Worst-case Performance Ratio Research Articles

The circular balancing problem

We propose a balancing problem with a minmax objective in a circular setting. This balancing problem involves the arrangement of an even number of items with different weights on a circle while minimizing the maximum total weight of items arranged on any half circle. Due to its generic structure, it may have applications in fair resource allocation schemes. We show the NP-hardness of the problem and develop polynomial-time algorithms when the number of distinct weights is a fixed constant. We propose for the general case a tight 7/6-approximation algorithm and show that it performs better than two existing algorithms designed for an equivalent problem in the literature. The worst-case performance ratio is derived through a linear combination of valid inequalities that are obtained from the problem definition, the properties of the proposed algorithm, and the optimal circular permutation structure. Furthermore, we formulate a more general problem of minimizing the maximum total weight of items on equally divided circular sectors and present its computational complexity and a tight approximation algorithm.

An improved approximation algorithm for scheduling monotonic moldable tasks

We are concerned with the problem of scheduling monotonic moldable tasks on identical processors to minimize the makespan. We focus on the natural case where the number m of processors as resources is fixed or relatively small compared with the number n of tasks. We present an efficient (3/2)-approximation algorithm with time complexity O(nmlog(nm)) (for m>n) and O(n2logn) (for m≤n). To the best of our knowledge, the best relevant known results are: (a) a (3/2+ϵ)-approximation algorithm with time complexity O(nmlog(n/ϵ)), (b) a fully polynomial-time approximation scheme for the case of m≥16n/ϵ, and (c) a polynomial-time approximation scheme with time complexity O(ng(1/ϵ)) when m is bounded by a polynomial in n, where g(·) is a super-exponential function. On the other hand, the novel general technique developed in this paper for removing the ϵ-term in the worst-case performance ratio can be applied to improving the performance guarantee of certain dual algorithms for other combinatorial optimization problems.

New exact and approximation algorithms for integrated production and transportation scheduling with committed delivery due dates and order acceptance

The problem of integrated production and transportation scheduling, commonly faced by make-to-order manufacturers under a commit-to-delivery business mode, is known to be strongly NP-hard. In this study, we propose two new exact algorithms to solve the problem with order acceptance decisions also taken into account. These algorithms can achieve polynomial or pseudo-polynomial running times in several practical situations. We also develop the first pseudo-polynomial time approximation scheme for this problem. It not only guarantees a worst-case performance ratio of (1+ϵ) for any fixed ϵ>0, but also achieves good computational performance.

Parallel batch scheduling: Impact of increasing machine capacity

Scheduling performance naturally improves with increased machine capacity, but the per-unit improvement typically decreases or even keeps unchanged with excessive capacity. We consider parallel batch scheduling on identical machines to minimize the makespan, with and without preemption. The machine capacity is the maximum number of jobs that a machine can process simultaneously. We quantitatively analyze the impact of capacity augmentation on the makespan in the form of increasing machine capacity. Considering machines costs, we need to determine the machine capacity that minimizes the weighted sum of the makespan and capacity costs. First, we obtain an upper bound of the ratio between the minimum makespans of two scheduling problems with different machine capacities. Second, noting the intractability of the scheduling problem without preemption, we analyze the upper bound of the ratio between the obtained makespans by heuristics of two scheduling problems with different machine capacities. Third, for the preemptive case, we develop a polynomial time algorithm to obtain the optimal machine capacity, and also present another polynomial time algorithm to obtain the optimal machine capacity as well as for bounding the approximation ratio of the non-preemptive case. Fourth, we design an approximation algorithm using machine capacity found in the preemptive case to yield a schedule for the non-preemptive case, and analyze the worst case performance ratio of the algorithm. This research provides new insights into performance improvement with increased machine capacity in parallel batch scheduling, and analyzes the trade-off between the makespan and capacity costs.

An improved algorithm for a two-stage production scheduling problem with an outsourcing option

We consider a two-stage production scheduling problem where each operation can be outsourced or processed in-house. For each operation in the same machine, the ratio of its outsourcing cost to its processing time is constant. The objective is to minimize the sum of the makespan and the total outsourcing cost. It is known that this problem is either polynomial time solvable or NP-hard according to the conditions of the ratios. Even though approximation algorithms for NP-hard cases had been developed, their tight worst-case performance ratios are still open. In this paper, we carefully analyze the approximation algorithms to identify their tight worst-case performance ratios for cases. In one case, we propose a new approximation algorithm with a better and tight worst-case performance ratio. In the process of analyzing the algorithm, we propose a technique utilizing nonlinear optimization.

Two-agent scheduling on mixed batch machines to minimise the total weighted makespan

This paper studies a two-agent scheduling problem on mixed batch machines in parallel. A mixed batch machine can process several jobs simultaneously as a batch, as long as the number of jobs in the batch does not exceed the machine capacity. The processing time of a mixed batch is the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective is to minimise the weighted sum of two agents' makespans. We present four approximation algorithms based on two strategies: the machine-centric strategy and the agent-centric strategy. For each strategy, a full batch longest processing time (FBLPT) rule and a longest processing time greedy (LPTG) rule are used. We conduct theoretical analyses based on the worst-case performance ratio to provide the provable guarantees on the performances of the algorithms, and simulation analyses based on randomly generated instances to evaluate the average performances of the algorithms. Furthermore, we verify the consistency between the theoretical and simulation results. The algorithms using agent-centric strategy perform better than ones using machine-centric strategy. Finally, we provide managerial insights for the problem by analysing the technological parameters of batches, importance of agents, and demand seasonality.

