Is My Linear Programming Model Formulated Correctly?

• BernardToh
In summary, your LP model accurately represents the problem with some minor suggestions for improvement.
BernardToh
Homework Statement
An interview is to be conducted and there are three family types that have been identified for the interview; 1-Child Family, 2-Child Family, and 3-Child Family.

At least 2200 families have to be interviewed for the entire interview exercise.

There are 3 interview timings; Weekdays Morning, Weekdays Evening, and Saturday Morning

The costs of the interviews are as follows:
Family Type Weekdays Morning Weekdays Evening Saturdays
One-Child $20$22 $20 Two-child$23 $25$23
Three-child $26$28 \$26

The total number of respondents during weekdays evening interviews must be at least equals to the total number of respondents during weekdays morning interviews.

The minimum number of respondents for each family types for weekdays interview are:
One child - 650
Two-child - 600
Three-child - 700

The combined total of two and three child households being interviewed on Saturday has to be at least 110.

Below is the allocation ratio for weekdays evening interviews to weekdays morning interviews per family type are:
One child - 55:45
Two child - 60:40
Three child - 60:40

Question - Do up a Linear Programming model based on the information given above to minimize the cost of the interview exercise.
Relevant Equations
Linear Programming
I've tried formulating the LP model for the question above and would like to check if I'm doing anything incorrectly.

Below is my LP model.

Let X1 be the number of one-child family interviewed on weekdays morning
Let X2 be the number of one-child family interviewed on weekdays evening
Let X3 be the number of one-child family interviewed on saturday
Let X4 be the number of two-child family interviewed on weekdays morning
Let X5 be the number of two-child family interviewed on weekdays evening
Let X6 be the number of two-child family interviewed on saturday
Let X7 be the number of three-child family interviewed on weekdays morning
Let X8 be the number of three-child family interviewed on weekdays evening
Let X9 be the number of three-child family interviewed on saturday
Let Z be the total cost of the interviews

Z = 20X1 + 22X2 + 20X3 + 23X4 + 25X5 + 23X6 + 26X7 + 28X8 + 26X9

Constraints
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 =>2200
X1 + X2 => 650
X4 + X5 => 600
X7 + X8 => 700
X6 + X9 => 110
X2 + X5 + X8 => X1 + X4 + X7
X1 = 0.45(X1 + X2)
X2 = 0.55(X1 + X2)
X4 = 0.4(X4 + X5)
X5 = 0.6(X4 + X5)
X7 = 0.4(X7 + X8)
X8 = 0.6(X7 + X8)

Last edited:

Overall, your LP model looks good and seems to accurately represent the problem. However, there are a few minor suggestions and corrections I would like to make.

1. In the constraints, you should use >= instead of =>. The >= symbol means "greater than or equal to" while => does not have a mathematical meaning.
2. In the constraint for total number of interviews, it should be <= 2200 instead of => 2200. This ensures that the total number of interviews does not exceed the maximum limit of 2200.
3. The last four constraints can be simplified. Instead of using separate constraints for X1, X2, X4, X5, X7, and X8, you can combine them into one constraint and use a single variable for each type of family. The constraints would look like this:
X1 + X2 = 0.45 * (X1 + X2 + X4 + X5 + X7 + X8)
X4 + X5 = 0.4 * (X1 + X2 + X4 + X5 + X7 + X8)
X7 + X8 = 0.4 * (X1 + X2 + X4 + X5 + X7 + X8)
This will make the model more concise and easier to read.

Overall, your LP model is well-formulated and correctly represents the problem. Good job!

1. What is linear programming?

Linear programming is a mathematical method used to optimize a linear objective function, subject to a set of linear constraints. It is commonly used in fields such as operations research, economics, and engineering to find the best solution to a problem with many variables.

2. What are the applications of linear programming?

Linear programming has a wide range of applications, including production planning, resource allocation, transportation and logistics, financial planning, and portfolio optimization. It can also be used in decision-making processes and to model real-world problems.

3. What are the key components of a linear programming problem?

The key components of a linear programming problem are the decision variables, the objective function, and the constraints. The decision variables represent the quantities that need to be optimized, the objective function defines the goal to be achieved, and the constraints limit the possible values of the decision variables.

4. How is a linear programming problem solved?

A linear programming problem is typically solved using algorithms and software programs. The most commonly used method is the simplex method, which involves iteratively improving a feasible solution until an optimal solution is found. Other methods include the interior-point method and the branch and bound method.

5. What are the limitations of linear programming?

Linear programming has some limitations, including the assumption of linearity in the objective function and constraints. It also requires a known and well-defined problem with a finite number of variables and constraints. Additionally, it may not always provide the most practical or realistic solution to a problem.

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