Given the Following All-Integer Linear Program:
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Max $3 x _ { 1 } + 2 x _ { 2 }$

question 51

Essay

Given the following all-integer linear program:
​
Max $3 x _ { 1 } + 2 x _ { 2 }$
s.t.
$\begin{array} { l l } \text { M.t. } & 3 x _ { 1 } + 2 x _ { 2 } \\ & 3 x _ { 1 } + x _ { 2 } \leq 9 \\ & x _ { 1 } + 3 x _ { 2 } \leq 7 \\ & - x _ { 1 } + x _ { 2 } \leq 1 \\ & x _ { 1 } , x _ { 2 } \geq 0 \text { and integer } \end{array}$

a.​Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂.
b.​What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c.What is the solution obtained by rounding down all fractions? Is it feasible?
d.​
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Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers,and show that the feasible solution obtained in (c)is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

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Definitions:

Demand Conditions

The factors that affect consumers' desire and ability to purchase a product, influencing its market demand.

Entry Barriers

Obstacles that make it difficult for new competitors to enter an industry, which can include high startup costs, regulatory requirements, and access to distribution channels.

Price-Searcher Market

A market in which sellers set their prices based on their assessment of market demand and conditions, allowing for some degree of price flexibility.

Competitive Price-Searcher

Competitive price-searchers are firms that actively seek out the optimal pricing strategies in markets characterized by some level of product differentiation, aiming to maximize their profits.