Mixed-integer Linear Programming Models in Supply-Chain and Reliability Management

这个项目的重点是双层优化。双层优化问题由两个问题组成，定义为领导者问题和跟随者问题，我们可以认为这两个问题是按顺序解决的。领导者应该先行一步，定义他们的变量，使他们的目标最大化。然后，跟随者将以领导者的变量作为参数来优化他们的问题。你可以认为这是跟随者对领导者的反应。追随者的变量包含在领导者的问题中，并将影响他们的目标。因此，当领导者进行优化时，他们必须确保他们问题中的跟随者变量与跟随者的反应相匹配，否则他们的解是不可行的。由于这种反应结构，双层优化在能源和交通部门等领域得到了应用。用于解决双层问题的最新技术之一是将跟随者的问题交换为最优所必需的约束。一种这样的方法称为值函数法，它确保领导者问题中的跟随者变量确实对跟随者是最优的。然而，值函数是一个复杂的函数，很难计算，并不总是显式地包含在领导者的问题中。该项目的目的是利用值函数方法来解决一个特定的两层问题，其中领导者的变量影响跟随者的目标，而不是他们的变量。本研究的目的是为这类问题设计一个基于值函数方法的有效算法，该算法应该比一般的双层解算器应用于这类问题的速度更快。

This project focusses on bilevel optimisation. Bilevel optimisation problems consist of two problems, defined as the leaders and the follower's problems and we can think of these two problems being solved sequentially. The leader shall go first, defining their variables such that they will maximise their objective. The follower, will then optimise their problem, with the leader's variables acting as parameters. One can think of this as the follower reacting to the leader. The follower's variables are contained within the leaders problem and will affect their objective. Hence, when the leader is optimising, they must ensure that the followers variables in their problem, will match the reaction of the follower, otherwise their solution is not a feasible one. Due to this reaction structure, bilevel optimisation has applications within, among others, the energy and transportation sectors. One of the state of the art techniques used to solve bilevel problems is to exchange the follower's problem for constraints which are necessary for optimality. One such method is known as the Value Function Approach, ensuring that the follower's variables in the leader's problem are indeed optimal for the follower.However, the value function is a complex function, difficult to evaluate, which cannot always be explicitly included in the leader's problem. The aim of this project is to utilise the Value Function Approach for a specific bilevel problem in which the leader's variables affect the follower's objective, but not their variables.The goal of this research is to conceive an efficient algorithm based on the Value Function Approach for this problem type, which shall be quicker than generic bilevel solvers applied to such problems.