Absolute Worst-case Ratio Research Articles

Two-agent scheduling on a single parallel-batching machine to minimize the weighted sum of the agents’ makespans

We schedule the jobs from two agents with equal processing times and non-identical job sizes on a single parallel-batching machine. The objective is to minimize the weighted sum of the two makespans of the jobs from two agents. We give a lower bound on the optimal solution, and present a polynomial-time algorithm H for solving the problem. Furthermore, we prove that both the absolute worst-case ratio and the asymptotic worst-case ratio of algorithm H are the functions of the weight of one agent, $$\alpha$$ . Specifically, (1) for the absolute worst-case ratio, $$H(I)\le f_1(\alpha )\times OPT(I)$$ , where $$f_1(\alpha )=(3-3\alpha +3\alpha ^2)/(2-4\alpha +4\alpha ^2)\le 9/4$$ ; (2) for the asymptotic worst-case ratio, $$H(I)\le f_2(\alpha ) \times OPT(I)+4/3$$ , where $$f_2(\alpha )=(11-11\alpha +11\alpha ^2)/(9-18\alpha +18\alpha ^2)$$ and $$11/9 \le f_2(\alpha )\le 11/6$$ . The effectiveness of algorithm H is demonstrated by using a set of computational experiments. The results show that algorithm H performs well in practice and tends to perform better when facing the large-scale instances.

Integrated scheduling of production and distribution for manufacturers with parallel batching facilities

We consider a class of integrated scheduling problems for manufacturers. The manufacturer processes job orders and delivers products to the customer. The objective is to minimize the service span, that is, the period lasting from the time when the order is received to the time when all the products have been delivered to the customer. In the production phase, parallel batch-processing facilities are used to process the jobs. Jobs have arbitrary sizes and processing times. Each facility has a fixed capacity and jobs are processed in batches with the restriction that the total size of jobs in a batch does not exceed the facility capacity. When all the jobs in a batch are completed, the batch is completed. In the distribution phase, the manufacturer uses a vehicle with a fixed capacity to deliver products. The transportation time from the manufacturer to the customer is a constant. Completed products can be delivered in one transfer if the total size does not exceed the vehicle capacity. We first consider the problem where jobs have the same size and arbitrary processing times. We propose approximation algorithms for the problem and we show that a worst-case ratio performance guarantee is respectively 2–1/m. Then we consider the problem where jobs have the same processing time and arbitrary sizes. An approximation algorithm is proposed with an absolute worst-case ratio of 13/7 and an asymptotic worst-case ratio of 11/9. Both the proposed algorithms can be executed in polynomial time.

Integrated scheduling on a batch machine to minimize production, inventory and distribution costs

We consider the problem of scheduling a set of jobs on a single batch-processing machine. Each job has a size and a processing time. The jobs are batched together and scheduled on the batch-processing machine, provided that the total size does not exceed the machine capacity. The processing time of the batch is the longest processing time among all the jobs in the batch. There is a single vehicle to deliver the final products to the customer. If the vehicle has not returned, completed batches will be put into the inventory. In this paper, we consider the problem of minimizing the production, delivery and inventory costs. We show that if the jobs have the same size, there is an O(nlog n)-time algorithm to find an optimal solution. If the jobs have the same processing time, there is a fast approximation algorithm with an absolute worst-case ratio less than 1.783 and an asymptotic worst-case ratio equal to 11/9. When the jobs have arbitrary sizes and arbitrary processing times, there is a fast approximation algorithm with absolute and asymptotic worst-case ratios less than or equal to 2, respectively.

Scheduling jobs with equal-processing-time on parallel machines with non-identical capacities to minimize makespan

We consider the problem of scheduling a set of equal-processing-time jobs with arbitrary job sizes on a set of batch machines with different capacities. A job can only be assigned to a machine whose capacity is not smaller than the size of the job. Our goal is to minimize the schedule length (makespan). We show that there is no polynomial-time approximation algorithm with an absolute worst-case ratio less than 2, unless P=NP. We then give a polynomial-time approximation algorithm with an absolute worst-case ratio exactly 2. Moreover, we give a polynomial-time approximation algorithm with asymptotic worst-case ratio no more than 3/2. Finally, we perform a computational experiment and show that our approximation algorithm performs very well in practice.

Two-machine flowshop scheduling with intermediate transportation under job physical space consideration

In this paper, we study a coordinated production–transportation scheduling problem in a two-machine flowshop environment where a single transporter may carry several jobs simultaneously as a batch between the machines. Each job has its own physical-space requirement while being loaded into the transporter. The goal is to minimize the makespan. For the jobs with the same size of physical space during transportation, we present a heuristic algorithm with an absolute worst-case ratio of 2 and a polynomial-time optimal algorithm for a special case with given job sequence. For the jobs having different size of physical storage space, a heuristic algorithm is constructed with an absolute worst-case ratio of 7/3 and asymptotic worst-case ratio of 20/9. Computational experiments demonstrate that the two heuristic algorithms developed are capable of generating near-optimal solutions quickly.

