# Piecewise Linear Approximations in MIP Models

In the past, I’ve written about piecewise linear approximations of functions of a single variable. (There are too many posts to list here. Just type “piecewise linear” in the search box of my blog if you want to find them.) Handling piecewise linear approximations of multivariable functions is a bit more intimidating. I’ll illustrate one …

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# Minimizing a Median

$$\def\xorder#1{x_{\left(#1\right)}} \def\xset{\mathbb{X}} \def\xvec{\mathbf{x}}$$A somewhat odd (to me) question was asked on a forum recently. Assume that you have continuous variables $$x_{1},\dots,x_{N}$$ that are subject to some constraints. For simplicity, I’ll just write $$\xvec=(x_{1},\dots,x_{N})\in\xset$$. I’m going to assume that $$\xset$$ is compact, and so in particular the $$x_{i}$$ are bounded. The questioner wanted to …

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# Rolling Horizons

I keep seeing questions posted by people looking for help as they struggle to optimize linear programs (or, worse, integer linear programs) with tens of millions of variables. In my conscious mind, I know that commercial optimizers such as CPLEX allow models that large (at least if you have enough memory) and can often solve …

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# Memory Minimization

As I grow older, I’m starting to forget things (such as all the math I ever learned) … but that’s not the reason for the title of this post. A somewhat interesting question popped up on Mathematics StackExchange. It combines a basic sequencing problem (ordering the processing of computational tasks) with a single resource constraint …

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# Enforcing Simultaneous Arrivals

I’m recapping here an answer to a modeling question that I just posted on a help forum. (Since it will now appear in two different places, let’s hope it’s correct!) The original poster (OP) was working on a routing model, in which vehicles (for which I will use and if needed as indices) are assigned …

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# Matching Ordering Is Not Always Easy

In some circumstances, you might want to build an optimization model containing two sets of variables, say and , and constrain them so that the sort order of each matches. That condition is easily expressed in logical terms: if and only if for all pairs with . Translating that into a mathematical programming model that …

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# Model Credibility

Someone asked an interesting question on a support forum recently. The gist was: “How do I confirm that my model is correct?” On the occasions that I taught simulation modeling, this was a standard topic. Looking back, I don’t recall spending nearly as much time on it when teaching optimization, which was a mistake on …

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# Formulating Optimization Models

Periodically, on OR Exchange and other forums, I encounter what are surely homework problems involving the construction of optimization models. “The Acme Anvil Corporation makes two types of anvils, blue ones and red ones. Blue anvil use 185 kg. of steel and have a gross revenue of \$325 each; red anvils …” Really? Does anyone …

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# Scheduling Instability

Fellow OR blogger Laura McLay recently wrote a post “in defense of model simplicity“, which is definitely worth the read. It contains a slew of links to related material. As I read it, though, my contrarian nature had me thinking “yes … as long as the model is not too simple”. A recent piece in …

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