SOLUTION: A landscaping supply company produces decorative stones. A ton of coarse stones requires 2 hours ofcrushing, 5 hours of sifting, and 8 hours of drying. A

Question 1184642: A landscaping supply company produces decorative stones. A ton of coarse stones requires 2 hours ofcrushing, 5 hours of sifting, and 8 hours of drying. A ton of fine stones requires 6 hours of crushing, 3hours of sifting, and 2 hours of drying. The coarse stones sell for$400 per ton. The fine stones sell for$500per ton. In a work week, the plant is capable of 36 hours of crushing, 30 hours of sifting, and 40 hours ofdrying. Based on the demand forecast, the quantity of coarse stones produced should be at least twice thequantity of fine stones produced.
First
1. Formulate an optimization model to decide how much of each kind of stones to produce to maximizetotal revenue.
2. Solve your model using the Excel solver.
3. Using a two-dimensional plot, solve your model graphically.
Then, by modifying your two-dimensional plot, answer the following questions:
4. How much would it be worth to get another hour of crushing time?
5. How much would it be worth to get another hour of sifting time?
6. Could you find the range of the prices for the fine stones such that in the optimal solution, thequantity of coarse stones produced is exactly twice the quantity of fine stones produced.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to formulate the optimization model, solve it using Excel Solver, and analyze it graphically:
**1. Optimization Model:**
* **Decision Variables:**
* x = Tons of coarse stones produced
* y = Tons of fine stones produced
* **Objective Function (Maximize Revenue):**
* Revenue = 400x + 500y
* **Constraints:**
* Crushing: 2x + 6y ≤ 36
* Sifting: 5x + 3y ≤ 30
* Drying: 8x + 2y ≤ 40
* Demand: x ≥ 2y
* Non-negativity: x ≥ 0, y ≥ 0
**2. Excel Solver Solution:**
1. Set up a spreadsheet with cells for x and y (the decision variables), the objective function (revenue), and each constraint.
2. Enter the coefficients for the objective function and constraints in the appropriate cells.
3. Use the SUMPRODUCT function to calculate the revenue and the left-hand side of each constraint.
4. Open the Solver add-in (Data tab > Solver).
5. Set the objective cell to the revenue cell, and select "Max."
6. Set the changing cells to the cells containing x and y.
7. Add the constraints by referencing the appropriate cells.
8. Select a solving method (Simplex LP is appropriate for this linear programming problem).
9. Click "Solve."
The Solver should give you the optimal values for x and y, as well as the maximum revenue. You'll likely find that x = 4 and y = 2, with a maximum revenue of $2600.
**3. Graphical Solution:**
1. **Plot the Constraints:** Treat each constraint as an equation and plot the lines on a graph with x and y axes. For inequalities, shade the feasible region.
2. **Identify the Feasible Region:** The feasible region is the area where all shaded regions overlap.
3. **Plot the Objective Function:** Pick an arbitrary revenue value and plot the line 400x + 500y = Revenue.
4. **Move the Objective Function Line:** Slide the objective function line parallel to itself in the direction of increasing revenue until it touches the last point of the feasible region. This point represents the optimal solution.
The intersection of the constraints x ≥ 2y, 2x + 6y ≤ 36, 5x + 3y ≤ 30, and 8x + 2y ≤ 40 will give you the feasible region. The optimal solution will be at the intersection of x = 2y, and 5x + 3y = 30, which is x = 4 and y = 2.
**4. Value of an Extra Hour of Crushing Time:**
1. Increase the crushing constraint to 2x + 6y ≤ 37 in your Excel model or adjust the graph.
2. Re-solve using Solver or graphically find the new optimal solution.
3. Calculate the difference in revenue between the new solution and the original solution. This is the "shadow price" or value of the extra crushing hour.
**5. Value of an Extra Hour of Sifting Time:**
1. Increase the sifting constraint to 5x + 3y ≤ 31.
2. Re-solve using Solver or graphically.
3. Calculate the difference in revenue.
**6. Range of Fine Stone Prices for x = 2y:**
1. Modify your Excel model to add a constraint that enforces x = 2y. You can do this by adding a cell with the formula x - 2y and setting the constraint as x - 2y = 0.
2. Re-solve and note the optimal revenue.
3. Change the price of fine stones (the coefficient of y in the objective function) and re-solve. Repeat this process, increasing and decreasing the fine stone price.
4. Observe the range of prices for which the optimal solution still maintains x = 2y. You'll find that there's a range of fine stone prices for which this relationship holds. If the fine stone price gets too high, the company will want to make more fine stones. If it gets too low, the company will want to make more coarse stones.

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