13.03 Application of linear programming
The following graph shows a region of feasible solutions for the objective function z = 4 x + 3 y:
Calculate the value of the function for each of the following corner points:
Calculate the maximum value of the objective function and state at what point it is obtained.
Calculate the minimum value of the objective function and state at what point it is obtained.
For each of the following graphs of feasible regions and corresponding objective functions:
Calculate the maximum value of the objective function and state at what point it is obtained.
Calculate the minimum value of the objective function and state at what point it is obtained.
The objective function is z = 5 x + 2 y.
The objective function is z = 3 x + 6 y.
The objective function is z = 3 x + 3 y.
The objective function is 10 y.
The objective function is z = 5 x - 2 y.
The following graph represents the feasible region for the objective function \\ T = 14 x + 31 yGiven that the coordinates of the corner vertices are \left(0, 9 \right),\left(\dfrac{36}{5}, \dfrac{21}{5}\right) \text{ and }\left(10, 0\right), determine the maximum value of the objective function T and state at what point it is obtained.
For each of the following systems of inequalities:
Plot the feasible region defined by the four inequalities.
Write the coordinates of the vertices of the region defined by the four inequalities.
For each of the following objective functions and their constraints:
Graph the region defined by the constraints in the first quadrant.
Calculate the maximum value of the function P and state at what point it is obtained.
Calculate the minimum value of the function P and state at what point it is obtained.
Objective function P = - 28 x + 10 y + 56, subject to the constraints 7 x + 5 y \leq 35, \\ 0 \leq x \leq 2 and 0 \leq y \leq 6.
Objective function P = 12 x + 46 y, subject to the constraints 4 x + 3 y \leq 24, \, 3 x + 4 y \leq 28, x \geq 0 and y \geq 0.
For each of the following objective functions and their constraints:
Graph the region defined by the constraints in the first quadrant.
Determine the maximum value of the objective function and the values of x and y for which the maximum occurs.
Check your answer using the sliding line method.
Objective function z = 6 x + 2 y, subject to the constraints 6 x + 2 y \leq 60, 2 x + 6 y \leq 54, x \geq 0 and y \geq 0.
Objective function z = 22 x + 7 y, subject to the constraints x + 2 y \geq 8, x + y \geq 6, \\ x + y \leq 11, x \geq 0, and y \geq 0.
Objective function z = 9 x + 14 y, subject to the constraints 2 x + y \geq 8, x + 2 y \geq 8, \\ x + y \leq 8, x \geq 0 and y \geq 0.
A manufacturer produces two types of tables. Each table requires a cabinet maker and a painter to build. The time taken for each worker varies according to the table below:
| Cabinet Maker | Painter | |
|---|---|---|
| Round Table | 2 | 2 |
| Square Table | 3 | 2 |
| Total time available in a week | 36 | 32 |
Let x represent the number of round tables built in a week, and let y represent the number of square tables built in a week.
Construct the set of four constraint inequalities.
Graph the region defined by the contraints in the first quadrant.
Hence, list the four vertices of the feasible region.
If a round table sells for a profit of \$240 and a square table makes a profit of \$280, write the objective function that models the weekly profit P.
Using the sliding line or corner point method, determine the vertex that maximises the manufacturer's profits.
Calculate the maximum weekly profit.
A shoe manufacturer produces shoes for both men and women. Each pair of shoes is made of rubber for the soles and leather for the upper. The amount of material required, in decimetres cubed, to make each type is shown in the table below:
| \text{Rubber (dm}^3) | \text{Leather (dm}^3) | |
|---|---|---|
| \text{Women's Shoe} | 8 | 8 |
| \text{Men's Shoe} | 16 | 12 |
| \text{Maximum amount of} \\\ \text{available materials in a week} | 3520 | 3360 |
Let x represent the number of pairs of women's shoes made in a week, and let y represent the number of pairs of men's shoes made in a week.
Construct the set of four constraint inequalities.
Graph the region defined by the constraints in the first quadrant.
List the four vertices of the feasible region.
If one pair of women's shoes sells for a profit of \$15 and one pair of men's shoes makes a profit of \$26, write the objective function that models the weekly profit P.
Using the sliding line or corner point method, determine the vertex that maximises the manufacturer's profit.
Hence state the maximum weekly profit.
The popular shop Bergner's Burgers makes and sells two types of burgers; beef and chicken. Each day, Mr Bergner knows that they need to make at least 190 beef burgers and 170 chicken burgers. The maximum number of burgers that can be made in one day is 460.
Let x represent the number of beef burgers made per day, and let y represent the number of chicken burgers made in a day.
Construct the set of three constraint inequalities.
Graph the region defined by the inequalities.
If Bergner's Burgers makes a profit of \$1.70 from each beef burger, and a profit of \$1.00 from each chicken burger, write the objective function that models the weekly profit P. Assume that every burger made is sold.
