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Exam 16: Linear Programming

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Question 1

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Determine the feasible region given the following constraints: 4x + 3y $\le$ 120 x $\le$ 20 y $\ge$ 10 x, y $\ge$ 0

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The feasible region is a trape...

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Question 2

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In a linear programming problem, there are 3 binding constraints. Constraint A has slope -1.5, constraint B has slope -0.5, and constraint C has slope -0.2. The objective function's slope is -1.2. Where would the optimal solution lie?

A) At the intersection of constraints A and B
B) At the intersection of constraints B and C
C) At the intersection of constraint A and the horizontal axis
D) At the intersection of constraint C and the vertical axis
E) Anywhere along constraint A

F) A) and D)
G) A) and B)

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Question 3

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A shadow price measures the:

A furniture manufacturer produces two types of tables. Table A sells for $430 and Table B for $300 per unit. Both types require 10 hours of labor. The hardwood requirement of Table A is $120 of per unit and that of Table B is $ per unit. The cost of labor is $10 an hour, and 400 labor hours are available per week. The firm's available supply of hardwood is $3,600 per week. Formulate, graph and solve the firm's linear programming problem.

In a linear programming problem, multiple optimal solutions are possible if:

The combination of decision variables optimizing a linear programming problem, occurs at:

Which of the following statements concerning sensitivity analysis is incorrect?

A firm is maximizing profit by producing goods X and Y, using resources A and B. The firm is fully utilizing its supply of resource A, while a surplus of resource B is available. The profit contributions from per units of goods X and Y are $5 and $4 respectively. The firm is considering expansion of its supply of resource A (at a cost of $8 per unit). Increasing A by one unit would allow the firm to produce 3 additional units of X, while producing 1 fewer units of Y. Should the firm expand its supply of A? Explain.

Determine the feasible region for the following linear programming problem: Maximize Z = 10x + 8y Subject to: 2x + 3y $\le$ 11 5x + 2y $\le$ 11; x, y $\ge$ 0

Graph and solve the following linear programming problem: Maximize Z = 100x + 50y Subject to: 10x + 10y $\le$ 50 y $\le$ 3, x, y $\ge$ 0

To identify the feasible region, one must graph:

Given, MB = Marginal benefit and MC = Marginal cost. Then, for a positive decision variable in the optimal solution, which of the following relations holds?

A manufacturer of leather goods produces two models of briefcases - the Executive (E) and the Student (S). Each unit of the E requires 1 square yard of leather, 2.5 hours of labor, and 1 hour of machine time. The S requires .75 square yard of leather, 2 hours of labor, and .5 hours of machine time. Each unit of E contributes $8 of profit while S contributes $5. The manufacturer has 500 square yards of leather available per week, 400 labor hours, and 180 machine hours. Formulate as a linear programming problem. The basic objective is to maximize profit.

Most of the large-scale linear programming problems are solved using:

An investor wishes to maximize the return on her portfolio while also maintaining certain liquidity and risk standards. The alternatives and their corresponding returns are: $\begin{array} { l r } \text { Alternative } & \text { Return } \\\text { Municipal Bonds (M) } & 6.2 \% \\\text { Certificates of Deposit (S) } & 5.1 \% \\\text { Treasury Bills (T) } & 6.9 \% \\\text { AA Bonds (B) } & 10.5 \%\end{array}$ The investor wishes to have at least 25% of the portfolio in Treasury Bills, no more than 20% in AA bonds; no more than 15% in Certificates of Deposit, and no more than 10% in municipal bonds. Formulate a linear programming problem for the investor seeking to maximize the expected return of a $200,000 portfolio.

Linear programming is useful for solving optimization problems involving:

A firm produces tires by utilizing machine-hours and labor-hours. It has the choice of producing through three separate processes using different combinations of inputs. The optimization can be done by undertaking a process singly or in combination. The combination matrix is provided below: \begin{array}{cccc} \text { } & \text { Process 1\left(X_{1}\right) } & \text {Process \( 2\left(X_{2}\right) \) } & \text { Process \( 3\left(X_{3}\right) \) } & \\ \text { Machine-hours } &1&2&3\\ \text { Labor-hours } &3&2&1\\\end{array} The firm can rent a machine at a price $10 and hire a labor at a wage $15. The firm needs to produce a minimum target of 50 tires per day. (a) Formulate and solve a linear programming problem which will minimize the firm's daily cost (C). (b) Find out the number of binding constraints in this problem.

In a linear programming problem, the objective function:

In an LP problem, the goal is to maximize the objective function S + 3T, subject to the binding constraints S + T ≤ 700 and S + 2T ≤ 1,000. The optimal solution is:

In an LP problem the inequalities 2X + Y ≤ 800 and X + 2Y ≤ 700, hold as binding constraints. The optimal solution is: