Shedz processes two varieties of kiwifruit – green and gold. Constraints on daily packing are
as follows:
• At most, 2 000 trays can be packed in total.
• At most, 1 200 trays of green can be packed.
• A minimum of 200 trays of gold need to be packed.
• The ratio of the number of packed trays of green to gold must be no more than 5:2.
The ratio of the profit per tray of the green variety to the gold is 12:13.
Three types of tray are available; “all cardboard”, “cardboard / wood” and “all wood”, and Shedz
wishes to determine how many of each type to purchase.
The cost per tray is $1 for “all cardboard”, $2 for “cardboard / wood” and $4 for “all wood”. Shedz
budgets $5 000 per day for purchasing trays. Shedz requires 2 000 trays per day, with between 10%
and 30% of them being of the “all cardboard” type and as many as possible (subject to the above
criteria) being of the “cardboard / wood” type.
Using these criteria, determine Shedz’s daily requirement for each type of tray.
HELP!!! i'm stuck
To determine Shedz's daily requirement for each type of tray, we can break down the problem into smaller parts and find solutions step by step.
1. Calculate the maximum number of trays of green kiwifruit that can be packed:
According to the constraints, at most, 1,200 trays of green kiwifruit can be packed.
2. Calculate the minimum number of trays of gold kiwifruit that need to be packed:
According to the constraints, a minimum of 200 trays of gold kiwifruit need to be packed.
3. Calculate the maximum number of trays that can be packed in total:
According to the constraints, at most, 2,000 trays can be packed in total.
4. Calculate the maximum number of trays of gold kiwifruit that can be packed:
Subtract the maximum number of trays of green kiwifruit from the maximum number of trays that can be packed in total.
Maximum trays of gold kiwifruit = Maximum trays in total - Maximum trays of green kiwifruit
5. Calculate the maximum profit for each variety of kiwifruit:
Multiply the number of trays by the profit per tray for each variety:
Profit from green kiwifruit = Number of green kiwifruit trays * profit per tray for green kiwifruit
Profit from gold kiwifruit = Number of gold kiwifruit trays * profit per tray for gold kiwifruit
6. Calculate the maximum profit for each type of tray:
Multiply the number of trays of each type by the cost per tray:
Profit from "all cardboard" trays = Number of "all cardboard" trays * cost per tray for "all cardboard" trays
Profit from "cardboard/wood" trays = Number of "cardboard/wood" trays * cost per tray for "cardboard/wood" trays
Profit from "all wood" trays = Number of "all wood" trays * cost per tray for "all wood" trays
7. Set up the objective function:
The objective is to maximize the profit per day, which is the sum of the profits from each type of tray.
Objective function = Profit from "all cardboard" trays + Profit from "cardboard/wood" trays + Profit from "all wood" trays
8. Set up the constraints:
- The number of trays packed in total should not exceed the maximum limit.
- The number of trays of green kiwifruit packed should not exceed the maximum limit.
- The number of trays of gold kiwifruit packed should not be less than the minimum requirement.
- The ratio of the number of packed trays of green kiwifruit to gold kiwifruit should not exceed 5:2.
- The budget for purchasing trays should not exceed $5,000.
Additionally, consider the requirement of having at least 10% of trays as "all cardboard" and maximize the number of "cardboard/wood" trays.
9. Use a linear programming solver:
Convert the objective function and constraints into a mathematical form and solve it using a linear programming solver or software.
These tools will provide the optimal solution that satisfies all the constraints and maximizes the profit.
By following these steps, Shedz can determine their daily requirement for each type of tray that fulfills all the constraints and maximizes their profit.