13

In general, chance constraints are difficult to handle computationally. Given a fixed $x$, is it often very difficult to even compute the value of $\Pr[F(x,\xi)\leq0]$, making the incorporation of such a constraint into an optimization model quite troublesome. One possible approach is outlined in the 2006 paper by Nemirovksi and Shapiro entitled Convex ...


12

In addition to David M's excellent answer, another approach which sometimes can be useful is to use integer programming. Specifically, if your distribution of $\xi$ is discrete (e.g. taking a sample-average-approximation of uncertainty is a reasonable approach) and you are willing to make a big-M assumption on $F(x, \xi)$ then another way to model a chance ...


8

These two types of constraint are totally different in terms of their applications in modeling. In fact, the way of using these constraint types (based on your modeling approach) end up in two totally distinct problems each of which can be solved different solution approaches. In the following, I will try to explain where we need to implement each of the ...


8

Logical constraints do not involve probability, except perhaps for the implicit probability of one or zero. Chance constraints specify conditions (constraints) which must hold with a(t least) specified probability, which generally would not be one or zero. Chance constraints could include logical conditions, and potentially even be specified in terms of ...


7

You may be interested in the following paper because it uses chance-constrained programming and bi-objective optimization together in a transportation application: https://link.springer.com/article/10.1007/s10288-019-00429-7 I would suggest to do the followings for your problem: 1- If you have bi-linear terms in your formulation then try to linearize them ...


4

To solve stochastic programming models with integer recourse, there are some methods. Most stochastic programming textbooks cover these methods. For example, chapter 7 of Introduction to Stochastic Programming by Birge and Louveux covers these techniques. In particular, I suggest either using the integer L-shaped method or the progressive hedging algorithm (...


3

Couldn't you just introduce binary variables $x_{ij}$ and consider the sum $$ Pr\left(\sum_{i \in I}\sum_{j=0}^{\text{max}} \tilde{w}_{ij} x_{ij} \le C \right)\ge 1-\alpha $$ where max is chosen with respect to the capacity and the minimal appearing weight? If you want the elements chosen "in order", you could introduce constraints like $x_{ij} \geq x_{...


2

You can try a master problem of the form \begin{alignat*}{1} \min & \quad \gamma\\ \textrm{s.t.} & \quad \sum_{s=1}^{S}P_{s}Y_{s}\le\alpha\\ & \quad \gamma\ge\gamma_{T}\left[\sum_{s\in T}(1-Y_{s})-|T|+1\right]\quad\forall T\in\mathcal{T}\\ & \quad Y_{s}\in\left\{ 0,1\right\} \quad\forall s\in\left\{ 1,\dots,S\right\} \end{alignat*} where $\...


1

How about \begin{align}\min&\quad\gamma\\\text{s.t.}&\quad(-r^s)^\top X\leq \gamma + M Y_s \qquad s=1,\ldots,S\\&\quad\sum_{s=1}^SP_sY_s \leq \alpha\\&\quad \sum_{i=1}^nx_i=1\\&\quad Y_s\in\{0,1\}\end{align}


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