Evaluate the profit corresponding to each vertex of the feasible region.
State the maximum daily profit.
State the number of beef burgers and number of chicken burgers that should be made each day to achieve the maximum profit.
If the prices of the burgers were switched, such that the profit from beef burgers was \$1.00 and the profit from chicken burgers was \$1.70, how many of each type of burger should now be made for a maximum profit?
A bakery produces two types of cake, each using the three main ingredients flour, sugar and butter in different proportions. The bakery has a certain amount of each type of ingredient delivered each week. The requirements for each cake and the amounts that can be stored are shown in the table below:
| Flour (kg) | Sugar (kg) | Butter (kg) | |
|---|---|---|---|
| Cake A | 1 | 1 | 1.2 |
| Cake B | 2 | 1 | 0.8 |
| Amount stored | 30 | 22 | 24 |
Let x represent the number of type A cakes baked in a week, and let y represent the number of type B cakes baked in a week.
Construct the set of five constraint inequalities.
Graph the region defined by the inequalities in the first quadrant.
Assume that every cake that is baked is also sold. If each type A cake sells for a profit of \$13 and each type B cake makes a profit of \$11, write the objective function that models the weekly profit P.
Evaluate the profit corresponding to each vertex of the feasible region.
State the maximum weekly profit for the bakery.
State the number of each type of cake that should be made to achieve the maximum profit.
A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \$75 for the rear-projection televisions and \$150 for the plasma televisions.
Let x be the number of rear-projection televisions sold in a month and y be the number of plasma televisions sold in a month.
Write the objective function that models the total monthly profit P.
The manufacturer is bound by the following constraints:
Equipment in the factory allows for making a maximum of 150 rear-projection televisions and 300 plasma televisions in one month.
The cost to manufacture per unit is \$400 for the rear-projection televisions, and \$300 for the plasma televisions.
Total monthly costs cannot exceed \$120\,000.
Express these three constraints as inequalities in terms of x and y .
Graph the region defined by the inequalities in the first quadrant.
Evaluate the objective function for total monthly profit at each of the corners of the feasible region. Round your answer to the nearest dollar.
Hence determine the maximum monthly profit.
State the number of each type of television that should be manufactured for maximum profit.
Dave is trying to save money to buy a car. He spends as much of his spare time as possible earning money from two activities: his parents pay him \$5 per hour to complete his homework, and he also works part-time at a cafe where he earns \$15 per hour.
Let x represent the number of hours spent doing homework, and y represent the number of hours spent working at the cafe.
Write down the objective function that models Dave's earnings, E.
Dave has 30 hours spare each week, and he must spend a minimum of 10 hours doing homework each week. He is also allowed to work at the cafe for no more than 5 hours more than he spends doing homework.
From this information, construct a set of four constraint inequalities.
Graph the region defined by the inequalities.
List the four vertices of the feasible region.
State the maximum amount that Dave can earn in a week.
State the number of hours Dave should spend doing homework and working at the cafe each week to earn the maximum possible income.
A camp organiser needs to transport 60 campers and 60 bags of luggage to a camp. The two types of transport available are buses and vans. Buses can carry 30 campers and 20 bags, while vans can carry 15 campers and 20 bags.
Let x represent the number of buses and y represent the number of vans to be used.
Construct the set of four constraint inequalities.
Graph the region defined by inequalities in the first quadrant.
If a bus costs \$32 to make the trip, while a van costs \$24, write the objective function that models the cost C.
Evaluate the cost corresponding to each vertex of the feasible region.
State the minimum cost for the transport.
State the number of buses and vans that will give the cheapest possible option.
Oliver is opening a new cross-state shipping company. He has the option to buy semi-trailer trucks or box trucks. To help in his decision, Oliver analysed the following data:
- The annual maintenance cost should not exceed \$562\,500.
- The total payload had to be at least 300 tonnes.
- Only 30 semi-trailer trucks are available.
| Semi-trailer truck | Box truck | |
|---|---|---|
| Annual maintainance cost | \$12\,500 | \$11\,250 |
| Payload | 20 \text{ tons} | 3 \text{ tons} |
The average profit made by a semi-trailer truck is \$12\,000 per year and the average profit made by a box truck is \$13\,000 per year.
Find how many of each type of truck Oliver has to buy in order to maximise his annual profit and state the amount of the annual profit.
Food and clothing are shipped to survivors of a hurricane. Each box of food will feed 4 people, while each box of clothing will help 6 people. Each 1.0 \text{ m}^3 box of food weighs 5 \text{ kg} and each 0.6 \text{ m}^3 box of clothes weighs 8 \text{ kg}. The planes transporting food and clothing are bound by the following constraints:
The total weight per plane cannot exceed 2000 \text{ kg}.
The total volume must be less than 300 \text{ m}^3.
Determine the number of boxes of food and clothing that should be sent with each plane shipment to maximise the number of people that are helped.