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General algebraic modeling system gams

What Is General Algebraic Modeling System (GAMS)?

The General Algebraic Modeling System (GAMS) is a high-level modeling system designed for mathematical optimization. It provides a powerful and comprehensive framework for formulating and solving complex optimization problems across various fields, including quantitative finance, engineering, economics, and operations research. GAMS allows users to express large-scale models in a concise, algebraic syntax, separating the model formulation from the underlying data and solution algorithms. This separation enhances model clarity, maintainability, and portability across different computing platforms and solvers.

History and Origin

GAMS originated in the 1970s as a research project at the World Bank by an economic modeling group, aiming to create a system that could combine mathematical algebra with traditional computer programming concepts to efficiently describe and solve optimization problems. This initiative was driven by the challenges of managing large, real-life economic models using existing tools. The development was funded by the International Bank for Reconstruction and Development, commonly known as the World Bank. In November 1987, the GAMS Development Corporation was founded, commercializing the system. The separation of model and data, along with the ability to handle sparse data structures, were key innovations that allowed GAMS to manage and solve large-scale problems effectively. The official website provides more details on the origins of the GAMS Development Corporation.7, 8

Key Takeaways

  • GAMS is a high-level modeling system for expressing and solving complex mathematical programming problems.
  • It supports various problem types, including linear programming, nonlinear programming, and mixed-integer programming.
  • GAMS separates the model definition from the data and the specific solver used, promoting flexibility and scalability.
  • It is widely applied in diverse sectors like finance, energy, agriculture, manufacturing, and supply chain.
  • The system facilitates sensitivity analysis and scenario planning for robust decision-making.

Formula and Calculation

GAMS itself is not a solver but a language for formulating optimization problems. These problems typically involve an objective function to be minimized or maximized, subject to a set of constraints. The general form of a mathematical optimization problem that GAMS can represent is:

Minimize or Maximize:

∑i∈Icixi\sum_{i \in I} c_i x_i

Subject to:

∑i∈Iajixi≤bj∀j∈J\sum_{i \in I} a_{ji} x_i \le b_j \quad \forall j \in J Li≤xi≤Ui∀i∈IL_i \le x_i \le U_i \quad \forall i \in I

Where:

  • (x_i) represents the decision variables to be determined.
  • (c_i) represents the coefficients of the objective function.
  • (a_{ji}) represents the coefficients in the constraints.
  • (b_j) represents the right-hand side values of the constraints.
  • (L_i) and (U_i) represent the lower and upper bounds for each variable (x_i).
  • (I) is the set of indices for variables.
  • (J) is the set of indices for constraints.

GAMS translates this algebraic formulation into a format readable by specialized external solvers, which then perform the complex calculations to find optimal solutions.

Interpreting the General Algebraic Modeling System (GAMS)

Interpreting GAMS involves understanding how a real-world problem is translated into its algebraic syntax and then deciphering the output from the chosen solver. GAMS provides a structured environment where users define sets, parameters, variables, and equations that collectively describe the optimization model. For instance, in a portfolio optimization scenario, sets might represent available assets, parameters could include expected returns and covariance, variables would represent the allocation to each asset, and equations would define the objective (e.g., maximizing return or minimizing risk) and constraints (e.g., budget, diversification limits).

Once a GAMS model is solved, the output typically includes the optimal value of the objective function, the optimal levels for all decision variables, and marginal values (or shadow prices) for the constraints. These marginal values indicate how much the objective function would improve if a particular constraint were relaxed by one unit, offering crucial insights for decision-making.

Hypothetical Example

Consider a company that produces two products, Product A and Product B, using two resources, Resource 1 and Resource 2.

  • Product A requires 2 units of Resource 1 and 1 unit of Resource 2. It yields a profit of $10 per unit.
  • Product B requires 1 unit of Resource 1 and 3 units of Resource 2. It yields a profit of $15 per unit.
  • The company has 100 units of Resource 1 and 120 units of Resource 2 available.

The company wants to maximize its total profit.

In GAMS, this problem could be formulated as follows:

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