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem. Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.

Absolute approximation ratios for packing rectangles into bins

We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an algorithm with an absolute worst-case ratio of 2 for the case where the rectangles can be rotated by 90 degrees. This is optimal provided $\mathcal{P}\not=\mathcal{NP}$ . For the case where rotation is not allowed, we prove an approximation ratio of 3 for the algorithm Hybrid First Fit which was introduced by Chung et al. (SIAM J. Algebr. Discrete Methods 3(1):66---76, 1982) and whose running time is slightly better than the running time of Zhang's previously known 3-approximation algorithm (Zhang in Oper. Res. Lett. 33(2):121---126, 2005).

A note on minimizing makespan on a single batch processing machine with nonidentical job sizes

In a relatively recent paper (G. Zhang, X. Cai, C.Y. Lee, C.K. Wong, Minimizing makespan on a single batch processing machine with nonidentical job sizes, Naval Research Logistics 48 (2001) 226–240), authors considered minimizing makespan on a single batching machine having unit capacity. For the restricted version of the problem in which the processing times of the jobs with sizes greater than 1 / 2 are not less than those of jobs with sizes not greater than 1 / 2 , they proposed an O ( n log n ) algorithm with absolute worst-case ratio 3 / 2 . We propose an algorithm with absolute worst-case ratio 3 / 2 and asymptotic worst-case ratio ( m + 1 ) / m ( m ≥ 2 and integer) for a more general version in which the processing times of the jobs of sizes greater than 1 / m are not less than the remaining (the case of m = 2 has been considered by Zhang et al.). This general assumption is particularly held for those problem instances in which the job sizes and job processing times are agreeable. We obtain an O ( n log n ) algorithm with asymptotic worst-case ratio 4 / 3 for these problems leading to a more dependable algorithm than that of Zhang et al.

Bin packing problems with rejection penalties and their dual problems

In this paper we consider the following problems: we are given a set of n items {u1, … , un} and a number of unit-capacity bins. Each item ui has a size wi ∈ (0, 1] and a penalty pi ⩾ 0. An item can be either rejected, in which case we pay its penalty, or put into one bin under the constraint that the total size of the items in the bin is no greater than 1. No item can be spread into more than one bin. The objective is to minimize the total number of used bins plus the total penalty paid for the rejected items. We call the problem bin packing with rejection penalties, and denote it as BPR. For the on-line BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio arbitrarily close to 1.75 while the lower bound is 1.540. For the off-line BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 1.5, and an algorithm with an asymptotic worst-case ratio of 1.5. We also study a closely related bin covering version of the problem. In this case pi means some amount of profit. If an item is rejected, we get its profit, or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1. The objective is to maximize the number of covered bins plus the total profit of all rejected items. We call this problem bin covering with rejection (BCR). For the on-line BCR problem, we show that no algorithm can have absolute competitive ratio greater than 0, and present an algorithm with asymptotic competitive ratio 1/2, which is the best possible. For the off-line BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound.

A 3-approximation algorithm for two-dimensional bin packing

In the classical two-dimensional bin packing problem one is asked to pack a set of rectangular items, without overlap and without any rotation, into the minimum number of identical square bins. We give an approximation algorithm with absolute worst-case ratio of 3.

An approximation algorithm for square packing

We consider the problem of packing squares into bins which are unit squares, where the goal is to minimize the number of bins used. We present an algorithm for this problem with an absolute worst-case ratio of 2, which is optimal provided P≠NP.

Linear time-approximation algorithms for bin packing

Simchi-Levi (Naval Res. Logist. 41 (1994) 579–585) proved that the famous bin packing algorithms FF and BF have an absolute worst-case ratio of no more than 7 4 , and FFD and BFD have an absolute worst-case ratio of 3 2 , respectively. These algorithms run in time O(n log n) . In this paper, we provide a linear time constant space (number of bins kept during the execution of the algorithm is constant) off-line approximation algorithms with absolute worst-case ratio 3 2 . This result is best possible unless P=NP. Furthermore, we present a linear time constant space on-line algorithm and prove that the absolute worst-case ratio is 7 4 .

SCHEDULING FILE TRANSFERS UNDER PORT AND CHANNEL CONSTRAINTS

The file transfer scheduling problem was introduced and studied by Coffman, Garey, Johnson and LaPaugh. The problem is to schedule transfers of a large collection of files between various nodes of a network under port constraint so as to minimize the overall finishing time. This paper extends their model to include communication channel constraint in addition to port constraint. We formulate the problem with both port and channel constraints as a new type of edge-coloring of multigraphs, called an fg-edge-coloring, and give an efficient approximation algorithm with absolute worst-case ratio 3/2.