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# 数学代写|运筹学代写Operations Research代考|IMSE56 TYPES OF INTEGER PROGRAMMING PROBLEMS

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## 数学代写|运筹学代写Operations Research代考|TYPES OF INTEGER PROGRAMMING PROBLEMS

Integer programming problems can be classified into two types:

1. Linear integer programming problem-Here both the objective function and the constraints are linear.
2. Non-linear integer programming problem-Here either the objective function or one or more constraints or both are non-linear.
Based on the type of variables, we can classify integer programming problems as:
3. Zero-one problems-Here all the decision variables take the value of zero or one.
4. Pure integer problems-Here all the variables take integer values.
5. Mixed problems-Here the variables can be zero-one, integer or continuous.
We start with understanding and solving zero-one problems.

## 数学代写|运筹学代写Operations Research代考|ZERO-ONE PROBLEMS

Let us consider the zero-one knapsack problem defined as follows:
A person going for a trek has the problem of identifying items to take along with him. There are, say, six items and each has a weight and a utility value. The person wishes to take items such that the sum of the utility is maximized and also satisfying a weight restriction. The problem is:
$$\text { Maximize } \sum_{j=1}^n C_j X_j$$

Subject to
$$\begin{gathered} \sum_{j=1}^n A_j X_j \leq B \ X_j=0,1 \end{gathered}$$
where $C_j$ represents the utility of item $j$ and $A_j$ represents the weight of item $j$. If we want to restrict the number of items to $N$, we can write
$$\sum_{j=1}^n X_j \leq N$$
If item 1 represents a torch and item 2 represents battery, the situation that the person takes the battery if he takes the torch is modelled as:
$$X_2 \geq X_1$$
If item 3 represents candle and the person takes either the candle or the torch, we can model it as:
$$X_1+X_3 \leq 1 \text { or }(=1)$$
if the person decides to take one of the two.
Thus, the binary variables are very useful to model allocation problems and to model specific conditions such as either/or $k$ out of $n$.
In fact, pure integer programming problems can be modelled as zero-one problems.

## 数学代写|运筹学代写Operations Research代考|ZERO-ONE PROBLEMS

$$\text { Maximize } \sum_{j=1}^n C_j X_j$$

$$\sum_{j=1}^n A_j X_j \leq B X_j=0,1$$

$$\sum_{j=1}^n X_j \leq N$$

$$X_2 \geq X_1$$

$$X_1+X_3 \leq 1 \text { or }(=1)$$

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