ECON 140 UCI Linear Programming the Objective Function & Maximum Value Worksheet

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Economics

econ 140

University of California Irvine

ECON

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HW (15 points) Answer the following questions on Linear Programming (use the class videos/notes to review the concepts). 1. A manager has formulated the following LP problem. Draw the graph and find the optimal solution. (In each, all variables are nonnegative). (3) Maximize: 10x+15y, subject to 2x+5y ≤ 40 and 6x+3y ≤ 48. 2. An athlete carefully watches her intake of calcium, protein, and calories. Her breakfast diet consists mainly of milk and cereal, whose prices and nutrient contents appear in the following table: Milk (1 oz.) Cereal (1 oz.) Calcium 2 2 Protein 2 6 Calories 6 2 Price $.10 $.15 She seeks a diet that supplies at least 50 units of calcium, 90 units of protein, and 66 calories at minimum cost. a. Formulate, graph, and solve this decision problem. What is the minimum cost of meeting the nutrient requirements? (9) b. Calculate and provide an economic interpretation of the shadow price associated with calcium. (3) An Introduction to Linear Pr0gramming What is Linear Programming? • Definition: A solution method for maximization or minimization decision problems subject to underlying constraints. • LP is a method of formulating and solving decision problems that involve explicit resource constraints. Examples • Analysts use the LP method to solve problems such as• How should a firm allocate its advertising expenditure among various media? • What quantities of two jointly manufactured goods should a firm produce with a fixed amount of labor and inputs? • How should a federal agency allocate its limited budget between two competing safety programs? What do all these problems have in common? • • • • All seek to find the best values of certain variables – • • • The right advertising mix The most profitable product quantities The appropriate budget allocation. Each decision has an explicit objective – maximum profit, minimum cost etc. Constraints limit the possible values of the decision variables (values the decision makers control). For example – limited labor supply may constrain the quantity of output. The problem is finding values for the decision variables that best meet the given objective while satisfying various constraints. Linear Programming -Examples • Key: Linear Programming can be applied only when the objective function and constraint conditions are linear. • Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be zero). • Linear constraints in LP are linear functions that are expressed in inequality form. • Examples on maximization and minimization problems (Notes)
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Explanation & Answer

Answer.

1) We have,
Maximize: 10x + 15y
s.t. constrains
2x+5y ≤ 40
6x+4y ≤ 48
x≥0
y≥0

Red line is constraint: 2x+5y ≤ 40
Green Line is constraint: 6x+4y ≤ 48
The maximum value of the objective function occurs at the corner solution: X = 3.636 and X2 =
6.545
The maximum value of the objective function is: 10*3.636 + 15*6.545 = 134.55

2-a) We have,
Milk (1 oz.) Cereal (1 oz.)
Calcium
2
2
Protein
2
6
Calories
6
2
Price
$0.10
$0.15

Milk Quantity = X1
Cereal Quantity = X2
Minimize $0.1X1 + $0.15X2
s.t. constraints
2X1 + 2X2 ≥ 50
2X1 + 6X2 �...


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