Bakery Production Plan Under Three Resource Limits
A Nagpur bakery makes two cake varieties: chocolate (x trays) and vanilla (y trays) per day. Flour limits production by 2x+y≤80 kg-equivalent units; eggs by x+y≤50; labour by x+2y≤80. All variables are in trays and x,y≥0. Profit is ₹40 per chocolate tray and ₹30 per vanilla tray. The owner wants the daily production plan that maximises profit. The supervisor has proposed making only chocolate because it earns more per tray, but the owner suspects the resource mix matters. The baker also wants to know how much labour will remain unused at the optimal plan, and whether the plan would change if the profit on chocolate rose to ₹60 per tray. The bakery cannot store more than 40 trays of either type, but the existing constraints already enforce that limit.
1. List the constraints and determine all feasible corner points of the region.
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Constraints: 2x+y≤80, x+y≤50, x+2y≤80, x≥0, y≥0. Axes: y=0 gives x≤min(40,50,80)=40, so (40,0). x=0 gives y≤min(80,50,40)=40, so (0,40). Origin is (0,0). Intersections: 2x+y=80 with x+y=50 gives (30,20); x+y=50 with x+2y=80 gives (20,30); 2x+y=80 with x+2y=80 gives (80/3,80/3), but that point has x+y≈53.3>50, so it is infeasible. Thus the feasible corners are (0,0), (40,0), (30,20), (20,30), (0,40).
2. Find the profit-maximising production plan and the unused labour at that plan.
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Evaluate P=40x+30y at the corners: (0,0):0; (40,0):1600; (30,20):1200+600=1800; (20,30):800+900=1700; (0,40):1200. Maximum is ₹1800 at x=30, y=20. Labour used = x+2y=30+40=70, so unused labour =80-70=10 units. Flour and eggs are fully used: flour 60+20=80, eggs 30+20=50. The supervisor's all-chocolate plan gives only ₹1600, so the mixed plan is better.
3. If the profit on chocolate rises to ₹60 per tray, does the optimal plan change? Explain.
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New P=60x+30y. Re-evaluate: (40,0):2400; (30,20):1800+600=2400; (20,30):1200+900=2100; (0,40):1200. The maximum value is ₹2400, attained at both (40,0) and (30,20). Thus at ₹60 there are multiple optimal plans; the bakery could choose all chocolate or the mixed plan. If chocolate profit rises strictly above ₹60, the all-chocolate corner (40,0) becomes uniquely optimal because its value rises faster with x. At exactly ₹60, the owner is indifferent between the two plans on profit grounds, so non-financial factors could decide.