1, a small firm builds two types of garden shed. Type A requires 2 hours of a machine time and 5 hours of crafts man time.type requires 3 hours of machine time and 5 hours of crafts man time. Each day there are 30 hours of machine time available and 60 hours of crafts man time.the profit on each type A shed is 60 and on each type B shed is 84
A,formulate the appropriate linear programming problem.
B, use graphic method to determined how many garden shed of each type build in order to maximize the total profit.
C,use simplex tableau method to determined how many garden shed of each type to build in order to maximize the total profit.
D, what is the amount of the scarce resource that is unused(slack)by the firm?
E, what do you recommend the firm?
A) The appropriate linear programming problem can be formulated as follows:
Let X be the number of Type A garden sheds to be built.
Let Y be the number of Type B garden sheds to be built.
Objective function: Maximize profit = 60X + 84Y
Subject to the following constraints:
2X + 3Y ≤ 30 (machine time constraint)
5X + 5Y ≤ 60 (craftsman time constraint)
X, Y ≥ 0 (non-negativity constraint)
B) Using the graphic method, we can plot these constraints on a graph and find the feasible region. The feasible region represents all the combinations of X and Y that satisfy the constraints.
C) To determine how many garden sheds of each type to build in order to maximize the total profit using the simplex tableau method, we would need the coefficients of the objective function, the constraint inequalities, and the right-hand side values of the constraints.
D) The amount of the scarce resource that is unused (slack) by the firm can be determined by finding the difference between the available resources and the resources used by the optimal solution. In this case, it would be the difference between the available machine time (30 hours) and the machine time used by the optimal solution, as well as the difference between the available craftsman time (60 hours) and the craftsman time used by the optimal solution.
E) The recommendation to the firm would depend on the optimal solution obtained from the linear programming problem. This solution would indicate the number of Type A and Type B garden sheds to build in order to maximize the total profit, considering the constraints and available resources.
A. The appropriate linear programming problem can be formulated as:
Maximize:
Profit = 60A + 84B
Subject to:
Machine Time constraint: 2A + 3B ≤ 30
Craftsman Time constraint: 5A + 5B ≤ 60
Non-negativity constraint: A ≥ 0, B ≥ 0
B. To determine how many garden sheds of each type should be built to maximize the total profit, we can use the graphical method.
First, plot the feasible region by graphing the constraints on a coordinate plane. Then, find the corner point(s) within the feasible region that yield the maximum profit.
C. To determine how many garden sheds of each type should be built to maximize the total profit using the simplex tableau method, we need to set up the simplex tableau and perform the simplex algorithm. However, in this case, since both the objective function and constraints are linear and the problem has only two variables, we can directly solve for the optimal solution using the graphical method as described in part B.
D. The amount of the scarce resource that is unused (slack) by the firm can be determined by calculating the difference between the available resource and the used resource in each constraint.
For the machine time constraint: 30 - (2A + 3B)
For the craftsman time constraint: 60 - (5A + 5B)
E. Based on the solution obtained from part B, the firm should produce the recommended number of garden sheds of each type to maximize their total profit.