ECON 19 Chapter Notes - Chapter 2: Lagrange Multiplier

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redchinchilla156

2 Dec 2018

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The second order conditions for a maximum are: The values of x may all have to be positive. Our choices are limited by the amount of resources/income available. One method used to solve constrained maximization problems is the lagrangian multiplier method. Suppose that we wish to find the values of x. ,, x n that maximize subject to a constraint that x. 2 permits only certain values of the x"s to be used. Lagrangian multiplier is an additional variable called a. Then apply the method used in absence of the constraint to l. Find the first-order conditions of the new objective function l: The first-order conditions can generally be solved for x. The x"s will obey the constraint: g(x. These x"s will make the value of l as large as possible. Since the constraint holds, l = f and f is also as large as possible.

Related Questions

Need assistance with these problems!

1. The following is a linear programming formulation of a labor planning problem. There are four overlapping shifts, and management must decide how many employees to schedule to start work on each shift. The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day. The variables X₁ - X₄ represent the number of employees starting work on each shift (shift 1 through shift 4).
Minimize X₁ + X₂ + X₃ + X₄
Subject to: X₁ + X₄%u2265 12 (shift 1)
X₁ + X₂%u2265 15 (shift 2)
X₂ + X₃%u2265 16 (shift 3)
X₃ + X₄%u2265 14 (shift 4)
all variables %u2265 0

Find the optimal solution using QM.
How many workers would be assigned to shift 1? (Points : 3) 12
13
0
none of the above

2. When appropriate, the optimal solution to a maximization linear programming problem can be found by graphing the feasible region and: (Points : 3)

finding the profit at every corner point of the feasible region to see which one gives the highest value.
moving the isoprofit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered.
locating the point that is highest on the graph.
none of the above.
all of the above.

3. Multiple optimal solutions can occur when the objective function is __________ a constraint line. (Points : 3)

unequal to
equal to
linear to
parallel to

4. When applying linear programming to diet problems, the objective function is usually designed to: (Points : 3)

maximize profits from blends of nutrients.
maximize ingredient blends.
minimize production losses.
maximize the number of products to be produced.
minimize the costs of nutrient blends.

5. For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the: (Points : 3)

same product mix, different total profit.
different product mix, same total profit as before.
same product mix, same total profit.
different product mix, different total profit.

6. The corner point solution method: (Points : 3)

will yield different results from the isoprofit line solution method.
requires that the profit from all corners of the feasible region be compared.
will provide one, and only one, optimum.
requires that all corners created by all constraints be compared.
will not provide a solution at an intersection or corner where a non-negativity constraint is involved.

7. Consider the following linear programming problem.
Maximize 4X + 10Y
Subject to: 3X + 4Y %u2264 480 and 4X + 2Y %u2264 3 360
all variables %u2265 0
The feasible corner points are (48,84), (0,120), (0,0), (90,0).

What is the maximum possible value for the objective function? (Points : 3)

1032
1200
360
none of the above

8. When using Excel's Solver to input and solve a linear programming problem, it is essential that one perform an additional task before submitting the formulation. That important additional function is: (Points : 3)

guessing the values of the variables.
putting in a value for the objective function.
choosing the options for assuming both a linear model and non-negative variables.
resetting the parameters.
none of the above.

9. Consider the following linear programming problem.
Maximize 5X + 6Y
Subject to: 4X + 2Y %u2264 420 and 1X + 2Y %u2264 120
all variables %u2265 0

Which of the following points (X,Y) is not feasible? (Points : 3)

(50,40)
(20,50)
(60,30)
none of the above

10. ____________ is used to analyze changes in model parameters. (Points : 3)

Optimal solution
Feasible solution
Sensitivity analysis
None of the above

Given prices for two goods PX = 2 and PY = 3 and a budget of $120 a consumer is trying to maximize the utility function U(X, Y ) = 2X0.3Y 0.7 . Use the Lagrange method to solve this problem, but donât worry about the second order conditions (if you wish, you can actually check that the MRS is decreasing and this utility function is well behaved). a) Write and solve the utility maximization problem subject to the budget constraint. What is the resulting value of the utility function? (U=49.0481) b) Write and solve the dual problem. Do you get the same optimal solution (the same optimal combination of X and Y)? What is the resulting expenditure? Does that match the budget of your original (primal) problem? (X=18, Y=28 should be the same as for a or very close due to rounding) c) Assume now general values for prices and income. Solve for the optimal amounts of X and Y as functions of PX, PY , and I. What conclusion can you draw regarding how you spend your budget when dealing with Cobb Douglas functions with exponents that add up to 1. You can actually try to solve the general case where we replace the exponents 0.3 and 0.7 with some Î± and Î², where Î± + Î² = 1. (Hint: you should find that you spend percent of your income on X and Î² percent of your income on Y. See if you can get there by solving!)

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ECON 19 Chapter Notes - Chapter 2: Lagrange Multiplier

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