Introduction to mathematical programming
Introduction to mathematical programming
4th Edition
ISBN: 9780534359645
Author: Jeffrey B. Goldberg
Publisher: Cengage Learning
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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,

x= Number of low income tax prayers.

y= Number of high income tax prayers.

Now, determine the below stated values

G= Per gallon tax rate

LTR= percentage tax rate charged on first $30,000 of income

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

C= Cut in spending

 Low IncomeHigh Income
Gas taxG0.5G
Tax on income up to $3000020LTR5LTR
Tax on income above $30000015HTR

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,

Gx+0.5y+20LTRx+5LTRy+15HTRy=1000

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

C150

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

0.5Gy+5LTRy+15HTRy550

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

Gx+20LTRx350

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

si-= Amount by which numerically under the ith goal.

si+= Amount by which numerically exceed the ith goal.

Thus the constraints become,

Gx+0.5y+20LTRx+5LTRy+15HTRy+s1+-s1-=1000C+s2+-s2-=1500

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Chapter 4 Solutions

Introduction to mathematical programming

Ch. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.10 - Prob. 1PCh. 4.10 - Prob. 2PCh. 4.10 - Prob. 3PCh. 4.10 - Prob. 4PCh. 4.10 - Prob. 5PCh. 4.11 - Prob. 1PCh. 4.11 - Prob. 2PCh. 4.11 - Prob. 3PCh. 4.11 - Prob. 4PCh. 4.11 - Prob. 5PCh. 4.11 - Prob. 6PCh. 4.12 - Prob. 1PCh. 4.12 - Prob. 2PCh. 4.12 - Prob. 3PCh. 4.12 - Prob. 4PCh. 4.12 - Prob. 5PCh. 4.12 - Prob. 6PCh. 4.13 - Prob. 2PCh. 4.14 - Prob. 1PCh. 4.14 - Prob. 2PCh. 4.14 - Prob. 3PCh. 4.14 - Prob. 4PCh. 4.14 - Prob. 5PCh. 4.14 - Prob. 6PCh. 4.14 - Prob. 7PCh. 4.16 - Prob. 1PCh. 4.16 - Prob. 2PCh. 4.16 - Prob. 3PCh. 4.16 - Prob. 5PCh. 4.16 - Prob. 7PCh. 4.16 - Prob. 8PCh. 4.16 - Prob. 9PCh. 4.16 - Prob. 10PCh. 4.16 - Prob. 11PCh. 4.16 - Prob. 12PCh. 4.16 - Prob. 13PCh. 4.16 - Prob. 14PCh. 4.17 - Prob. 1PCh. 4.17 - Prob. 2PCh. 4.17 - Prob. 3PCh. 4.17 - Prob. 4PCh. 4.17 - Prob. 5PCh. 4.17 - Prob. 7PCh. 4.17 - Prob. 8PCh. 4 - Prob. 1RPCh. 4 - Prob. 2RPCh. 4 - Prob. 3RPCh. 4 - Prob. 4RPCh. 4 - Prob. 5RPCh. 4 - Prob. 6RPCh. 4 - Prob. 7RPCh. 4 - Prob. 8RPCh. 4 - Prob. 9RPCh. 4 - Prob. 10RPCh. 4 - Prob. 12RPCh. 4 - Prob. 13RPCh. 4 - Prob. 14RPCh. 4 - Prob. 16RPCh. 4 - Prob. 17RPCh. 4 - Prob. 18RPCh. 4 - Prob. 19RPCh. 4 - Prob. 20RPCh. 4 - Prob. 21RPCh. 4 - Prob. 22RPCh. 4 - Prob. 23RPCh. 4 - Prob. 24RPCh. 4 - Prob. 26RPCh. 4 - Prob. 27RPCh. 4 - Prob. 28RP
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