Linear Optimisation

A video chain has been conducting research for a new outlet. The outlet can stock old classic films only, new releases only, or a combination of these.The maximum number of old classics available is 4500; the maximum number of new releases available is 8000. There is shelf space in the outlet for no more than 9000 videos.The cost of stocking each old classic is $50, and for each new release it is $30. The maximum budget for stocking is $300,000.The profit on each old classic is $40 and on each new release it is $60.The video chain wants to find how many of each type of video it should stock to maximise its profit.

Example 1: Bursary 1991 Question 2

Step 1: Define your variables

x = Old Classics

y = New Releases

Step 2: Write the inequations for the constraints.

Step 3: Draw the graph and establish the feasible region

Step 4: Write the objective function

Step 5: List coordinates of vertices in the feasible region and substitute values into objective function to find the maximum profit.

Conclusion: Buy 1000 Old Classics and 8000 New Releases for a maximum profit of $520,000

Example 2: 1995 Bursary

Louise makes and sells two types of herbal hand-cream: rosemary and lavender. Each jar of rosemary cream takes 4 minutes to make and requires 60 grams of lanolin. Each jar of lavender cream takes 12 minutes to make and requires 100 grams of lanolin. Each day Louise can spend at most 180 minutes making hand-cream and can obtain up to 1800 grams of lanolin. Her friend Camille does the packing and can pack at most 25 jars a day.The rosemary cream sells at a profit of $4 a jar and the lavender cream sells at a profit of $5 a jar. Find the solution to the linear programming problem that maximises the total profit.

Step 1: Define your variables

Let x represent the number of jars of rosemary cream, and y represent the number of jars of lavender cream.

Step 2: Write inequations for the constraints

Step 3: Graph constraints and identify the feasible region.

Step 4; Write the objective function

Step 5: List coordinates of vertices in the feasible region and substitute values into objective function to find the maximum profit.

Conclusion: make 17.5 and 7.5 resp. of each for a maximum profit of $107.50

Problem: What if Louise can make only whole numbers of jars each day

Solution: Analyse all integer values around the solution to find the maximum. Remember to eliminate those that do not meet the constraints.

Conclusion: Make 18 jars of Rosemary and 7 of Lavendar for maximum profit of $107

Example 3: Bursary 1994

The school principal has arranged for a day out and needs transport for the journey. Provision must be made to transport at least 400 students, 36 teachers and 120 boxes of supplies. The local rental company can provide vans at $50 each and buses at $160 each. Each van can take 10 students, 1 teacher and 4 boxes; each bus can take 40 students, 3 teachers and a certain number of boxes.

Step 1: Define the variables

-vans (x) and

- buses (y)

Step 2: Write the inequations for the constraints.

Notes:

We do not need the non-negatives as everything is greater than.

We do not know how many boxes a bus can take- information will come from the graph

Graph is given.

Step 3: Identify the feasible region (Region 1) and

check the intercepts of “boxes” line (y-intercept is 15 and hence bus can take 120/15 = 8 boxes.

Step 4: Write the objective function (Cost to be minimised in this case)

Step 5: Identify the vertices in the feasible region and evaluate.

Conclusion: Minimise cost by using 24 vans and 4 buses

Problem: The day before the trip the rental company informs the principal that while the cost of a van was correctly given as $50, there was a mistake in giving the cost of a bus.What range of costs for bus hire will still give the same optimum solution found.

Solution: For the same solution the gradient of the objective function must lie between the gradients of the two adjacent lines.