
Concept explainers
Explanation of Solution
Formulating preemptive goal
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,
Now, determine the below stated values
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,
Goal 2: Cut spending by at most $150 billion. The constraint formed is given below,
Goal 3: Raise at most $550 billion in taxes from rich. The constraint formed is given below,
Goal 4: Raise at most $350 billion in taxes from the poor.
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
Thus the constraints become,

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Chapter 4 Solutions
Introduction to mathematical programming
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