{"id":5065,"date":"2024-01-05T09:33:31","date_gmt":"2024-01-05T09:33:31","guid":{"rendered":"https:\/\/nag.com\/?page_id=5065"},"modified":"2024-01-11T10:29:04","modified_gmt":"2024-01-11T10:29:04","slug":"mixed-integer-linear-programming","status":"publish","type":"page","link":"https:\/\/nag.com\/mixed-integer-linear-programming\/","title":{"rendered":"Mixed Integer Linear Programming (MILP) Solver"},"content":{"rendered":"\n
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Mixed Integer Linear Programming (MILP)<\/h1>\n \n
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At Mark 29.3, n<\/i><\/span>AG introduces a cutting-edge solver (nag_mip_handle_solve_milp<\/mtext><\/math><\/a>) designed specifically for addressing large-scale mixed-integer linear programming (MILP) problems. This marks a significant stride in n<\/i><\/span>AG\u2019s commitment to enhancing and broadening its offerings in the field of mathematical optimization.<\/p>\n

MILP finds widespread application across diverse industries, including but not limited to finance, manufacturing, logistics, transportation, and telecommunications. By accommodating both continuous and discrete decision variables, the solver empowers organizations to model practical and challenging problems, including resource allocation, scheduling, and network flow.<\/p>\n

Large-scale MILP problems of the form\u00a0<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

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\n<\/p>\n<\/mstyle> minimize<\/mtext><\/mstyle><\/mrow>x<\/mi> \u2208<\/mo> \u211d<\/mi><\/mrow>n<\/mi><\/mrow><\/msup><\/mrow><\/munder><\/mtd>c<\/mi><\/mrow>T<\/mi><\/mrow><\/msup>x<\/mi> <\/mtd> <\/div>\n<\/mtr> <\/mtr> subject to<\/mtext><\/mstyle> <\/mtd>l<\/mi><\/mrow>A<\/mi><\/mrow><\/msub> \u2264<\/mo> Ax<\/mi> \u2264<\/mo> u<\/mi><\/mrow>A<\/mi><\/mrow><\/msub>,<\/mo><\/mtd>\n<\/mtr> <\/mtr> <\/mtd>l<\/mi><\/mrow>x<\/mi><\/mrow><\/msub> \u2264<\/mo> x<\/mi> \u2264<\/mo> u<\/mi><\/mrow>x<\/mi><\/mrow><\/msub>,<\/mo> <\/mtd>\n <\/mtd><\/mtr><\/mtable>\n<\/mrow><\/math><\/td>

(1)<\/td><\/tr><\/table>\n\n\n
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\\( x \\) \u2208<\/mo><\/mo>\u00a0\\(P \\),<\/mo><\/p>\n <\/div>\n <\/div>\n<\/div>\n\n

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where \\( A \\in \\mathbb{R}^{m\\times n} \\), \\(l_{A}, u_{A} \\) \u2208 \u211d\\(^m\\), \\( c \\), \\( l_{x} \\), \\(u_{x}\\) \u2208 \u211d\\(^n\\)<\/mo><\/mo><\/mo>\u00a0 are the problem data, and \\( P = \\) \u2124\\(^{n_{int}}\\) \u00d7 \u00a0\u211d \\(^{n_{l}}\\) are quite ubiquitous in real-life applications. It\u2019s important to highlight that solving MILP problems poses a considerable challenge owing to their combinatorial nature, and in many practical scenarios, finding an exact solution within reasonable time may be very difficult. We are happy to present the latest addition to the n<\/i><\/span>AG Library and aim to help users to make decisions efficiently and accurately.<\/span><\/mo><\/p>\n

MILP Modelling and Solving<\/h3>\n

Model (1) covers many practical use cases. One of the distinctive features of MILP is its capability to model logical conditions such as implications or dichotomies.<\/p>\n

Take fixed charge modelling as an example. Economic activities frequently involve both fixed and variable costs. In these cases, a fixed cost \\(c_{fix} \\) only occurs when variable \\( y \\) is positive. Given a variable cost \\(c_{var}\\), the total cost when \\( y \\) ><\/mo>0<\/mn><\/mrow><\/math> is \\(c_{fix} \\) + \\(c_{var}y\\) and 0 otherwise. It\u2019s easy to observe that the cost function is nonlinear. By introducing binary variable \\(x\\) we can model it as the linear expression<\/p>\n