Minimizing maximum delivery completion time for order scheduling with rejection

We study an order scheduling problem with rejection, in which each order consists of multiple product types and each product type should be manufactured on a dedicated machine. The aim is to find a solution to minimize a linear sum of the maximum delivery completion time of the accepted orders and the total penalty of the rejected orders. Even if the delivery times of all orders are zero, the problem is shown to be binary $$\mathcal{NP}$$ -hard in the two-machine case and it is shown to be unary $$\mathcal{NP}$$ -hard when the number of machines is arbitrary. Three approximation algorithms are proposed and their worst-case performance ratios are analyzed. For the scenario where the number of machines is fixed, a pseudo-polynomial dynamic programming algorithm and a fully polynomial time approximation scheme are devised for it.

Approximation algorithms for makespan minimization on identical parallel machines under resource constraints

The problem of minimizing the makespan on parallel identical machines is considered in the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We establish a lower bound on the worst-case performance of any group technology algorithm, which schedules the composite jobs formed of the original jobs that require the same resource. A simple group technology algorithm is given such that in the worst case no group technology algorithm performs better. An algorithm for the two-machine case is presented which guarantees a tight worst-case performance ratio of 6/5.

Approximation Algorithms for a New Truck Loading Problem in Urban Freight Transportation

Motivated by urban freight transportation practices in China, we study an optimal truck loading problem in which a fixed cost and an additional cost that depends on the number of unloading points are associated with each truck used. The truck loading problem is modeled as a one-dimensional bin packing problem, where the cost of each bin is a convex fixed-plus-linear function of the number of items in the bin. The objective is to minimize the total cost of bins used. We develop an asymptotic polynomial time approximation scheme and an efficient approximation algorithm with an asymptotic worst-case performance ratio of 1.5 to tackle the problem. Our computational experiments indicate that the approximation algorithm performs well for certain order patterns, and also reveal several important insights on using the freight charge scheme.

Iterative job splitting algorithms for parallel machine scheduling with job splitting and setup resource constraints

This paper examines a parallel machine scheduling problem with job splitting and setup resource constraints for makespan minimization. Jobs can be split into multiple sections, and such sections can be processed simultaneously on different machines. It is necessary to change setups between the processes of different jobs on a machine, and the number of setups that can be performed simultaneously is restricted due to limited setup operators. To solve this problem, we propose a mathematical programming model and develop iterative job splitting algorithms that improve a feasible initial solution step by step, taking into account job splitting, setup times, and setup resources. We derive a worst-case performance ratio of the algorithms and evaluate the performance of the proposed heuristics on a large number of randomly generated instances. We finally provide a case study of piston manufacturing in Korea.

Two Parallel Machines Scheduling with Two-Vehicle Job Delivery to Minimize Makespan

A problem of parallel machine scheduling with coordinated job deliveries is handled to minimize the makespan. Different jobs call for dissimilar sizes of storing space in the process of transportation. A range of jobs of one customer in the problem have priority to be processed on two identical parallel machines without preemption and then delivered to the customer by two vehicles in batches. For this NP-hard problem, we first prove that it is impossible to have a polynomial heuristic with a worst-case performance ratio bound less than 2 unless P = NP. Thereafter, we develop a polynomial heuristic for this problem, the worst-case ratio of which is bounded by 2.

Mixed batch scheduling on identical machines

This paper studies a new mixed batch scheduling problem that arises in vacuum heat treatment. A mixed batch machine can process at most a given number of jobs simultaneously. The processing time of a batch is the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective is to minimize the makespan. We first prove that the problem on a single machine can be solved in polynomial time, while the problem on multiple identical machines is NP-hard. Then, we develop a pseudopolynomial time exact algorithm when the number of machines is fixed. Further, we analyze the worst-case performance ratio of a full batch longest processing time algorithm and design Algorithm LPT-Greedy with improved worst-case performance.

Environmental efficiency in the agricultural sector of Latin America and the Caribbean 1990–2015: Are greenhouse gas emissions reducing while agricultural production is increasing?

The paper explores the relationship between production and greenhouse gas (GHG) emissions in the agricultural sector of Latin America and the Caribbean (LAC) between 1990 and 2015. By appraising how GHG emissions have changed with respect to agricultural production (elasticity), we are able to classify countries into six decoupling states. While the decoupling elasticity provides information for assessing the evolution within countries, we introduce a performance ratio for country comparison by calculating GHG emissions over agricultural production for 2015 and using the global distribution as benchmark. The information from both measurements serves to identify various country and sub regional situations. The findings illustrate that countries with performance ratios in the top 25% and under the best-case elasticity scenario (of strong decoupling) are not necessarily the same as the ones usually identified under factor productivity analyses, suggesting that environment-specific policy incentives and tools play a key role in enhancing sustainable agricultural production. Another major finding is that very few countries in the region (and worldwide) are able to offset agriculture emissions with land use-related carbon sequestration practices. Lastly, the evidence on small Caribbean island-states reporting the worst-case elasticity scenario (of strong negative decoupling) and performance ratios close to the world median, further supports the need for development plans related to climate change mitigation and adaptation in this sub region.

