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    <p>This is a formulation of your problem that ends up a mixed integer and non linear programming problem:</p> <p>For all building type <code>i</code> let's define:</p> <ul> <li><code>Pi</code> : price of building type <code>i</code></li> <li><code>Ci</code> : Price increment of building type <code>i</code></li> <li><code>Mi</code> : income for building type <code>i</code></li> <li><code>B</code> : Budget</li> <li><code>Ni</code> : Number of buildings of type <code>i</code> purchased.</li> </ul> <p>Purchasing Ni building of type i equals:</p> <pre><code>Sum for j=1 to Ni Pi + (j - 1) × Ci = Ni (Pi + ( Ni - 1) / 2 × Ci) </code></pre> <p>Mixed integer non linear programming formulation:</p> <pre><code>Maximise Sum for i Ni × Mi Subject to : sum for i Ni (Pi + (Ni - 1) / 2 ×Ci) &lt;= B </code></pre> <p>Note that Pi, Ci, Mi and B are constants. Decision variables are Ni</p> <p>An other solution is to purchase a building at a time selecting the one with the maximum income per invested money as per the following ratio:</p> <pre><code>Mi / (Ni (Pi + ( Ni - 1) / 2 × Ci)) </code></pre> <p>At each step you calculate the building with the maximum ratio, purchase a building, deduct the price of the budget and repeat until the budget is exhausted. I do not have a proof that you will get an optimum my following this algorithm.</p> <p>Third solution pseudocode (brute force):</p> <pre><code>(income, index_list) function maximize_income(i, b) if i &gt; last_building_type return 0, [] endif max_income = 0 max_income_index = 0 while b &gt;= P(i) - (j-1) * C(i) b = b - P(i) - (j-1) * C(i) (income, index_list) = maximize_income(i+1, b) income = income + j * M(i) if income &gt; maximum_income maximum_income = income maximum_income_index = j endif endwhile add maximum_income_index to index_list return maximum_income, index_list end function </code></pre> <p>index_list is an array containing the number of each type of building </p>
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    1. POAlgorithm to compare multiple values
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    1. COI have no knowledge about linear programming. How do i go about doing this?
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    2. COAs you said, this is non-linear. Your last paragraph still states linearity.
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    3. CO@G. Bach I initially thought it was linear but after formulating, I found out that it was a mixed integer non linear problem. Thanks for pointing this out.
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