# Linear Programming and Using Solver

Working with chemists at Virginia Tech and George Washington Universities, landscape contractor Kenneth Golding blended his own fertilizer, calling it 'Golding-Grow.' It consists of four chemical compounds, C-30, C-92, D-21, E-11. The cost per pound for each compound is indicated as follows:

Chemical Compound Cost Per Pound ($)

C-30 .12

C-92 .09

D-21 .11

E-11 .04

The specifications for Golding-Grow are as follows: (1) E-11 must constitute at least 15% of the blend, (2) C-92 and C-30 must together constitute at least 45% of the blend, (3) D-21 and C-92 can together constitute no more than 30% of the blend; and (4) Golding-Grow is packaged and sold in 50 pound bags.

A) Formulate an LP problem to determine what blend of the four chemicals will allow Golding to minimize the cost of a 50 pound bag of the fertilizer.

B) Solve using the computer LP approach.

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#### Solution Preview

Working with chemists at Virginia Tech and George Washington Universities, landscape contractor Kenneth Golding blended his own fertilizer, calling it 'Golding-Grow.' It consists of four chemical compounds, C-30, C-92, D-21, E-11. The cost per pound for each compound is indicated as follows:

Chemical ...

#### Solution Summary

This solution provides the student with guidelines for formulating the LP program and using Excel Solver to solve. Answer, sensitivity and limits reports are included.

Non-Linear Programming using Excel Solver Add-in

Problem:

A company makes products C and D from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. One unit of C costs $30 to make and demand is estimated to be 50 - .09 * Price of C. One unit of D costs $20 to make and demand is estimated to be 30 - .14 * Price of D. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.

Product C D Available resources

Labor (hr/unit) 2 4 150

Material (ounces/unit) 2 8 220

Manufacturing cost($/unit) 30 20

Demand (units) 50 - 0.09*P1 30 - 0.14*P2

(Part a) Let X1 = demand for C's and X2 =demand for D's. Let P1 = price for C's and P2 = price for D's. Provide an algebraic formulation for the NLP.

(Part b) Implement and solve the problem in Excel. Describe the optimal solution.

(Part c) Comment on the values of the Lagrange multipliers for the Labor and Material availability constraints. What is worth noting?

The following problem requires the use to the Solver add-in within Excel. No other software application(s) will suffice for the purposes of this learning exercise. Please show all work and label the answers, so the student can benefit from context. Note: The problem is also posted in the attached MS Word document, so that it is sure to be legible in transmission. The formatting is disrupted in this text field, but the MS Word document clearly conveys the columns, etc.

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