Iterative Integer Research Articles

Iterative integer linear programming-based heuristic for solving the multiple-choice knapsack problem with setups

This paper studies the multiple-choice knapsack problem with setup (MCKS) which is a generalization of the knapsack problem with setup (KPS) where the items can be processed in multiple periods. The Integer Linear Programming (ILP) of the MCKS shows its limitations when solved using CPLEX 12.7 solver, and the existing algorithms VNS&IP and VND-LB from the literature outperform the ILP and provide the best-known solutions but solve to optimality only 29 out of 120 instances, respectively. This paper is dedicated to exposing an Iterative Integer linear programming-based heuristic (IILP-H) that deals with the MCKS. The IILP-H has been experimented on the MCKS benchmark instances. A sensitivity analysis of the MCKS parameters is provided. A comparison of the IILP-H with the upper bound obtained by the CPLEX 12.7 solver and the best state-of-the-art algorithms has been conducted. The numerical results show that the IILP-H outperforms all the existing solving techniques when it comes to solution quality and computation time. It reaches 120 out of 120 best solutions, including 77 out of 120 optimal and 69 new best solutions. The complexity of the MCKS and the nature of the solutions obtained by the IILP-H are studied regarding the number of periods to which a family is assigned.

On the Design of Iterative Approximate Floating-Point Multipliers

approximate multipliers provide power and area-saving for error-resilient applications. In this paper, we first propose two approximate floating-point multipliers based on two-dimensional pseudo-Booth encoding: floating-point pseudo-Booth (PB), and floating-point iterative pseudo-Booth (IPB). The accuracy of proposed multipliers can be tuned by three parameters: iteration, encoder's radix (R), and word length after truncation (W). Next, we developed the conventional iterative multipliers with a simplified steering circuit for their correction part to eliminate the power consumption of multipliers. The proposed iterative multipliers are compared with conventional iterative integer multipliers implemented by a simplified steering circuit for the floating-point area. The results reveal that the proposed PB-R4-W4 and IPB-R16-W19, compared to the exact floating-point multiplier, provide up to 98.9% and 67.5% reductions in power consumption, respectively, in TSMC 180nm CMOS technology. Also, their MRED values are, respectively, 2.9% and ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$7.4\times10^{-4}$</tex-math></inline-formula> )%. Finally, we evaluated the functionality of the proposed multipliers for real-life applications, including a hyper-plane classifier and two image processing applications of smoothing and sharpening.

A global satisfaction degree method for fuzzy capacitated vehicle routing problems

There are several uncertain capacitated vehicle routing problems whose delivery costs and demands cannot be estimated using deterministic/statistical methods due to a lack of available and/or reliable data. To overcome this lack of data, third–party information coming from experts can be used to represent those uncertain costs/demands as fuzzy numbers which combined to an iterative–integer programming method and a global satisfaction degree is able to find a global optimal solution. The proposed method uses two auxiliary variables α,λ and the cumulative membership function of a fuzzy set to obtain real–valued costs and demands prior to find a deterministic solution and then iteratively find an equilibrium between fuzzy costs/demands via α and λ. The performed experiments allow us to verify the convergence of the proposed algorithm no matter the initial selection of parameters and the size of the problem/instance.

Solving the Whistler-Blackcomb Mega Day Challenge

The Whistler-Blackcomb (WB) Mega Day challenge requires a skier to ride all 24 lifts at the resort in a single day. Among over two million people who ski annually at WB, only 313 completed the challenge in the 14 months following the introduction of a system that tracks lift use by skier. Apart from the physical challenge of skiing, a key difficulty is finding a route that matches one’s skill level while accounting for variable lift opening and closing times. We use data from WB’s radio-frequency identification (RFID) ticketing system to estimate ski times between lifts for skiers of various skill levels. We then formulate and solve the problem by a combined, iterative integer programming and heuristic approach, up to the highest feasible skier skill level. The problem’s distinctive features preclude the use of known solution methods for similar problems; therefore, we use a practical, staged-solution approach. Our results include a recommended route that enables the greatest number of skiers, roughly the fastest quartile, to achieve the challenge. We also provide a benchmark that skiers who can ski a particular common run in 12 minutes or less should be able to complete the challenge. In the three months following communication of our recommended solution, the rate at which skiers completed the Mega Day challenge increased by two-thirds over the previous seven skiing months.

Minimal functional routes in directed graphs with dependent edges

AbstractGraphs are mathematical structures used to model a set of objects and the relations between them. One of the basic concepts of graph theory, the path, has wide real‐world applications. In classic graph models, edges ending at a node are assumed to be independent. However, many real graphs/networks can only be correctly described by considering a dependency among nodes or edges. Paths in such graphs may not be functional if the conditional dependency is ignored. In this study, we investigate the routing problem in directed graphs with dependent edges represented by general graph models as alternatives to hypergraphs. We define a minimal functional route (MFR) as a minimal set of nodes and edges that can independently perform information transfer between two given nodes, and formulate the determination of MFRs as a graph search problem. A depth‐first‐search (DFS) top‐down algorithm, an iterative integer linear programming (ILP) bottom‐up algorithm, and a subgraph‐growing bottom‐up algorithm are devised subsequently to solve this problem. Numerical experiments verify the effectiveness of the algorithms. The defined MFR problem and the proposed algorithms are expected to find many practical applications.

Fast division using accurate quotient approximations to reduce the number of iterations

A class of iterative integer division algorithms is presented based on look-up table and Taylor-series approximations to the reciprocal. The algorithm iterates by using the reciprocal to find an approximate quotient and then subtracting the quotient multiplied by the divisor from the dividend to find a remaining dividend. Fast implementations can produce an average of either 14 or 27 b per iteration, depending on whether the basic or advanced version of this method is implemented. Detailed analyses are presented to support the claimed accuracy per iteration. Speed estimates using state-of-the-art ECL components show that this method is faster than the Newton-Raphson technique and can produce 53-b quotients of 53-b numbers in about 25 ns using the basic method and 21 ns using the advanced method. In addition, these methods naturally produce an exact remainder, which is very useful for implementing precise rounding specifications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>