Question

The management of Hatman Toy Company is trying to determine the best production and overtime schedule for the coming month to attain maximum profit. The company makes two toy items. Each of the two toys needs to go through three departments (A: Casting, B: Painting, C: Costume) to be completed. The following table shows the amount of time (in minutes) each department needs to process each toy item: Product (minutes/unit) Department Toy 1 Toy 2 A 60 21 B 18 12 C 12 30 Toy item 1 is sold at $30 profit/unit and toy item 2 is sold at $15 profit/unit. Departments A, B, and C have a total of 100, 36, and 50 hours of (regular) labor time available each month for production. However, the company can schedule some overtime in each department at an additional cost. Each hour of overtime costs $18 in department A, $22.50 in department B, and $12 in department C. Up to a maximum of 10, 6, and 8 hours of overtime can be scheduled in departments A, B, and C, respectively. a) Write a mathematical formulation of the problem, which simultaneously determines how many units of each toy item to make, and how many hours of overtime should be used in each department (Hint: So we are trying to figure out 5 numbers in total). Clearly define your decision variables (full sentence!), and then write the objective function and constraints algebraically?

Answer 1

Solution

Let

x1 = number of units to be produced of Toy item 1

x2 = number of units to be produced of Toy item 2

x3 = number of hours of overtime to be used in department A

x4 = number of hours of overtime to be used in department B

x5 = number of hours of overtime to be used in department C

Toy item 1 is sold at $30 profit/unit and toy item 2 is sold at $15 profit/unit. =>

Total profit generated is: 30x1 + 15x2 .......................................................................................... (1)

Each hour of overtime costs $18 in department A, $22.50 in department B, and $12 in department C. =>

Total cost of overtime = 18x3 + 22.5x4 + 12x4 ..............................................................................(2)

(1) - (2) gives: Net profit, z = 30x1 + 15x2 - 18x3 - 22.5x4 - 12x4 ............................................... (3)

Total number of hours required from department A is: 60x1 + 21x2 ..............................................(4)

Total number of hours available in department A is: 100 (regular) + x3 (overtime) ........................(5)

(4) and (5) gives the time constraint of department A as: 60x1 + 21x2 ≤ 100 + x3 or

60x1 + 21x2 – x3 ≤ 100 ................................................................................................................. (6)

Similarly,

time constraint of department B is: 18x1 + 12x2 – x4 ≤ 36............................................................ (7)

time constraint of department C is: 12x1 + 30x2 – x5 ≤ 50............................................................ (8)

Up to a maximum of 10, 6, and 8 hours of overtime can be scheduled in departments A, B, and C, respectively. =>

x3 ≤ 10 ........................................................................................................................................... (9)

x4 ≤ 6 ........................................................................................................................................... (10)

x5 ≤ 8 ........................................................................................................................................... (11)

Thus, vide (3), (6), (7), (8), (9), (10), (11) in that order, the mathematical formulation of the problem is:

Maximize z = 30x1 + 15x2 - 18x3 - 22.5x4 - 12x4

Subject to

60x1 + 21x2 – x3 ≤ 100

18x1 + 12x2 – x4 ≤ 36

12x1 + 30x2 – x5 ≤ 50

x3 ≤ 10

x4 ≤ 6

x5 ≤ 8

x1, x2, x3, x4, x5 ≥ 0. Answer

DONE