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Problem 1
Mine scheduling
Last two ID digits: —7 points
Universal Mine Inc. operates three mines in West Virginia. The ore from each mine is separated into high-grade and low-grade ore before shipment. The table gives the number of tons produced during one operating day and the corresponding daily operating cost. The company has one week to meet its personalized delivery requirements at minimum total cost.
Mine
High grade tons/day
Low grade tons/day
Cost $/day
I
4
4
2,000
II
6
4
2,500
III
1
6
1,500
Universal Mine has committed to deliver at least tons of high-grade ore and at least tons of low-grade ore by the end of the week.
a · Formulate the complete LP
Formulate a linear programming model for Universal Mine's weekly operating decision. Use the separate modeling spaces below.
Decision variables
VariableMeaning and units
Objective function
Constraints and their meanings
ConstraintOperational meaning
Sign restrictions
b · Add one new constraint
Suppose crew availability allows at most total mine-days across the three mines. Write only the additional constraint that represents this new restriction.
Problem 2
Paint production
Last two ID digits: —11 points
Sarah runs a small family business that produces premium and standard custom-blended paint. Each gallon of premium paint earns a profit of $, and each gallon of standard paint earns a profit of $. Her brother can produce at most gallons of premium paint per day, and her sister can produce at most gallons of standard paint per day. Sarah has at most gallons of base material available each day. One gallon of premium paint requires 3 gallons of base material, while one gallon of standard paint requires 4 gallons. Sarah wants to determine the daily production quantities that maximize total profit.
a · Formulate the complete LP
Formulate a linear programming model for Sarah's daily production decision. Use the separate modeling spaces below.
Decision variables
VariableMeaning and units
Objective function
Constraints and their meanings
ConstraintOperational meaning
Sign restrictions
The page connects each pair of points and extends the boundary across the graph. Click to mark your proposed optimum. Selected point: none.
b · Solve graphically
For every non-axis constraint in your formulation, enter two distinct points on its boundary line. The coordinate axes already represent the nonnegativity constraints. Add as many boundary rows as you need.
For each personalized LP, select the plot that represents its feasible region. Then provide one isoprofit or isocost line, decide whether the objective line must move right or left, and use the full-plot sweep to explore what happens. Finally identify the solution type, optimal solution, and optimal value. The four models represent four different possible outcomes, and each model is worth 2 points.
Problem 4
Write an LP in matrix form
Last two ID digits: —4 points
Consider the personalized linear program below. Rewrite it in the form min cᵀx subject to Ax ≤ b and x ≥ 0. Define each of the four objects x, c, A, b explicitly and preserve the order of the variables and constraints shown.
First identify m and n. The page will then create correctly sized entries for x, c, A, b. Enter one number in each box.
Finish
Prepare your submission
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