students are reminded that submitted assignments must be typed (i.e. **can NOT be hand written**), neat, readable, and well-organized. However, GRAPHS are ok to plot them by hand as long as they are large, legible, and properly labeled and that their calculations are typed within the rest of the assignment. Assignment marks will be adjusted for sloppiness, poor grammar, spelling, for technical errors as well as if you submit a PDF file.

**Problem 1**

The manager of a food processing plant that specializes in potato chips has developed an LP model to reflect processing times.

x1 = boxes of regular chips

x2 = boxes of crinkle cut chips

**Winter 2019**

**Page 1**

Maximize subject to

Cutting Frying Packing Crinkle Crinkle

Z = .40×1 + .30×2 (profit)

3.6×1 + .8×2 ≤ 144 minutes 3.2×1 + 1.6×2 ≤ 160 minutes 4.8×1 + 7.2×2 ≤ 576 seconds

x2 ≤ 80 boxes

x2 ≥ 20 boxes x1, x2 ≥ 0

**ADM2302 Section M, N, P and Q**

**Assignment 1**

Briefly explain or define each of these parts of the model:

- The .40 in the objective function.
- The product of the .3 and x2 in the objective function.
- The 144 minutes in the cutting constraint.
- The 4.8 in the packing constraint.
- x2≥20.
- The product of 3.2 and x1 in the frying constraint.
- What two key questions can be answered using this model?

The optimal solution is to produce 15 boxes of x1 and 70 boxes of x2 for a total maximal profit of $27.00. Determine each of the following:

h. The amount of each resource that will be used. i. The amount of slack for each resource.

**Problem 2**

The new manager of a food processing plant hopes to reduce costs by using linear programming to determine the optimal amounts in kilograms of two ingredients it uses, x1 and x2. The manager has constructed this model:

**Winter 2019**

**Page 2**

Minimize Subject to

Protein Carbohydrates

Z = .40×1 + .40×2

3×1 + 5×2 ≥ 30 grams 6×1 + 4×2 ≥ 48 grams

x1, x2 ≥ 0

- Graph the constraints and identify the feasible region.
- Determine the optimal solution and the minimum cost (Show your work).
- There is a single optimal solution to part b above. However, if the objective

Maximize Subject to x1+x2 ≤6 x1 -x2 ≥0 x1 +x2 ≥3 x1, x2 ≥ 0

Z = x1 + 2×2

function had been parallel to one of the constraints, there would have been two equally optimal solutions. If the cost of x2 remains at $.40, what cost of x1 would cause the objective function to be parallel to the carbohydrate constraint? Explain how you determined this.

**Problem 3:**

Given the linear programming problem:

**ADM2302 Section M, N, P and Q Assignment 1**

- Graph the constraints and identify the feasible region.
- Determine the optimal solution (Show your work).
- Are any constraints binding? If so, which one?

**Problem 4**

The Oak Works is a family-owned business that makes handcrafted dining room tables and chairs. They obtain the oak from a local farm tree farm, which ships them 2,500 pounds of oak each month. Each table uses 50 pounds of oak while each chair uses 25 pounds of oak. The family builds all the furniture itself and has 480 hours of labor available each month. Each table or chair requires six hours of labor. Each table nets Oak Works $400 in profit, while each chair nets $100 in profit. Since chairs are often sold with tables, they want to produce at least twice as many chairs as tables.

The Oak Works would like to decide how many tables and chairs to produce so as to maximize profit.

**Winter 2019**

**Page 3**

a. b.

Formulate algebraically the linear programming model of this problem.

Formulate this same linear programming problem on a spreadsheet and SOLVE using Excel solver (Provide a printout of the corresponding “Excel Spreadsheet” and the “Answer Report”).