Focus : A Constraint for Concentrating High Costs

Abstract

Abstract. Many Constraint Programming models use integer cost variables aggregated in an objective criterion. In this context, some constraints involving exclusively cost variables are often imposed. Such constraints are complementary to the objective function. They characterize the solutions which are acceptable in practice. This paper deals with the case where the set of costs is a sequence, in which high values should be concentrated in a few number of areas. Representing such a property through a search heuristic may be complex and overall not precise enough. To solve this issue, we introduce a new constraint, FOCUS(X, yc, len, k), where X is a sequence of n integer variables, yc an integer variable, and len and k are two integers. To satisfy FOCUS, the minimum number of distinct sub-sequences of consecutive variables in X, of length at most len and that involve exclusively values strictly greater than k, should be less than or equal to yc. We present two examples of problems involving FOCUS. We propose a complete filtering algorithm in O(n) time complexity.

Three generalizations of the FOCUS constraint

“…B Solution satisfying FOCUS(X, [1,1], 5, 3), with a new machine rented for 5 days. C Practical solution violating FOCUS(X, [1,1], 5, 3), with a new machine rented for 3 days but not used on the second day bound max(x l ) of the domain D(x l ) of each x l ∈ X (this observation was made for FOCUS [12]). Thus, we propose a BC algorithm.…”

“…We thus have for each quantity two values for each variable x l . To aggregate them, we implement regret mechanisms directly derived from Propositions 1 and 2,, according to the parameters len and h. Line 4 is optional but it avoids some work when the variable y c is fixed, thanks to the same property as FOCUS (see [12]). Algorithm 2 performs a constant number of traversals of the set X.…”

“…We use the following notations from [12]. Definition 7 ([12]) Given an integer k, a variable x l ∈ X is:…”

“…∀x l ∈ X, x l > k ⇔ ∃s i,j ∈ S X such that l ∈ s i,j 3. ∀s i,j ∈ S X , j − i + 1 ≤ len FOCUS can be used in various contexts including cumulative scheduling problems where some excesses of capacity can be tolerated to obtain a solution [12]. In a cumulative scheduling problem, we are scheduling activities, and each activity consumes a certain amount of some resource.…”

“…In particular, several constraints have been proposed for obtaining well-balanced solutions [9,11,17]. Recently, the FOCUS constraint [12] was introduced to express the opposite notion. It captures the concept of concentrating the high values in a sequence of variables to a small number of intervals.…”