Measuring the outcome of movement-based three-way decision using proportional utility functions

The trisecting-acting-outcome (TAO) model of three-way decisions includes trisecting a universal set into three separate and closely connected regions, devising, and applying efficient strategies on the three regions, furthermore evaluating the outcome. This paper introduces the proportional utility function (PUF), representing the ratio between an object's initial and final quantity, to measure the outcomes from two different perspectives for movement-based three-way decision. The first perspective, is that if each object produces the same benefits or costs when they have the same movement (call region-independent), we sum up the three regions' utility as the overall outcome. The second scenario, is that each object generates a different cost or benefit, even if they have the same movements (call region-dependent). Here, finding an optimal investment plan is an essential matter based on their equivalent classification according to their specific characteristics. For a single equivalence class, we design a strategy to reach the goal under a specific investment, although this investment may be conservative. For all equivalence classes, we give one-time optimal investment plans using PUF based on invested or reserved resources in the movement-based three-way decision. Based on the program, we adopt a series of strategies in action under these investment budgets. The experimental results show that our strategy choices have practical significance and are in line with expected results.