# Οperational Research And linear Programming-The Houses Builder Problem

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:

and this is the message when I use Solver to solve it and find MaxProf:

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!

• I'm voting to close this question as off-topic because the question is not math related. Better site maybe super user meta.stackexchange.com/questions/155487/… – mathcounterexamples.net Dec 26 '18 at 13:03
• I´ve done the same in excel. It gave me the same message. It seems that the program is not feasible. Have you tried to sketch the feasible solution space? – callculus Dec 26 '18 at 13:17
• Υeah, but why? Did you use the same mathematical modelling as me? It seems impossible to me that the program is not feasible, because it's only the first question in my project. All the other questions depend on this question! Strange ha? – Paris K. Patsogiannis Dec 26 '18 at 13:26
• @callculus yeah and I didn't find any feasible solution! – Paris K. Patsogiannis Dec 26 '18 at 13:31
• It is clear that the problem has no solution. With a minimum of 5 houses of each, you don’t have enough land. – mathcounterexamples.net Dec 26 '18 at 13:48

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:

Excel sheet with formulas

• This is after using Solver?I use Sumproduct function,not Sum!Is this wrong?And what is these $1050000$ and $600000$??From where did you get these? – Paris K. Patsogiannis Dec 26 '18 at 14:07
• It is $30\cdot 35000=1,050,000$ for instance. And similar for y: $10\cdot 60,000=600,000$ – callculus Dec 26 '18 at 14:10
• So,i made mistake with the variables.I use 1 and 1 but that's wrong because i have a constraint that tells me that i have $Y \geq 5$ and $X \geq 5$.Right? – Paris K. Patsogiannis Dec 26 '18 at 14:12