2. A company makes and sells two different widgets, A and B. The demand for widgets is unlimited, but the company is constrained by the machine capacity and government enforced emissions restrictions. Production of A requires 3 machine hours/widget, and production of B requires 4 machine hours/widget. There are 20,000 machine hours available in the current production period. The factory produces 3 lbs. of CO2 for each A produced, and 1 lb. of CO2 for each B produced. The government has imposed a limit of the total CO2 produced by the factory to be less than or equal to 12000 lbs. The production costs are $3/widget for A and $2/widget for B. The sale price of A is $6/widget, and the sale price of B is $5.40/widget.
(a) Formulate a linear program (LP) to maximize profit, subject to the machine capacity and emissions restrictions. The LP should be in the standard form
min x c T x Subject to Ax ≤ b
where x is the vector of decision variables in R n , c is the cost vector in R n , A is a matrix in R m×n and b is a vector in R m.
Note: You do not need to turn anything in for the following items i - vi, but will need to determine them to complete part b, c, and d of this problem. Consider the following as you proceed:
i. What are the decision variables?
ii. What is the inequality constraint imposed by the machine capacity?
iii. What is the inequality constraint imposed by the emission restrictions?
iv. Are there other constraints? If so, what are they?
v. What quantity should be maximized? vi. Write the LP in standard form, including a description of the meaning of the decision variable x.
(NOTE: how do you transform a maximization into a minimization?)
(b) Solve the problem graphically (hand graded)
i. Using the plot command in MATLAB, plot the inequality constraints of the Linear Program problem. Use the fill command to define the feasibility set. Use appropriate axis limits when plotting the constraints.
ii. On the same graph, plot the α-levelset of the objective function to be minimized for α = −18000, −20000, −22000. Definition: For any function f, the α-levelset is the set of all x in the domain of f such that f(x) = α. For linear functions of n-variables, level sets are (n−1)-dimensional planes. So, for functions of 2 variables, the α-levelsets are straight lines in the 2-dimensional plane.
iii. What is the optimal solution of the Linear Program? HINT: The optimal solution is the coordinate of some vertex of the feasibility set.
iv. Plot the α-levelset of the objective function to be minimized corresponding to the optimal solution on the same graph of the inequality constraints.
(c) Solve the problem using a MATLAB LP solver. The file lpsolver.m (along with lpLV.m) is available at the Linear Programming module. The function defined in this file is a wrapper around an LP solver written by Prof. Lieven Vandenberghe of UCLA. We can use this function to solve for x, the optimal solution of a linear program in standard form, with the command:
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