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Integer linear programming for mining systems
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Although the MIP formulation is easy to implement, it is the most intractable, and finding a feasible solution is NP-hard (Lerchs and Grosmann, 1965). Since the production scheduling problem is NP-hard, we have discretized it into a set of small sub-problems, and each sub-problem was solved sequentially, as discussed in Chatterjee and Dimitrakopulos (2020) and Chowdhury and Chatterjee (2014). Finally, the results of all sub-problems were combined to get the final solution of the main problem. Each sub-problem was solved sequentially by branch-and-cut algorithm, which is a combined method of branch-and-bound and cutting plane, discussed in this Chapter. The large optimization problem was written using ZIMPL software (https://zimpl.zib.de/), which creates the lp file, which can be used in any standard optimization solver. The IBM CPLEX solver (www.ibm.com/analytics/cplex-optimizer) is used in this case study; however, any other commercial or free solvers can be used for solving the problem. The list of available solvers can be found here (https://neos-server.org/neos/solvers/). Appendix 5 shows the steps involved in solving the MIP problem using ZIMPL and CPLEX.
More MILP models for hybrid flow shop scheduling problem and its extended problems
Published in International Journal of Production Research, 2020
Leilei Meng, Chaoyong Zhang, Xinyu Shao, Biao Zhang, Yaping Ren, Wenwen Lin
Mathematical model is the basis of understanding scheduling problems. It can describe the objectives, constraints and all the other characteristics of the scheduling problem clearly and serves as the key to mine scheduling knowledge and heuristic rules (Unlu and Mason 2010). Mathematical models can be solved with branch and bound method, dynamic search, and branch-and-cut among others (Naderi, Gohari, and Yazdani 2014). The branch-and-cut method is implemented in CPLEX of this paper and is used to solve MILP problems. The branch-and-cut method is the combination of cutting plane and branch-and-bound methods, which works by running a branch-and-bound algorithm and using cutting planes to tighten the linear programming relaxations (Meng et al. 2019b). The efficiency of the mathematical model depends on many factors such as the number of decision variables, the dimensionality of decision variables, the number of constraints and constraints’ tightness. As efficient mathematical models can solve problems fast and get better results within the same time, the construction of such models is of great significance.
Mathematical modelling and optimisation of energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines
Published in International Journal of Production Research, 2019
Leilei Meng, Chaoyong Zhang, Xinyu Shao, Yaping Ren, Caile Ren
Even though MILP models are not efficient solution algorithms, they are very useful in having insight into the scheduling problems and exploring effective heuristic methods for scheduling problems (Naderi, Gohari, and Yazdani 2014). The solution of MILP models depends on computer capacity and specified software. Owing to recent advances in computer capacity and advent of efficient software (e.g. CPLEX and GUROBI), the MILP model is attracting growing attention. MILP models can be solved by many methods such as cutting plane, branch-and-bound, dynamic programming, branch-and price-and branch-and-cut (Naderi, Gohari, and Yazdani 2014). The Cplex solver used in this paper is based on branch-and-cut method for solving MILP models. The branch-and-cut method is the combination of cutting plane and branch-and-bound methods, which works by running a branch-and-bound algorithm and using cutting planes to tighten the linear programming relaxations.
Reliability assessment for economic dispatch problem in the energy hub concept
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2018
Seyed Meisam Ezzati, Faramarz Faghihi, Hosein Mohammadnezhad Shourkaei, Seyed Babak Mozafari, Soodabeh Soleymani
MINLP is thus the simultaneous combination of The Mixed Integer Programming (MIP) and Nonlinear Programming (NLP). The LINDOGlobal as the MINLP solver in GAMS supports most mathematical functions such as non-smooth and discontinues in nonconvex, nonlinear and integer mathematical models. The solver procedure employs branch-and-cut methods to determine the optimal solution and breaks the problem into a number of sub-problems. It contains a multi-start feature that restarts solver from several intelligently generated points (choosing some distinct initial points to quickly find an appropriate local optimum). Moreover, its main settings are as follows: 1) Feasibility tolerance for nonlinear constraints is equal to 10−6, 2) Default number of iterations is 2 × 109, 3) Default running time is 1000 sec and 4) Tolerance for the gradients of nonlinear functions is equal to 10−7. The detailed data and options can be accessed in GAMS (2018).