{"id":180,"date":"2021-01-20T23:55:34","date_gmt":"2021-01-20T23:55:34","guid":{"rendered":"https:\/\/www.kindsonthegenius.com\/data-science\/?p=180"},"modified":"2021-01-21T11:44:26","modified_gmt":"2021-01-21T11:44:26","slug":"solving-an-optimization-problem-with-python-step-by-step","status":"publish","type":"post","link":"https:\/\/www.kindsonthegenius.com\/data-science\/solving-an-optimization-problem-with-python-step-by-step\/","title":{"rendered":"Solving an Optimization Problem with Python – Step by Step"},"content":{"rendered":"

In this example, we are going to solve a typical Constraint Optimization problem. In the previous tutorial, we tool a simple example <\/a>of just one constraint.<\/p>\n

Example 1: Maximize the function 2x + 2y + 3z with respect to the following constraints:<\/strong><\/p>\n\n\n\n\n\n\n\n
x<\/em>\u00a0+\u00a07<\/sup>\u20442<\/sub>\u00a0y<\/em>\u00a0+\u00a03<\/sup>\u20442<\/sub>\u00a0z<\/em><\/td>\n\u2264<\/td>\n25<\/td>\n<\/tr>\n
3x<\/em>\u00a0– 5y<\/em>\u00a0+ 7z<\/em><\/td>\n\u2264<\/td>\n45<\/td>\n<\/tr>\n
5x<\/em>\u00a0+ 2y<\/em>\u00a0– 6z<\/em><\/td>\n\u2264<\/td>\n37<\/td>\n<\/tr>\n
x<\/em>,\u00a0y<\/em>,\u00a0z<\/em><\/td>\n\u2265<\/td>\n0<\/td>\n<\/tr>\n
x<\/em>,\u00a0y<\/em>,\u00a0z<\/em>\u00a0integers<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Note:<\/strong> Since we are performing integer optimization, we must make sure the coefficients of the first constraint are all integer. We can achieve this by multiplying through by 2.<\/p>\n

So let’s now take the steps<\/p>\n

Step 1:<\/strong> Declare the model<\/p>\n

 # Declare the Model<\/span>\r\nfrom<\/span> ortools.sat.python<\/span> import<\/span> cp_model\r\nmodel =<\/span> cp_model.<\/span>CpModel()\r\n<\/pre>\n
 <\/p>\n
Step 2:<\/strong> Define the variables<\/p>\n
# Define the variables<\/span>\r\nnum_vars =<\/span> 3<\/span>\r\nx =<\/span> model.<\/span>NewIntVar(0<\/span>, 50<\/span>, 'x'<\/span>)\r\ny =<\/span> model.<\/span>NewIntVar(0<\/span>, 50<\/span>, 'y'<\/span>)\r\nz =<\/span> model.<\/span>NewIntVar(0<\/span>, 50<\/span>, 'z'<\/span>)\r\n<\/pre>\n
 <\/p>\n
Step 3:<\/strong> Create the constraints<\/p>\n
# Set up the constraints<\/span>\r\nmodel.<\/span>Add(2<\/span>*<\/span>x +<\/span> 7<\/span>*<\/span>y +<\/span> 3<\/span>*<\/span>z <=<\/span> 50<\/span>)\r\nmodel.<\/span>Add(3<\/span>*<\/span>x -<\/span> 5<\/span>*<\/span>y +<\/span> 7<\/span>*<\/span>z <=<\/span> 45<\/span>)\r\nmodel.<\/span>Add(5<\/span>*<\/span>x +<\/span> 2<\/span>*<\/span>y -<\/span> 6<\/span>*<\/span>z <=<\/span> 37<\/span>)\r\n<\/pre>\n
 <\/p>\n
Step 4:<\/strong> Define the cost function<\/p>\n
# Define the objective function 2x + 2y + 3z<\/span>\r\nmodel.<\/span>Maximize(2<\/span>*<\/span>x +<\/span> 2<\/span>*<\/span>y +<\/span> 3<\/span>*<\/span>z)\r\n<\/pre>\n
 <\/p>\n
Step 5:<\/strong> Invoke the solver<\/p>\n
# Invoke the solver<\/span>\r\nsolver =<\/span> cp_model.<\/span>CpSolver()\r\nstatus =<\/span> solver.<\/span>Solve(model)\r\n<\/pre>\n
 <\/p>\n
Step 6:<\/strong> Display the results<\/p>\n
# Display the results<\/span>\r\nif<\/span> status ==<\/span> cp_model.<\/span>OPTIMAL:\r\n    print<\/span>('Value of objective function: %i'<\/span> %<\/span> solver.<\/span>ObjectiveValue())\r\n    print<\/span>('x = %i'<\/span> %<\/span>solver.<\/span>Value(x))\r\n    print<\/span>('y = %i'<\/span> %<\/span>solver.<\/span>Value(y))\r\n    print<\/span>('z = %i'<\/span> %<\/span>solver.<\/span>Value(z))\r\n<\/pre>\n
The output would be as shown below:
\n<\/p>\n
Value of objective function: 35\r\nx = 7\r\ny = 3\r\nz = 5\r\n<\/pre>\n
 <\/p>\n
Complete program is given below:<\/p>\n
\"Constraint<\/a>
Constraint Optimization Problem in Python<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"
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