Maximization Problem: Production Optimization A metallurgical company produces three types of metal products: A, B and C. Thes Products require different amounts of resources to manufacture. The company has limitations on the number of labor hours, machines available and raw Materials. The goal is to maximize total profit. Variables X₁: Quantity of products A to produce. X₂: Quantity of products B to produce. X₂: Quantity of products C to be produced. X₂: Hours of labor used. X: Hours of machines used. X: Amount of raw material used for A. x₁: Amount of raw material used for B. Xg: Amount of raw material used for C. X₂: Upper limit on total product production. X10: Upper limit on the number of work hours available. Objective Function Maximize Z = 5x₁+8x₂+6x Restrictions 2x₁+4x₂+3x3 ≤ x₂ (Labor restriction) 3x₂+2x₂+5x3 ≤X (Machine restriction) X ≤ 5x₂ (Raw material for A) 2x, ≤ 6x₂ (Raw material for B) 3x ≤ 4x3 (Raw material for C) X₂+x₂+x3 ≤ x₂ (Limit on total production) X4 SX₁0 (Limit on the number of work hours) Additional Restrictions X₁, X₂, X3, X4, X5, X₁, X₁, X, X, X1020 (Quantities cannot be negative) Solve this exercise using the scipy.optimize Python library. Define the function target correctly, along with the constraints and then implement the solution to find the optimal values of the variables.
Maximization Problem: Production Optimization A metallurgical company produces three types of metal products: A, B and C. Thes Products require different amounts of resources to manufacture. The company has limitations on the number of labor hours, machines available and raw Materials. The goal is to maximize total profit. Variables X₁: Quantity of products A to produce. X₂: Quantity of products B to produce. X₂: Quantity of products C to be produced. X₂: Hours of labor used. X: Hours of machines used. X: Amount of raw material used for A. x₁: Amount of raw material used for B. Xg: Amount of raw material used for C. X₂: Upper limit on total product production. X10: Upper limit on the number of work hours available. Objective Function Maximize Z = 5x₁+8x₂+6x Restrictions 2x₁+4x₂+3x3 ≤ x₂ (Labor restriction) 3x₂+2x₂+5x3 ≤X (Machine restriction) X ≤ 5x₂ (Raw material for A) 2x, ≤ 6x₂ (Raw material for B) 3x ≤ 4x3 (Raw material for C) X₂+x₂+x3 ≤ x₂ (Limit on total production) X4 SX₁0 (Limit on the number of work hours) Additional Restrictions X₁, X₂, X3, X4, X5, X₁, X₁, X, X, X1020 (Quantities cannot be negative) Solve this exercise using the scipy.optimize Python library. Define the function target correctly, along with the constraints and then implement the solution to find the optimal values of the variables.
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter18: Deterministic Dynamic Programming
Section: Chapter Questions
Problem 6RP
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Transcribed Image Text:Maximization Problem:
Production Optimization
A metallurgical company produces three types of metal products: A, B and C. These
Products require different amounts of resources to manufacture. The company
has limitations on the number of labor hours, machines available and
raw Materials. The goal is to maximize total profit.
Variables
X₁: Quantity of products A to produce.
X₂: Quantity of products B to produce.
X3: Quantity of products C to be produced.
X4: Hours of labor used.
X5: Hours of machines used.
X: Amount of raw material used for A.
X7: Amount of raw material used for B.
X₂: Amount of raw material used for C.
Xg: Upper limit on total product production.
X10: Upper limit on the number of work hours available.
Objective Function
Maximize Z = 5x, +8x₂+6x3
Restrictions
2x₁+4x₂+3x3 ≤ x₂ (Labor restriction)
3x₁+2x₂+5x3 ≤ X5 (Machine restriction)
X ≤ 5x₂ (Raw material for A)
2x, ≤ 6x₂ (Raw material for B)
3x ≤ 4x, (Raw material for C)
X₂+x₂+x3 ≤ x₂ (Limit on total production)
X4 SX₂0 (Limit on the number of work hours)
Additional Restrictions
X1, X2, X3, X4, X5, X6, X7, XB, X₁, X10 20 (Quantities cannot be negative)
Solve this exercise using the scipy.optimize Python library. Define the function
target correctly, along with the constraints and then implement the solution to
find the optimal values of the variables.
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