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Widely available waste cooking oil offers a sustainable feedstock for biodiesel...
May 6, 2024
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Solution by Steps
step 1
To find the optimal conditions for biodiesel yield using the steepest descent method, we first need to calculate the gradient of the yield function with respect to the variables T, M, and C
step 2
The gradient of the function Y=79.57+9.82T−5.85M+15.71C−4.94T2−20.57M2−7.51C2+17.3TM−10.03TC+11.83MCY = 79.57 + 9.82T - 5.85M + 15.71C - 4.94T^2 - 20.57M^2 - 7.51C^2 + 17.3TM - 10.03TC + 11.83MC is given by the vector of partial derivatives with respect to T, M, and C
step 3
The partial derivatives are: ∂Y∂T=9.82−2(4.94)T+17.3M−10.03C\frac{\partial Y}{\partial T} = 9.82 - 2(4.94)T + 17.3M - 10.03C, ∂Y∂M=−5.85−2(20.57)M+17.3T+11.83C\frac{\partial Y}{\partial M} = -5.85 - 2(20.57)M + 17.3T + 11.83C, and ∂Y∂C=15.71−2(7.51)C−10.03T+11.83M\frac{\partial Y}{\partial C} = 15.71 - 2(7.51)C - 10.03T + 11.83M
step 4
Assuming we start at a point (T0,M0,C0)(T_0, M_0, C_0), we calculate the gradient at this point and then take a step in the opposite direction of the gradient to find a new point (T1,M1,C1)(T_1, M_1, C_1). The step size needs to be chosen appropriately
step 5
After the first iteration, we calculate the gradient at the new point (T1,M1,C1)(T_1, M_1, C_1) and take another step in the opposite direction to find the next point (T2,M2,C2)(T_2, M_2, C_2). This process is repeated until convergence or for the specified number of iterations
step 6
For part (b), with T and M fixed at 1, we minimize the function Y=79.57+9.82−5.85+15.71C−4.94−20.57−7.51C2+17.3−10.03C+11.83CY = 79.57 + 9.82 - 5.85 + 15.71C - 4.94 - 20.57 - 7.51C^2 + 17.3 - 10.03C + 11.83C with respect to C using the fixed-step size search method
step 7
We choose a step size and evaluate the function at various points C, looking for the minimum value of Y. If no global minimum is found within the range of C, we may need to adjust the step size or the range
[question a] Answer
The steepest descent method involves calculating the gradient of the yield function, taking steps in the opposite direction of the gradient, and iterating this process. The exact steps and resulting values depend on the starting point and step size chosen.
[question b] Answer
The fixed-step size search method involves evaluating the yield function at various points for C while keeping T and M fixed, and looking for the minimum value of Y. The step size and range of C are critical for finding the optimal value.
Key Concept
Steepest Descent and Fixed-Step Size Search Methods
Explanation
The steepest descent method is used to find the minimum of a function by moving in the direction opposite to the gradient. The fixed-step size search method evaluates the function at intervals to find the minimum value within a range.
Solution by Steps
step 1
Define the decision variables for the linear programming problem. Let xAx_A, xBx_B, xCx_C, and xDx_D represent the number of units produced for Gadgets A, B, C, and D, respectively
step 2
Formulate the objective function. The objective is to maximize profit, which is the total revenue minus the total costs (labor and materials). The revenue for Gadget A is 7.20xA7.20x_A, for the first 50 units of Gadgets B, C, and D it is 8.00xB8.00x_B, 5.00xC5.00x_C, and 5.00xD5.00x_D respectively, and for units beyond 50, it is 4.00(xB−50)4.00(x_B - 50), 4.00(xC−50)4.00(x_C - 50), and 4.00(xD−50)4.00(x_D - 50) respectively. The costs include labor costs and material costs
step 3
Write the constraints based on machine availability and processing times. For example, for Machine Alpha, the total processing time cannot exceed its availability: 10xA+4xB+7xC+4xD≤138×6010x_A + 4x_B + 7x_C + 4x_D \leq 138 \times 60. Similar constraints are written for Machines Beta and Gamma
step 4
Include non-negativity constraints: xA,xB,xC,xD≥0x_A, x_B, x_C, x_D \geq 0. Also, since the selling price changes after the first 50 units, include constraints to handle the piecewise nature of the pricing for Gadgets B, C, and D
step 5
Set up the linear programming problem in Microsoft Excel Solver with the objective function, constraints, and decision variables defined in the previous steps
step 6
Run the Solver to find the optimal solution that maximizes profit. Ensure that the Solver settings are correct, such as choosing 'Simplex LP' for the solving method if the problem is linear
[question a] Answer
The mathematical statement of the problem as a linear programming problem has been formulated with decision variables, an objective function, and constraints.
[question b] Answer
The optimal solution to the problem can be found using Microsoft Excel Solver with the setup from the previous steps.
Key Concept
Linear Programming
Explanation
Linear programming involves formulating an objective function to maximize or minimize, subject to a set of linear constraints. Decision variables represent quantities to determine, the objective function represents the goal (e.g., profit maximization), and constraints represent limitations or requirements (e.g., machine availability). Microsoft Excel Solver is a tool that can solve linear programming problems by finding the optimal values for the decision variables.
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