Here is a word problem dealing with The Simplex Method: Maximization, Minimization
A dietitian is designing a daily diet that is to contain at least 60 units of protein, 40 units of carbs, and 120 units of fat. The diet is to consist of two types of foods. One serving of food A contains 30 units of protein, 10 units of carbs, and 20 units of fat and costs $3. On serving of food B contains 10 units of protein, 10 units of carbs, and 60 units of fat and costs $1.50. Design the diet that provides the daily requirements at the least cost.
To solve this problem using the Simplex Method, we need to set up a linear programming model.
Step 1: Define the decision variables.
Let's define:
x = number of servings of food A in the diet
y = number of servings of food B in the diet
Step 2: Write the objective function.
The objective is to minimize the cost, so the objective function is:
Cost = 3x + 1.5y
Step 3: Write the constraints.
Based on the given requirements, we have the following constraints:
Protein constraint: 30x + 10y ≥ 60
Carbs constraint: 10x + 10y ≥ 40
Fat constraint: 20x + 60y ≥ 120
Non-negativity constraint: x ≥ 0, y ≥ 0
Step 4: Convert the inequalities to equalities.
We need to convert the inequalities to equalities by introducing slack variables:
Protein constraint: 30x + 10y + s1 = 60
Carbs constraint: 10x + 10y + s2 = 40
Fat constraint: 20x + 60y + s3 = 120
Non-negativity constraint: x ≥ 0, y ≥ 0
Step 5: Set up the initial Simplex tableau.
Construct the initial Simplex tableau using the coefficients from the objective function and the constraints.
| C | x | y | s1 | s2 | s3 | rhs |
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z | 1 | -3 | -1.5| 0 | 0 | 0 | 0 |
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s1 | 0 | 30 | 10 | 1 | 0 | 0 | 60 |
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s2 | 0 | 10 | 10 | 0 | 1 | 0 | 40 |
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s3 | 0 | 20 | 60 | 0 | 0 | 1 | 120 |
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Step 6: Apply the Simplex Method.
Continue applying the Simplex Method by performing row operations until an optimal solution is found.
In each iteration, find the pivot column (the one with the most negative coefficient in the objective row) and the pivot row (the one with the smallest non-negative ratio of the right-hand side to the pivot column coefficient). Then perform row operations to make the pivot element equal to 1 and all other elements in the pivot column equal to 0.
Repeat this process until there are no negative coefficients in the objective row.
In this specific problem, you can continue using the Simplex Method to find the solution that provides the daily requirements at the least cost.