Οperational Research And linear Programming-The Houses Builder Problem

linear programmingmathematical modelingoptimization

EXERCISE

A builder has purchased $21,000$ square meters of land on which it is planned to build two types of houses, X(detached) and Y(a combination of town-house units), within an overall budget of $2100000$ euros.

A type X house costs $35000$ euros to build and requires $600$ square meters of land. A type Y house cost $60000$ euros to build and requires $300$ square meters of land.

To comply with local planning regulations, not more than $40$ buildings may be constructed on this land, but there must be at least five of each type. From past experience, it is known that the profit per type X house is about $10000$ euros and per type Y house to be about $6000$ euros.Profit is to maximized

QUESTION

The problem must be formulated in the form of linear programming and has to be solved in EXCEL by SOLVER. Based on this output, you have to state what the optimal solution is.

ATTEMPT

  • Definition Of The Variables

X: Type X houses(Detached)

Y: Type Y houses(a combination of town-house units)

  • Objective Function

Maximize problem(Profit is to maximized)
$$\textrm{Max profit} =10000X + 6000Y$$

  • Constraints

$X+Y \leq 40$ (Not more than $40$ Buildings)

$600X+300Y \leq 21000$(Square meters Constraint)

$35000X+60000Y \leq 2100000$(Land requirements)

$X \geq 5$ (At Least Five Of Each Type)

$Y \geq 5$ (At Least Five Of Each Type)

So, when I get these constraints in my excel, Solver can't find a feasible solution and I don't know why. Is there any problem with my mathematical modelling? I believe that the constraints are obvious from the exercise!

This is my Formulation in Excel:
enter image description here

and this is the message when I use Solver to solve it and find MaxProf:
enter image description here

I would be really grateful if somebody can help me with this as it's important for the next questions of my project! Any thorough solution or explanation would be really helpful!

Thanks in advance!

Best Answer

You have made a typo at the exercise. $\text{It is "A builder has purchased 21,00}$ $\color{red}{0}$ $\text{square meters of land}$ $\text{on which it is planned to build... "}$

But you have written it at your constraint: $600X+300Y \leq 21,000 \ \ $ (Square meters Constraint)

If you use this constraint the problem is feasible. This is what I got by using the Excel solver:

enter image description here

Excel sheet with formulas

enter image description here

Input mask

enter image description here

Related Question