Question

A manufacturer makes two types of tractors, the X-Harvester and the Y-Combine. The time for assembling is 3 hours for the X and 4 hours for the Y. The total assembly time allotted is 8,400 hours. Finishing the machines takes 3 hours for the X, 2.5 hours for Y. Total finishing time allotted is 5,700 hours. Packing takes 0.8 hour for X, 0.4 hour for Y. Total packing time is 1,340 hours. Profit (P) is $30,000 for each X-Harvester and $37,500 for each Y-Combine. b) List all 5 constraints c) List the corners of the constraint graph. (A manufacturer can produce a fraction of a machine in this period, to be finished and sold in the next period.) d) Evaluate the objective function at the corners. i) What is the maximum profit? ii) How many of each type of tractor should this manufacturer produce? X-Harvesters Y-Combines

Answer 1

The linear programming problem involves finding the maximum profit for a manufacturer by evaluating the objective function at the corners of the constraint graph, which are determined by the time constraints for assembling, finishing, and packing two types of tractors.

Step-by-step explanation:

Given a linear programming problem for a manufacturer producing two types of tractors, we can identify the constraints, the corners of the constraint graph, and the maximum profit by using the given information about the assembly, finishing, and packing times, as well as the profits for each type of tractor:

• Constraint for assembly time: 3X + 4Y ≤ 8,400 hours
• Constraint for finishing time: 3X + 2.5Y ≤ 5,700 hours
• Constraint for packing time: 0.8X + 0.4Y ≤ 1,340 hours

The constraints represent limitations on the number of X-Harvester (X) and Y-Combine (Y) tractors the manufacturer can produce based on time constraints for assembly, finishing, and packing.

The corners of the constraint graph

To find the corners, we would graph the constraints on an XY plane and identify where the lines intersect. The corners are where the feasible region bounded by these constraints meets or where the constraints intersect. Each corner (X,Y) represents a possible combination of tractors that maximize profit without violating any constraints.

Evaluating the objective function at the corners

The objective function is the profit, P, which is P = 30,000X + 37,500Y. Evaluating P at each corner of the constraint graph will give us different profit values. To find the maximum profit, we choose the corner that yields the highest P value. The number of X-Harvesters and Y-Combines produced at that point will be the manufacturer's optimal production levels.