Western Pulp (WP) produces recycled paperboard for box manufacturers by combining four grades of recycled paper stock. Each grade of stock has a different strength, color, and texture. The strength, color, and texture of the paperboard are approximately a weighted average of those characteristics of the paper inputs. The table gives the characteristics of the paper stocks and their cost per ton. WP has
received an order for 500 tons of paperboard with a strength rating of at least 7, a color of at least 5, and texture of at least 6. WP would like to determine the least costly mix required to produce this paperboard.
Paper Grade Strength Color Texture Cost/Ton
1 8 9 8 $150
2 6 7 5 $110
3 5 5 6 $90
4 3 4 5 $50
a)Formulate the problem as a linear program.
b) Solve this problem on a computer and explain the optimal solution.
mọi ngưởi chỉ em giải bài này với ạ
To formulate the problem as a linear program, we need to define the decision variables, objective function, and constraints.
Decision variables: Let x1, x2, x3, x4 represent the number of tons of each grade of paper stock to be used.
Objective function: We want to minimize the cost of producing the paperboard, so the objective function is:
Minimize Z = 150x1 + 110x2 + 90x3 + 50x4
Constraints:
1) Strength constraint: The paperboard should have a strength rating of at least 7:
8x1 + 6x2 + 5x3 + 3x4 >= 7
2) Color constraint: The paperboard should have a color rating of at least 5:
9x1 + 7x2 + 5x3 + 4x4 >= 5
3) Texture constraint: The paperboard should have a texture rating of at least 6:
8x1 + 5x2 + 6x3 + 5x4 >= 6
4) Non-negativity constraint: The number of tons of each paper grade cannot be negative:
x1, x2, x3, x4 >= 0
Now, to solve this problem on a computer, you can use a linear programming solver such as Excel Solver, MATLAB's linprog function, or PuLP library in Python. These solvers will find the optimal solution that minimizes the cost while satisfying the given constraints.
The optimal solution will provide the values of x1, x2, x3, and x4 that minimize the cost. In this case, it will tell you how many tons of each paper stock (grade) you should use to produce 500 tons of paperboard with the desired strength, color, and texture.
Note: Since you mentioned solving it on a computer, you should input the problem into the chosen solver with the objective function, constraints, and variable bounds, and let the solver do the calculations for you.