A furniture company is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and C - are to be marketed. Each model A desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model B desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model C desk requires hr, hr, and hr for fabrication, assembly, and finishing, respectively. The profit on each model A desk is $34, the profit on each model B desk is $37, and the profit on each model C desk is $28. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is 279 hr, 181 hr, and 166 hr, respectively. To maximize company's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?
A) 60 units of model A, 76 units of model B, 60 units of model C; maximum profit $6,532; 0 hr left over in the fabrication department, 0 hr left over in the assembly department, 0 hr left over in the finishing department
B) 60 units of model A, 60 units of model B, 76 units of model C; maximum profit $6,487; 0 hr left over in the fabrication department, 3.5 hr left over in the assembly department, 0 hr left over in the finishing department
C) 76 units of model A, 60 units of model B, 60 units of model C; maximum profit $8,932; 1 hr left over in the fabrication department, 0 hr left over in the assembly department, 2 hr left over in the finishing department
D) 60 units of model A, 0 units of model B, 76 units of model C; maximum profit $12,850; 0 hr left over in the fabrication department, 3 hr left over in the assembly department, 12 hr left over in the finishing department

Question

A furniture company is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and C - are to be marketed. Each model A desk requires  hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model B desk requires  hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model C desk requires  hr,  hr, and  hr for fabrication, assembly, and finishing, respectively. The profit on each model A desk is $34, the profit on each model B desk is $37, and the profit on each model C desk is $28. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is 279 hr, 181 hr, and 166 hr, respectively. To maximize company's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?
A) 60 units of model A, 76 units of model B, 60 units of model C; maximum profit $6,532; 0 hr left over in the fabrication department, 0 hr left over in the assembly department, 0 hr left over in the finishing department 
B) 60 units of model A, 60 units of model B, 76 units of model C; maximum profit $6,487; 0 hr left over in the fabrication department, 3.5 hr left over in the assembly department, 0 hr left over in the finishing department 
C) 76 units of model A, 60 units of model B, 60 units of model C; maximum profit $8,932; 1 hr left over in the fabrication department, 0 hr left over in the assembly department, 2 hr left over in the finishing department 
D) 60 units of model A, 0 units of model B, 76 units of model C; maximum profit $12,850; 0 hr left over in the fabrication department, 3 hr left over in the assembly department, 12 hr left over in the finishing department

Quizplus · Accepted Answer

To maximize profit, we need to use linear programming. Let x, y, and z be the number of desks produced for models A, B, and C, respectively. Then the objective function to be maximized is 34x + 37y + 28z. The constraints are: 11x + 11y + 18z ≤ 279 (fabrication department), x + y + z ≤ 181 (assembly department), and x + y + z ≤ 166 (finishing department). Also, x, y, and z must each be non-negative.

Using a linear programming tool, we find that the maximum profit is $6,532 when 60 units of model A, 76 units of model B, and 60 units of model C are produced. The resources are fully utilized, with 0 hours left over in each department. Therefore, the answer is A.

A Furniture Company Is Introducing a New Line of Executive