform the ruler line method to maximise or minimise P = 5x + 2y

Answer 17

1) Replace P with any non-zero positive number e.g. 10, 20, 30, etc
2) More your ruler perpendicular to the line in the equation created
3) First vertex is the minimal solution, last vertex is the maximum solution

Question 18

With the ruler-line method, what solution is the first vertex reached

Answer 18

Minimise point

Question 19

With the ruler-line method, what solution is the last vertex reached

Answer 19

Maximise point

Question 20

How do you perform the vertex-testing method to find the minimising or maximising point

Answer 20

1) Solve the intersection of lines simultaneously to find the coordinates of all the vertices
2) Substitute all these solutions into the objective function
3) Lowest answer is minimised points, biggest answer is maximised point

Question 21

What letter denotes maximising profit on sales

Answer 21

P

Question 22

What letter denotes minimising production costs

Answer 22

C

Question 23

What happens if the objective line is parallel to the final constraint line it will hit

Answer 23

There will be two vertices it reaches last at the same time
All points on the line the objective line is parallel to will be optimal points

Question 24

Why may we need integer solutions to linear programming problems

Answer 24

Non-integer solutions may not be practical or feasible e.g. cannot have a fraction of a person

Question 25

How would you find the integer solution if the optimal point is (16 and a third, 20 and 2 fifths)

Answer 25

1) Draw a box with each integer point around the optimal point (16,20)(17,20)(16,21)(17,21)
2) Reject any integer points that are clearly not in the feasible region
3) If it is not clear test each point against the constraints to see if it in feasible region, if not reject it
4) Test the remaining points in the objective function to find which integer point maximises (or minimises)