Goal programming (GP) is a powerful method to solve multi-objective programming problems. In GP the preferential weights are incorporated in different ways into the achievement function. The problem becomes more complicated if the preferences are imprecise in nature, for example ‘Goal A is slightly or moderately or significantly important than Goal B’. Considering such type of problems, this paper proposes standard goal programming models for multi-objective decision-making, where fuzzy linguistic preference relations are incorporated to model the relative importance of the goals. In the existing literature, only methods with linear preference relations are available. As per our knowledge, nonlinearity was not considered previously in preference relations. We formulated fuzzy preference relations as exponential membership functions. The grades or achievement function is described as an exponential membership function and is used for grading levels of preference toward uncertainty. A nonlinear membership function may lead to a better representation of the achievement level than a linear one. Our proposed models can be a useful tool for different types of real life applications, where exponential nonlinearity in goal preferences exists. Finally, a numerical example is presented and analyzed through multiple cases to validate and compare the proposed models. A distance measure function is also developed and used to compare proposed models. It is found that, for the numerical example, models with exponential membership functions perform better than models with linear membership functions. The proposed models will help decision makers analyze and plan real life problems more realistically.

中文翻译：

---

具有线性和指数模糊偏好关系的目标规划模型

目标规划（GP）是解决多目标规划问题的强大方法。在 GP 中，优先权重以不同的方式合并到成就函数中。如果偏好本质上不精确，问题就会变得更加复杂，例如“目标 A 比目标 B 稍微或中度或显着重要”。考虑到此类问题，本文提出了用于多目标决策的标准目标规划模型，其中结合了模糊语言偏好关系来对目标的相对重要性进行建模。在现有文献中，只有具有线性偏好关系的方法可用。据我们所知，以前在偏好关系中没有考虑非线性。我们将模糊偏好关系表述为指数隶属函数。成绩或成就函数被描述为指数隶属函数，用于对不确定性的偏好水平进行分级。非线性隶属函数可能比线性隶属函数更好地表示成就水平。我们提出的模型可以成为不同类型的现实生活应用的有用工具，在这些应用中，目标偏好存在指数非线性。最后，给出了一个数值例子，并通过多个案例进行分析，以验证和比较所提出的模型。还开发了距离测量函数并用于比较建议的模型。发现，对于数值示例，具有指数隶属函数的模型比具有线性隶属函数的模型性能更好。