Integer Linear Programming (ILP) is required to solve optimization problems with integer values as the solution.
Integer Linear Programming is a subset of Linear Programming (LP). It has all the characteristics of an LP, an attempt to find a maximum or minimum solution to a function given certain constraints, except for some or all of the variables to the LP solution must be restricted to integers.
There are many events we need to solve the real world optimization problems with Integer Linear Programming, instead of Linear Programming. It doesn't make sense to use a continuous variable to represent the number of airplanes to produce because there is no point in manufacturing a partial airplane. The production of large indivisible items is one of the use case of ILP.
We showed how to load boxes to a set of lorries in our previous article Graphical Method for Integer Linear Programming (ILP), in which we cannot rent a partial lorry (7.5 lorries). It has to be a whole lorry (7 or 8 lorries), leaving the empty space unoccupied.
The variables take only integer values, instead of a solid area of feasible region. The optimal solution is always on the edge of the feasible region, however, rounding off the floating point solution can lead to infeasible solution.
So, the integer solution may not lie on an extreme point of continuous feasible region, and LP cannot be used. From the graphic, the collection of dots/points indicate the new feasible solutions.
Types of Integer Programming
According to the nature of the variables, we can distinguish three types of IP models, which are Pure IP, Mixed IP and Binary IP.
Pure IP requires that all decision variables have integer values in the final solution.
Mixed IP requires some, not all, of the decision variables to have integer solutions.
Binary IP, also 0-1 IP, involves problems in which the variables are restricted to be either 0 or 1, such as state or mode decisions, yes/no decisions or logical decisions.
Example:
Unit commitment problem with the state or mode decisions. A particular stage 0-1 binary variables (boolean), i.e. on or off.
Planning of investments as a yes/no decisions. It takes a value 1 to invest in a warehouse (yes) and 0 to ignore it (no).
A given tax break is only applicable (true/false) if a certain investment is made. This is a logical decisions, the logic constraints between different decision variables, i.e. AND, OR, NOT, IMPLY, ... etc.
The solution complexity increases with the number of possible combinations of integer variables (combinatorial problem). Even the fastest computer can take excessively long time to solve a big integer programming problem.
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
There are some methods for solving ILP problems, such as:
Rounding off a non-integer solution
Cutting Plane method
Branch and Bound method
The addictive algorithm for 0-1 IP
which we will explain in details in the coming lessons.
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality
Anya is LIVE right now
FREE
Free to watch • No registration required • HD streaming
Graphical Method for Integer Linear Programming (ILP)
Revisiting the lorry loading puzzle as an integrer linear programming problem with the graphical method.
Consider the following problem from Linear Programming with Minizinc Introduction:
300 boxes need to be loaded and shipped to a supermarket from a warehouse. We may rent lorries that can accommodate 30 boxes and 40 boxes, costing us RM 400 and RM 500 respectively. How many lorries of each type to load all boxes while minimizing the cost?
Our goal is to minimize the cost of loading these boxes by selecting the right amount of the different lorries.
Previously from Linear Programming with Graphical Method - Part 1, we concluded that with 7.5 lorries of 40 boxes (total 300 boxes), we can load all the boxes and pay only RM 3750 (7.5 x RM 500).
However, in reality, we cannot rent a 0.5 lorry.
It has to be a whole lorry, leaving the empty space unoccupied.
So, based on our graph, we will have to go for the 10 lorries of 30 boxes with the cost of RM 4000.
If we really think so, we are leaving money on the table!
Any point under the shaded area is the possible solution.
Luckily, not many are integer coordinates, i.e. a pair of integer numbers.
In order to find the coordinates, simply insert an integer value for either nLorry30 or nLorry40.
For example, we can find the value of nLorry30 when nLorry40 = 0 by inserting zero in place of nLorry40 and solving the equation as follows:
For solution of this Assignment Contact Us at We deliver original Work. [email protected] Integer Programming 1. Explain how the applications of Integer programming differ from those of linear programming. Give specific instances in which you would use an integer programming model rather than an LP model. Provide real-world examples. 2. Identify any challenges you have in setting up an…
Overviews: Definition of Integer Programming PROTOTYPE EXAMPLE Decision Variables Objective function Constraints Mathematical model of the problem Solving the problem Some BIP applications Package Problem Formulate Mathematical Model Fixed Investment Formulate Mathematical Model Site selection Formulate Mathematical Model Job Assignment Formulate Mathematical Model Innovative Uses of Binary…
After doing the MIT SICP lectures over the summer, I'm always amazed at how language development frequently attempts to implement functionality implementable in LISP. It seems like LISP really was a consummately expressive language, and it's frustrating sometimes to not have simple things I got used to when learning the functionals like Scheme and Haskell.
That aside, this macro capability is amazing, and it's illustrated here to quite the effect. The stupidly simple knapsack I implemented is done here in three lines that are much more semantically meaningful than my larger program. They macros are extended at run time into full expressions, and their data structure is much better for the solution of general knapsack problems.
My implementation would not quite win out in a diff battle.
This paper by Miles Lubin at MIT really demonstrates the ability of these macros to reduce general overhead and waste for IP/MIP. I'd really recommend a read, if you're interested in math programming. The more I learn about this language, the more I'm convinced it's posed, from an infrastructural perspective, to replace at least R. But, then again, I'm not getting my degree in telling futures to the present.
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality
Anya is LIVE right now
FREE
Free to watch • No registration required • HD streaming
TRANS- Max is a mathematical model that maximizes the overall service value of a route rather than attempting to minimize cost. Cost or distance is considered as a constraint on the extent of the route. The service value of a route is a function of the service values on nodes which are intersections of the street network, and the service values of the arcs connecting those nodes. Service values can be a function of the population, employment opportunities, and other measures of route attractiveness given their access to the bus route. The mathematical formulation of the TRANSMax model borrows from the structure of linear programming problems designed for both the traveling salesman problem and the vehicle routing problem. [...] This method serves to not only make the transit routing problem tractable by dividing the problem into smaller sub-problems, but it also provides a range of high-quality alternate routes for consideration in the decision making process.
The Transit Route Arc-Node Service Maximization problem, Kevin M. Curtin & Steve Biba (2010)
Final project topic selected finally. Closing all those tabs felt good.