To formulate a real-world optimization problem, it is sometimes necessary to adopt a set of non-linear terms in the mathematical formulation to capture specific operational characteristics of that decision problem. However, the use of non-linear terms generally increases computational complexity of the optimization model and the computational time required to solve it. This motivates the scientific community to develop efficient transformation and linearization approaches for the optimization models that have non-linear terms. Such transformations and linearizations are expected to decrease the computational complexity of the original non-linear optimization models and, ultimately, facilitate decision making. This study provides a detailed state-of-the-art review focusing on the existing transformation and linearization techniques that have been used for solving optimization models with non-linear terms within the objective functions and/or constraint sets. The existing transformation approaches are analyzed for a wide range of scenarios (multiplication of binary variables, multiplication of binary and continuous variables, multiplication of continuous variables, maximum/minimum operators, absolute value function, floor and ceiling functions, square root function, and multiple breakpoint function). Furthermore, a detailed review of piecewise approximating functions and log-linearization via Taylor series approximation is presented. Along with a review of the existing methods, this study proposes a new technique for linearizing the square root terms by means of transformation. The outcomes of this research are anticipated to reveal some important insights to researchers and practitioners, who are closely working with non-linear optimization models, and assist with effective decision making.
Ride-sharing has the potential of addressing many socioeconomic challenges related to transportation. The rising popularity of ridesharing platforms (e.g., Uber, Lyft, DiDi) in addition to the emergence of new applications like food delivery and grocery shopping which use a similar platform, calls for an in-depth and detailed evaluation of various aspects of this problem. Auction frameworks and mechanism design, have been widely used for modeling ride-sharing platforms. A key challenge in these approaches is preventing the involving parties from manipulating the platform for their personal gain which in turn, can result in a less satisfactory experience for other parties and/or loss of prot for the platform provider. We introduce a latent space transition model for ride-sharing platforms which drivers can exploit and predict the future supply of the drivers (i.e., available drivers) to their own advantage. Following, we propose a pricing model for ride-sharing platforms which is both truthful and individually rational based on Vickery auctions and show how we can manage the loss of revenue in this approach. We compare our predicting model and pricing model with competing approaches through experiments on New York City's taxi dataset. Our results show that our model can accurately learn the transition patterns of people's ride requests. Furthermore, our pricing mechanism forces drivers to be truthful and takes away any unfair advantage the drivers can achieve by bidding untruthfully. More importantly, our pricing model forces truthfulness without sacricing much prot unlike what is typical with second-price auction schemes.
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