Practice: linear programming
Practice: linear programming
A flower shop makes small bouquets and large bouquets.
A small bouquet has 10 roses, and a large one 20 roses.
400 roses reach the shop each day.
The florist has time to make at most 30 bouquets a day.
is the number of small bouquets a day, and the number of large ones.
Write the constraint of the roses and the constraint of the number of bouquets on the graph board.
In the flower shop of question 1:
Can the florist make 25 small bouquets and 5 large ones in one day?
And 10 small bouquets and 16 large ones?
Check each plan in both constraints.
On the graph board, the flower shop’s boundary lines: – the roses, and – the number of bouquets (the dashed one).
Mark the corner points of the feasible region, apart from the origin.
The shop makes a profit of 15 ₪ on a small bouquet and 40 ₪ on a large one.
Work out the profit at each corner point.
How many bouquets of each kind should the florist make a day? What is the profit?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
A leather workshop makes wallets and belts.
A wallet takes an hour of work, and a belt 2 hours; there are 40 hours of work a week.
A wallet needs 2 units of leather, and a belt one unit; there are 50 units of leather a week.
The profit: 60 ₪ on a wallet, and 50 ₪ on a belt.
is the number of wallets a week, and the number of belts.
Write the two constraints on the graph board.
Solve the system of equations of the two boundary lines.
Write all the corner points of the feasible region.
Write the objective function and work it out at each corner point.
What is the best plan, and what is its profit?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
In the workshop of question 5 the price of wallets has gone up: now the profit is 100 ₪ on a wallet and 30 ₪ on a belt.
The constraints have not changed.
What is the best plan now, and what is its profit?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
A farmer mixes two kinds of feed for his cows: feed A and feed B.
A kilogram of feed A gives 3 units of protein and one unit of fibre.
A kilogram of feed B gives one unit of protein and 2 units of fibre.
The daily mix needs at least 18 units of protein and at least 16 units of fibre.
A kilogram of feed A costs 8 ₪, and of feed B 5 ₪.
is the number of kilograms of feed A a day, and of feed B.
The constraints are and , and the feasible region lies above both boundary lines: and (the dashed one).
Mark the three corner points of the feasible region.
Write the objective function – the cost of the mix – and work it out at each corner point.
Which mix is the cheapest? What does it cost a day?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power
Noa said about the flower shop: “The plan of 30 small bouquets and 30 large ones gives a profit of 1,650 ₪ – more than any corner point.”
Her profit sum is right. What is her mistake?
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power
/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power