Chapter 4.16, Problem 13P

Explanation of Solution

Formulating preemptive goal programming model:

Consider the linear programming problem of new president deciding the tax rate to achieve the following goals:

Goal 1: Balance the budget (this means revenues are at least as large as costs).

Goal 2: Cut spending by at most $150 billion.

Goal 3: Raise at most $550 billion in taxes from rich.

Goal 4: Raise at most $350 billion in taxes from the poor.

Let,

$\text{x}$= Number of low income tax prayers.

$\text{y}$= Number of high income tax prayers.

Now, determine the below stated values

$\text{G}$= Per gallon tax rate

$\text{LTR}$= percentage tax rate charged on first $30,000 of income

$\text{HTR}$= Percentage tax rate charged on any income earned more than $30,000

$\text{C}$= Cut in spending

	Low Income	High Income
Gas tax	G	0.5G
Tax on income up to $30000	20LTR	5LTR
Tax on income above $30000	0	15HTR

From the given information, the following constraints are formed.

Goal 1: Balance the budget. Amount spend (1000 billion = amount collected as tax). The constraint formed is given below,

$\text{Gx+0}\text{.5y+20LTRx+5LTRy+15HTRy=1000}$

Goal 2: Cut spending by at most $150 billion. The constraint formed is given below,

$\text{C}\le \text{150}$

Goal 3: Raise at most $550 billion in taxes from rich. The constraint formed is given below,

$\text{0}\text{.5Gy+5LTRy+15HTRy}\le \text{550}$

Goal 4: Raise at most $350 billion in taxes from the poor.

$\text{Gx+20LTRx}\le \text{350}$

Here, the user can observe that the above set of constraints there is no feasible region. That is all constraints cannot be met. Hence the user should assign a cost value incurred if any of the goal is not met. So, introduce the deviational variables as follows

${\text{s}}_{\text{i}}{}^{\text{-}}$= Amount by which numerically under the ${\text{i}}^{\text{th}}$ goal.

${\text{s}}_{\text{i}}{}^{\text{+}}$= Amount by which numerically exceed the ${\text{i}}^{\text{th}}$ goal.

Thus the constraints become,

$\begin{array}{c}\text{Gx+0}{\text{.5y+20LTRx+5LTRy+15HTRy+s}}_1{}^{\text{+}}{\text{-s}}_1{}^{\text{-}}\text{=1000}\\ {\text{C+s}}_2{}^{\text{+}}{\text{-s}}_2{}^{\text{-}}\text{=150}\\ \text{0}{}_{}\end{array}$