Bakery Fundraiser: Linear Programming and the Rounding Trap
A youth group in a small town runs a bakery to raise money for a school library. They bake two kinds of loaf: large (x) and small (y). Each large loaf uses 0.5 kg of flour and 20 minutes of oven time and earns a profit of 6 birr. Each small loaf uses 0.3 kg of flour and 10 minutes of oven time and earns a profit of 3.5 birr. Each baking day the group has 30 kg of flour and 1200 minutes of oven time available. A regular customer has agreed to buy at least 20 small loaves every day, and because of tray size the group never bakes more than 40 large loaves in a day. Nothing can be stored overnight. One member argues: 'Large loaves earn almost double, so we should cut the small ones and bake only large loaves.' Another member says the group should simply solve the problem on paper and round the answer to whole loaves.
1. Formulate the constraints and determine the daily plan that maximises profit. Show the corner points you check.
Read the worked answer
Let x = large loaves, y = small loaves, in birr.
- Flour: 0.5x + 0.3y ≤ 30
- Oven: 20x + 10y ≤ 1200, i.e. 2x + y ≤ 120
- Contract: y ≥ 20; tray limit: 0 ≤ x ≤ 40
- Maximise P = 6x + 3.5y
Corner points of the feasible region:
- (0, 20): P = 70
- (40, 20): P = 240 + 70 = 310; flour 20 + 6 = 26 ≤ 30, oven 100 ≤ 120 ✓
- (0, 100): flour 30 ✓, oven 100 ✓, P = 350
- Where x = 40 meets the flour line: 0.5(40) + 0.3y = 30 gives y = 33⅓, P = 240 + 116.67 ≈ 356.67; oven 80 + 33⅓ ≈ 113 ≤ 120 ✓
The intersection of the flour and oven lines gives x = 60, y = 0, which violates x ≤ 40, so it is outside the region.
The maximum is at (40, 33⅓): 40 large and 33⅓ small loaves, profit approximately 356.67 birr. Both the tray limit and the flour supply are fully used; oven time is not, with about 7 minutes to spare.
2. The group can only bake whole loaves. Is rounding the paper answer down to 40 large and 33 small the best whole-loaf plan? Justify with numbers.
Read the worked answer
No. Rounding down gives P = 6(40) + 3.5(33) = 240 + 115.5 = 355.5 birr, but a slightly different plan beats it.
Try 39 large: the flour used is 19.5 kg, leaving 10.5 kg, so y ≤ 10.5/0.3 = 35. Oven time is 2(39) + 35 = 113 ≤ 120 ✓. Profit = 6(39) + 3.5(35) = 234 + 122.5 = 356.5 birr.
Why does this work? At the paper optimum, flour is the binding resource, and a large loaf earns 6/0.5 = 12 birr per kg of flour against 3.5/0.3 ≈ 11.67 for a small loaf. Giving up one large loaf frees 0.5 kg of flour, enough for one extra small loaf plus 0.2 kg of slack, and it also frees 20 minutes of oven time, which was not fully used. The trade is therefore +3.5 − 6 = −2.5 for the loaf given up, recovered by the extra loaf the freed flour allows; across the boundary the net gain is 1 birr.
So the correct whole-number answer is 39 large and 35 small, 356.5 birr. Rounding a linear-programming answer to the nearest whole number is not automatically optimal; whole-number problems must be checked at neighbouring integer points.
3. Evaluate the member's claim that the group should bake only large loaves, using the idea of profit per unit of each scarce resource.
Read the worked answer
The claim confuses profit per loaf with profit per unit of a limiting resource.
- Per loaf: large 6 birr, small 3.5 birr - the member's figure.
- Per minute of oven time: large 6/20 = 0.30; small 3.5/10 = 0.35. Small loaves are better.
- Per kg of flour: large 6/0.5 = 12; small 3.5/0.3 ≈ 11.67. Large loaves are marginally better.
Which ratio matters depends on which resource actually binds. At the optimum, flour and the 40-loaf tray limit bind, while oven time has slack, so the flour ratio is the relevant one - which is why the best plan leans towards large loaves, but only slightly.
Baking only large loaves is also infeasible on two counts: the contract requires at least 20 small loaves, and 40 large loaves would use only 20 kg of the 30 kg of flour, leaving capacity and profit unused. The all-large corner (0, 100) in oven terms yields 350 birr, which is less than 356.5.
The sound rule is: rank products by profit per unit of the binding constraint, then check the non-binding ones and any contractual minimums.