The Goal Programming Problem Below Was Solved with the Management $\ge$

question 2

Essay

The goal programming problem below was solved with the Management Scientist.
Min
P₁(d₁^-) + P₂(d₂⁺) + P₃(d₃^-)
s.t.
72x₁ + 38x₂ + 23x₃ - 20,000
.72x₁ -.76x₂ - .23x₃ + d₁^- - d₁⁺ = 0
x₃ + d₂^- - d₂⁺ = 150
38x₂ + d₃^- -d₃⁺ = 2000
x₁, x₂, x₃, d₁^-, d ₁⁺, d₂^-, d₂⁺, d₃^-, d₃⁺ $\ge$ 0
Partial output from three successive linear programming problems is given. For each problem, give the original objective function expression and its value, and list any constraints needed beyond those that were in the original problem. $$\begin{array}{l}\text { a. Objective Function Value } = 0.000\\\begin{array} { l c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { D1MINUS } & 0.000 & 1.000 \\\text { X1 } & 52.632 & 0.000 \\\text { X2 } & 0.000 & 0.000 \\\text { X3 } & 150.000 & 0.000 \\\text { D1PLUS } & 3.395 & 0.000 \\\text { D2MINUS } & 0.000 & 0.000 \\\text { D2PLUS } & 0.000 & 0.000 \\\text { D3MINUS } & 0.000 & 0.000 \\\text { D3PLUS } & 0.000 & 0.000\end{array}\end{array}$$ $$\begin{array}{l}\text { b. Objective Function Value } = 0.000\\\begin{array} { l c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { D2PLUS } & 0.000 & 1.000 \\\text { X1 } & 52.632 & 0.000 \\\text { X2 } & 0.000 & 0.000 \\\text { X3 } & 150.000 & 0.000 \\\text { D1MINUS } & 0.000 & 0.000 \\\text { D1PLUS } & 3.395 & 0.000 \\\text { D2MINUS } & 0.000 & 0.000 \\\text { D3MINUS } & 0.000 & 0.000 \\\text { D3PLUS } & 0.000 & 0.000\end{array}\end{array}$$ $$\begin{array}{l}\text { c. Objective Function Value } = 0.000\\\begin{array} { l c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { D3MINUS } & 0.000 & 1.000 \\\text { X1 } & 52.632 & 0.000 \\\text { X2 } & 0.000 & 0.000 \\\text { X3 } & 150.000 & 0.000 \\\text { D1MINUS } & 0.000 & 0.000 \\\text { D1PLUS } & 3.395 & 0.000 \\\text { D2MINUS } & 0.000 & 0.000 \\\text { D2PLUS } & 0.000 & 0.000 \\\text { D3PLUS } & 0.000 & 0.000\end{array}\end{array}$$

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Definitions:

Constructive Dilemma (CD)

A form of argument where, given premises that jointly imply two conclusions, if one of them is true, then so is the other.

Atomic Sentences

Simple statements in logic that contain no logical connectives or quantifiers, expressing a basic claim.

Truth Table

A mathematical table used in logic to determine the veracity of various propositions under all possible scenarios.

Hypothetical Syllogism (HS)

A form of deductive reasoning that connects two conditional statements to arrive at a conclusion, if P then Q; if Q then R, therefore if P then R.