Bakery production plan with a labour offer
A cooperative bakery makes country loaves (x per day) and rye loaves (y per day). Each country loaf uses 0.5 oven-hour and 0.25 labour-hour; each rye loaf uses 0.75 oven-hour and 0.25 labour-hour. The bakery has 12 oven-hours and 5 labour-hours available each day. Demand limits are at most 30 country loaves and at most 12 rye loaves. Profit is EUR 3 per country loaf and EUR 4 per rye loaf. A neighbouring bakery offers one additional labour-hour for EUR 3.50. Separately, a wholesaler offers a higher profit of EUR 5 per rye loaf on all rye loaves, with no change for country loaves. The owner wants to know the current best plan, whether the extra labour-hour is worth buying, and whether the wholesaler's offer changes the production plan and by how much daily profit rises.
1. Formulate the linear programming model for the current situation and solve it. State the optimal numbers of country and rye loaves and the maximum daily profit.
Read the worked answer
Let x be country loaves and y rye loaves. The constraints are
0.5x+0.75y≤12 (oven hours), 0.25x+0.25y≤5 (labour hours), 0≤x≤30, 0≤y≤12.
The objective is P=3x+4y. The labour constraint simplifies to x+y≤20. The feasible region has vertices (0,0), (20,0), (12,8), (6,12) and (0,12). The point (12,8) is the intersection of x+y=20 and 0.5x+0.75y=12; the point (6,12) comes from y=12 and the oven constraint.
Evaluate the profit: (0,0):0 (20,0):60 (12,8):3(12)+4(8)=36+32=68 (6,12):18+48=66 (0,12):48
The maximum is EUR 68 per day at x=12 country loaves and y=8 rye loaves. At this point the oven is exactly full: 0.5(12)+0.75(8)=6+6=12, and labour is also exactly full: 0.25(12)+0.25(8)=5.
2. The bakery can buy one extra labour-hour for EUR 3.50. Determine the value of that extra hour and decide whether to buy it.
Read the worked answer
With one extra labour-hour, the labour constraint becomes x+y≤24 instead of x+y≤20. The oven constraint and demand limits are unchanged. A new vertex is formed by the intersection of x+y=24 and the oven constraint:
x=24-y, 0.5(24-y)+0.75y=12, 12-0.5y+0.75y=12, 0.25y=0, so y=0 and x=24.
This point is feasible because 24≤30 and the oven use is 0.5(24)=12. Its profit is 3(24)+4(0)=72. The previous maximum was 68, so the extra labour-hour raises profit by EUR 4. Since it costs EUR 3.50, the net gain is EUR 0.50. The bakery should buy the hour, provided the offer really applies to exactly one hour and no other constraints change. The value is positive because the extra labour relaxes the binding labour constraint while the oven still has enough capacity.
3. Now suppose the wholesaler pays EUR 5 profit per rye loaf instead of EUR 4, with country loaves unchanged. Re-solve the model and compare the new optimal plan and profit with the current plan.
Read the worked answer
The new objective is P=3x+5y with the original constraints. Re-evaluate the vertices of the same feasible region:
(0,0):0 (20,0):60 (12,8):3(12)+5(8)=36+40=76 (6,12):3(6)+5(12)=18+60=78 (0,12):60
The new maximum is EUR 78 at x=6, y=12. This uses the full rye demand and the full oven: 0.5(6)+0.75(12)=3+9=12; labour use is 0.25(6)+0.25(12)=1.5+3=4.5≤5. The plan therefore changes from 12 country and 8 rye to 6 country and 12 rye. Daily profit rises from EUR 68 to EUR 78, an increase of EUR 10. The extra labour-hour offer should be reconsidered after this change: with the new objective, the original labour constraint is no longer binding at the optimum, so buying an extra hour would not necessarily increase profit.