Scheduling in the Random-Order Model

Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive [Graham, 1966]. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201 [Fleischer and Wahl, 2000]. No deterministic online strategy can obtain a competitiveness smaller than 1.88 [Rudin III, 2001]. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting [Osborn and Torng, 2008]. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability

Minimizing value-at-risk in single-machine scheduling

The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the problem parameters is low. However, as variability increases accounting for this randomness explicitly in the model becomes crucial in order to counteract the ill effects of the variability on the system performance. In this paper, we consider single-machine scheduling problems in the presence of uncertain parameters. We impose a probabilistic constraint on the random performance measure of interest, such as the total weighted completion time or the total weighted tardiness, and introduce a generic risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) of the random performance measure at a specified confidence level. We propose a Lagrangian relaxation-based scenario decomposition method to obtain lower bounds on the optimal VaR and provide a stabilized cut generation algorithm to solve the Lagrangian dual problem. Furthermore, we identify promising schedules for the original problem by a simple primal heuristic. An extensive computational study on two selected performance measures is presented to demonstrate the value of the proposed model and the effectiveness of our solution method

Variable retort temperature optimization benefit in scheduling for retorts of different capacities in food canneries

In the majority of small- to medium-sized canneries, retorting is carried out in a battery of retorts as a batch process. For such canneries, the unloading and reloading operations for each retort are labor-intensive; therefore, a well-designed and well-managed plant should be utilized in order to optimize the whole sterilization process. In other words, it is necessary to develop a suitable mathematical model for the operation of the whole plant and to determine the optimal values of its decision variables. The result of such a model involves the quantities of each product to be loaded onto the retorts for each of the batches, and the optimal solution provides an optimum scheduling. On the other hand, it is well-known that variable retort temperature processing can be used for reducing the sterilization processing time required for sterilization using the traditional constant retort temperature processing. Therefore, the objective of this research consisted of utilizing a variable retort temperature processing in developing a mathematical model for scheduling at food canneries for the case of retorts of different capacities. The developed model was based on mixed-integer linear programming and simultaneous sterilization based on variable retort temperature processing. The adaptive random search algorithm coupled with penalty functions approach, and the finite difference method with cubic spline approximation are utilized in this study to obtain the simultaneous sterilization vectors to be processed under time-variable retort temperature. The proposed in this study methodology can be useful for small- and medium-sized food canneries, which work with many different products simultaneously

Compute-and-forward on a line network with random access

Signal superposition and broadcast are important features of the wireless medium. Compute-and-Forward, also known as Physical Layer Network Coding (PLNC), is a technique exploiting these features in order to improve performance of wireless networks. More precisely, it allows wireless terminals to reliably de- code a linear combination of all messages, when a superposition of the messages is received through the physical medium.\ud In this paper, we propose a random PLNC scheme for a local interference line network in which nodes perform random access scheduling. We prove that our PLNC scheme is capacity achieving in the case of one symmetric bi-directional session with terminals on both ends of this line network model. We demonstrate that our scheme significantly outperforms any other scheme. In particular, by eligibly choosing the access rate of the random access scheduling mechanism for the network, the throughput of our PLNC scheme is at least 3.4 and 1.7 times better than traditional routing and plain network coding, respectively