### Refine

#### Document Type

- Article (7)

#### Language

- English (7)

#### Keywords

- Scheduling (2)
- COVID-19 (1)
- Concave programming (1)
- Demand elasticity (1)
- Dynamic pricing (1)
- Fixed interval scheduling (1)
- Healthcare (1)
- Hotel revenue management (1)
- Interval graphs (1)
- Knapsack problem (1)

A single machine scheduling problem with assignable job due dates to minimize total late work has recently been introduced by Mosheiov, Oron, and Shabtay (2021). The problem was proved NP-hard in the ordinary sense, and no solution algorithm was proposed. In this note, we present two pseudo-polynomial dynamic programming algorithms and an FPTAS for this problem. Besides, we introduce a new single machine scheduling problem to minimize maximum late work of jobs with assignable due dates. We develop an O(n log n) time algorithm for it, where is the number of jobs. An optimal solution value of this new problem is a lower bound for the optimal value of the total late work minimization problem, and it is used in the FPTAS.

Braverman et al. [Math. Oper. Res. 41(1), (2016), pp. 352–376], introduce the problem Provision-after-Wait which is to find a stable (ey free) assignment of n patients to m hospitals, and their waiting times before admission, such that the social welfare is maximized, subject to a limited budget. Chan et al. [ACM Trans. Econ. Comput. 5(2), (2017), Article 12, pp. 12:1–12:36] focus on a natural case of d-ordered preferences, in which patients are ordered according to the differences of their values between consecutive hospitals. For this case, they provide a sophisticated proof of ordinary NP-hardness, reduce it to the problem called Ordered Knapsack, and develop a fully polynomial time approximation scheme for Ordered Knapsack. We present a simple proof that Ordered Knapsack is NP-hard, which implies NP-hardness of a more restrictive case of the original problem, and present an alternative fully polynomial time approximation scheme with a reduced run time by a quadratic factor of n, for a fixed m. A similar algorithm is developed to find a solution for which the social welfare is as high as for the optimal solution of Ordered Knapsack, and the budget limit can be exceeded by at most 1-ε times. We also present polynomial algorithms for the cases of Ordered Knapsack, in which the number of distinct input parameters is fixed.

In this paper we present a novel approach to the dynamic pricing problem for hotel businesses. It includes disaggregation of the demand into several categories, forecasting, elastic demand simulation, and a mathematical programming model with concave quadratic objective function and linear constraints for dynamic price optimization. The approach is computationally efficient and easy to implement. In computer experiments with a hotel data set, the hotel revenue is increased by about 6% on average in comparison with the actual revenue gained in a past period, where the fixed price policy was employed, subject to an assumption that the demand can deviate from the suggested elastic model. The approach and the developed software can be a useful tool for small hotels recovering from the economic consequences of the COVID-19 pandemic.

We study a problem of scheduling n jobs on machines of two types: in-house machines and third-party machines. Scheduling on in-house machines incurs no additional costs, while using third-party machines implies costs depending on their number and the time of usage. Each job has a fixed time interval for being processed which can be divided and allocated among several machines, as long as there is only one machine processing the job at any time. No machine can process more than one job at a time. Jobs can be rejected, and they are of different importance that is reflected in the weight of each job. The objective is to find a subset of the jobs and the number of third-party machines for any period of time so that the accepted jobs can be feasibly scheduled, the total weight of the accepted jobs is maximized, and the total machine usage costs does not exceed a given upper bound. We also study a similar problem in which the objective is to maximize the total time at which at least one job is processed. Both problems are encountered in situations in which certain activities with given start and completion times have to be serviced by human operators. Examples are air traffic control and the monitoring safe vehicle unloading. Other examples are the employment of subcontractors in agriculture, construction or transportation. We will present NP-hardness proofs, polynomial and pseudo-polynomial optimal algorithms and an approximation algorithm for these problems and their special cases. These problems admit graph-theoretical interpretations associated with finding independent sets and a proper vertex coloring in interval graphs.

A parallel machine schedule updating game with compensations and clients averse to uncertain loss
(2019)

There is a finite number of non-cooperating clients, who are averse to uncertain loss and compete for execution of their jobs not later than by their respective due dates in a parallel service eironment. For each client, a due date violation implies a cost. In order to address the minimization of the total scheduling cost of all clients as a social criterion, a game mechanism is suggested. It is designed such that no client has an incentive to claim a false due date or cost. The game mechanism allows the clients to move their jobs to complete earlier in a given schedule. However, they must compensate costs of those clients whose jobs miss their due dates because of these moves. Algorithmic aspects are analyzed. Furthermore, that determines an equilibrium of the considered game is suggested and embedded into the game mechanism. Computational tests analyze the performance and practical suitability of the resulting game mechanism.

Basic concepts and brief description of revenue management models and decision tools in the hotel business are presented. An overview of the relevant literature on dynamic pricing, forecasting methods and optimization models is provided. The main ideas of the authors’ customized revenue management method for the hotel business are presented.

We study a scheduling problem for a container block, in which there are incoming containers only. Container placement is served by two non‐passing stacking cranes based at the opposite sides of the container block. The same time is required for any crane to move between two adjacent bays of the container block and the same different time is required for any crane to perform any down‐and‐up operation related to container lifting at the pick‐up point or container lowering at the storage point. Containers are assigned to the cranes according to one of the following policies: (1) two fixed sequences policy where a container processing sequence is given for each crane, (2) dedicated crane policy where containers are preassigned to the cranes, (3) one fixed, one arbitrary sequence policy where a container processing sequence is given for one crane and it can be arbitrary for the other crane, (4) flexible policy where any container can be assigned to any crane at any time, and (5) global fixed sequence policy where the container sequence is given and the relative processing order of containers in this sequence must be preserved by any crane. The objective is to minimize the completion time of the latest operation. We show that the problem is polynomially solvable for policy 1 and, if the number of containers to be placed in the same bay is no more than a half of all containers, for policy 4. It is NP‐hard in the strong sense for policies 2 and 3. Approximation algorithms with guaranteed absolute and relative deviations from the optimum are devised for policies 4 and 5. The results translate for the case of outgoing containers only.