Minimizing the makespan on two identical parallel machines with mold constraints

This paper deals with the scheduling problem of minimizing the makespan on two identical parallel machines with molds as resource constraints. Two or more jobs with the same mold requirement cannot be processed on the same or different machines at the same time. While quite common in practice, like in wafer fabrication, such problems have not received much attention in the literature. Since this problem is NP-hard, a mathematical model for solving it is derived and tested. Moreover, three simple, fast, and effective heuristics are proposed with a worst-case performance ratio of 3/2. Computational results show that the proposed heuristics can efficiently obtain good solutions for problem instances containing a large number of jobs. In addition, the use of these proposed algorithms as initial solution can improve the performance of metaheuristics. Furthermore, the solutions obtained by these proposed heuristics can be improved by metaheuristics.

Better Algorithm of Ordinal Online Schedule for Jobs with Similar Sizes on Two Machines

Ordinal online schedule for jobs with similar sizes in on two parallel machines system is considered. Firstly it is proved that the worst case performance ratio of the existing algorithm P2 cannot be improved even if the job processing times are known in for any . Then a better algorithm named S is developed and its worst case performance ratio is given for .

<i>LPT</i> Algorithm for Jobs with Similar Sizes on Three Machines

In this paper, LPT (largest processing time) algorithm is considered for scheduling jobs with similar sizes on three machines. The objective function is to minimize the maximum completion time of all machines. The worst case performance ratio of the LPT algorithm is given as a piecewise linear function of r if job sizes fall in [1, r]. Our result is better than the existing result. Furthermore, the ratio given here is the best. That means our result cannot be improved any more.

Two-machine flow-shop scheduling with equal processing time on the second machine for minimizing total weighted completion time

We consider the classical two-machine flow-shop scheduling for minimizing the total weighted completion time. For this problem, the computational complexity of a version in which the jobs have a common processing time on the second machine, has not been addressed. We show that the problem is unary NP-hard, answering an open problem posed in Zhu et al. (2016). Then we present an approximation algorithm for the problem with worst-case performance ratio at most 2.

Network Resilience and the Length-Bounded Multicut Problem

Motivated by networked systems in which the functionality of the network depends on vertices in the network being within a bounded distance T of each other, we study the length-bounded multicut problem: given a set of pairs, find a minimum-size set of edges whose removal ensures the distance between each pair exceeds T . We introduce the first algorithms for this problem capable of scaling to massive networks with billions of edges and nodes: three highly scalable algorithms with worst-case performance ratios. Furthermore, one of our algorithms is fully dynamic, capable of updating its solution upon incremental vertex / edge additions or removals from the network while maintaining its performance ratio. Finally, we show that unless NP ⊆ BPP, there is no polynomial-time, approximation algorithm with performance ratio better than Omega (T), which matches the ratio of our dynamic algorithm up to a constant factor.

Fast algorithms for fragmentable items bin packing

Bin packing with fragmentable items is a variant of the classic bin packing problem where items may be cut into smaller fragments. The objective is to minimize the number of item fragments, or equivalently, to minimize the number of cuts, for a given number of bins. Models based on packing fragmentable items are useful for representing finite shared resources. In this article, we present improvements to approximation and metaheuristic algorithms to obtain an optimality-preserving optimization algorithm with polynomial complexity, worst-case performance guarantees and parametrizable running time. We also present a new family of fast lower bounds and prove their worst-case performance ratios. We evaluate the performance and quality of the algorithm and the best lower bound through a series of computational experiments on representative problem instances. For the studied problem sets, one consisting of 180 problems with up to 20 items and another consisting of 450 problems with up to 1024 items, the lower bound performs no worse than 5 / 6. For the first problem set, the algorithm found an optimal solution in 92 % of all 1800 runs. For the second problem set, the algorithm found an optimal solution in 99 % of all 4500 runs. No run lasted longer than 220 ms.

Network Resilience and the Length-Bounded Multicut Problem

Motivated by networked systems in which the functionality of the network depends on vertices in the network being within a bounded distance T of each other, we study the length-bounded multicut problem: given a set of pairs, find a minimum-size set of edges whose removal ensures the distance between each pair exceeds T . We introduce the first algorithms for this problem capable of scaling to massive networks with billions of edges and nodes: three highly scalable algorithms with worst-case performance ratios. Furthermore, one of our algorithms is fully dynamic, capable of updating its solution upon incremental vertex / edge additions or removals from the network while maintaining its performance ratio. Finally, we show that unless NP ⊆ BPP , there is no polynomial-time, approximation algorithm with performance ratio better than $Ømega (T)$, which matches the ratio of our dynamic algorithm up to a constant factor.