Question

12. Using the information provided below, construct a linear model to maximize the profits for Puck and Pawn Company that produces hockey sticks and chess sets.

Product	Labor (Hr./Unit)	Wood (Lb./Unit)	Profit ($/Unit)
Hockey Sticks (x1)	1	4	40
Chess Sets (x2)	2	3	50

Resource Availability:   40 hrs of labor per day; 120 lbs of wood

Decision Variables:       x₁ = number of Hockey Sticks to produce; x₂ = number of Chess Sets to produce

Identify the Objective Function:    Z = ___40____x₁ + __50_____x₂

Identify the Constraint Equation for Labor:    ___1____x₁ + __2_____x₂ <= ____40___

Identify the Constraint Equation for Wood:    _____4__x₁ + ____3___x₂ <= 120_______

Identify any other Constraint Equation(s):                           

Point A: X₁ = ___30______ X₂ = ______0___ Z = ____1200_____

Point B: X₁ = _____24____ X₂ = ____8_____ Z = _____1360____

Point C: X₁ = ______0___ X₂ = ___20______ Z = _____1000____

A.) Optimal point is (Circle one)   Point A       Point B       Point C

B.) Is there any slack associated with labor? Yes No, If yes, how many hours? _________

C.) Is there any slack associated with wood? Yes No, If yes, how many pounds? _________

Answer 1

Objective Function: Z = 40x1 + 50x2

Constraints:

x1 + 2x2 <= 40 [Constraint on Wood]

4x1 + 3x2 <= 120 [Constraint on labor]

x1 >= 0 [number of hockey sticks must be greater than zero]

x2 >= 0 [number of chess sets must be greater than zero]

We need to maximize objective function Z with the above constraints

a) Optimal Point will be the point B producing maximum profit of $1360

b) Number of labor hour used = 24 * 1 + 2 * 8 = 40

Available hours = 40

Hence there is no labor hours left, so the slack is (40-40) which is zero

No slack

c) Number of wood used = 24 * 4 + 3 * 8 = 120

Available hours = 120

Hence there is no wood hours left, so the slack is (120-120) which is zero

No slack