\u00a0<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>\n\n\n\n c<\/mi><\/mrow>fix<\/mi><\/mrow><\/msub>x<\/mi> +<\/mo> c<\/mi><\/mrow>var<\/mi><\/mrow><\/msub>y<\/mi>,<\/mo>\n<\/mrow><\/math>\n\n\n\n
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Where we add constraints<\/mrow><\/math><\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>\n\n\n\n 0<\/mn> \u2264<\/mo> y<\/mi> \u2264<\/mo> Mx<\/mi>,<\/mo><\/mtd>\n<\/mtr> <\/mtr> x<\/mi> \u2208<\/mo>{<\/mo>0<\/mn>,<\/mo>1<\/mn>}<\/mo>,<\/mo> <\/mtd><\/mtr><\/mtable>\n<\/mrow><\/math><\/td><\/tr><\/table>\n\n\n\n
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Where \\(M \\) is an upper bound of \\(y\\) and should be chosen as the tightest known bound, instead of an arbitrarily large \\(M \\). This modelling technique appears in applications such as facility location and network design.<\/p>\n

Another great usage of MILP is to model disjunctions. For example, when scheduling jobs on a machine, we might want to allow job \\(i\\) to be scheduled before job \\(j\\) or vice versa. Let \\(p_{i} \\) and \\(p_{j} \\) denote the processing time of job \\(i\\) and \\(j\\), while \\(t_{i}\\) and \\(t_{j}\\) represent their starting times. Then we will require at least one of the following constraints is satisfied.<\/p>\n

\u00a0<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n t<\/mi><\/mrow>j<\/mi><\/mrow><\/msub> \u2265<\/mo> t<\/mi><\/mrow>i<\/mi><\/mrow><\/msub> +<\/mo> p<\/mi><\/mrow>i<\/mi><\/mrow><\/msub>,<\/mo><\/mtd>\n<\/mtr> <\/mtr> t<\/mi><\/mrow>i<\/mi><\/mrow><\/msub> \u2265<\/mo> t<\/mi><\/mrow>j<\/mi><\/mrow><\/msub> +<\/mo> p<\/mi><\/mrow>j<\/mi><\/mrow><\/msub>,<\/mo><\/mtd><\/mtr><\/mtable>\n<\/mrow><\/math><\/td><\/tr><\/table>\n\n\n\n
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By introducing binary variables \\(x_{1} \\) and \\(x_{2} \\), the problem can be modelled as<\/mrow><\/msub><\/math><\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n

<\/div>\n\n\n\n x<\/mi><\/mrow>1<\/mn><\/mrow><\/msub> +<\/mo> x<\/mi><\/mrow>2<\/mn><\/mrow><\/msub> \u2265<\/mo> 1<\/mn>,<\/mo> <\/mtd>\n<\/mtr> <\/mtr> x<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>,<\/mo> <\/mi>x<\/mi><\/mrow>2<\/mn><\/mrow><\/msub> \u2208<\/mo>{<\/mo>0<\/mn>,<\/mo>1<\/mn>}<\/mo>,<\/mo> <\/mtd>\n<\/mtr> <\/mtr> t<\/mi><\/mrow>j<\/mi><\/mrow><\/msub> \u2265<\/mo> t<\/mi><\/mrow>i<\/mi><\/mrow><\/msub> +<\/mo> p<\/mi><\/mrow>i<\/mi><\/mrow><\/msub> \u2212<\/mo> M<\/mi>(<\/mo>1<\/mn> \u2212<\/mo> x<\/mi><\/mrow>1<\/mn><\/mrow><\/msub>)<\/mo>,<\/mo><\/mtd>\n<\/mtr> <\/mtr> t<\/mi><\/mrow>i<\/mi><\/mrow><\/msub> \u2265<\/mo> t<\/mi><\/mrow>j<\/mi><\/mrow><\/msub> +<\/mo> p<\/mi><\/mrow>j<\/mi><\/mrow><\/msub> \u2212<\/mo> M<\/mi>(<\/mo>1<\/mn> \u2212<\/mo> x<\/mi><\/mrow>2<\/mn><\/mrow><\/msub>)<\/mo>,<\/mo><\/mtd><\/mtr><\/mtable>\n<\/mrow><\/math><\/td><\/tr><\/table>\n\n\n\n
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where \\( M \\) again is positive and serves as a known bound.<\/span><\/p>\n

There are many other types of constraints that are very useful in the field of operations research, including but not limited to<\/p>\n