Asia-Pacific Journal of Operational Research

This paper presents a mathematical model, linking the classical Markov models for brand switching and models for product life cycles, to forecast competition analysis and market share. This integrated model can be used to forecast market shares of all competitors, and their market shares, including customers retained, customers gained from market growth, and customers gained from competitors over the product life cycle. Such information provides forecasters with valuable insight about their market positions. The model is generic and can be applied to different types of products and services, under different types and patterns of product life cycle curves. A numerical example on a typical mobile telecommunication industry is used to illustrate the application of the proposed approach.

This paper discusses vehicle routing problem involving split deliveries and pickups, in which the customer has pickup and delivery demands that can exceed the capacity of the vehicle, and one customer can be visited by no more than one vehicle or by one vehicle for no more than one time. The objective of the problem is to minimize total travel cost. A new heuristic algorithm, the competitive decision algorithm, is proposed to solve the problem. Computational experiments on the benchmark provided by Mitra are performed to evaluate our algorithm against the previously proposed method. The experiment shows that our algorithm provides the best solutions to a total of 110 problems whether or not the number of vehicles used is restricted to the minimum.

In this paper, we consider the problem of finding the sparsest solution to underdetermined linear systems. Unlike the literatures which use the ℓ₁ regularization to approximate the original problem, we consider the ℓ_2/3 regularization which leads to a better approximation but a nonconvex, nonsmooth, and non-Lipschitz optimization problem. Through developing a fixed point representation theory associated with the two thirds thresholding operator for ℓ_2/3 regularization solutions, we propose a fixed point iterative thresholding algorithm based on two thirds norm for solving the k-sparsity problems. Relying on the restricted isometry property, we provide subsequentional convergence guarantee for this fixed point iterative thresholding algorithm on recovering a sparse signal. By discussing the preferred regularization parameters and studying the phase diagram, we get an adequate and efficient algorithm for the high-dimensional sparse signal recovery. Finally, comparing with the existing algorithms, such as the standard ℓ₁ minimization, the iterative reweighted ℓ₂ minimization, the iterative reweighted ℓ₁ minimization, and iterative Half thresholding algorithm, we display the results of the experiment which indicate that the two thirds norm fixed point iterative thresholding algorithm applied to sparse signal recovery and large scale imageries from noisy measurements can be accepted as an effective solver for ℓ_2/3 regularization.

In this paper, we present a semi-proximal alternating direction method of multipliers (sPADMM) for solving 3-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed sPADMM for the step-length and the penalty parameter σ ∈ (0, +∞). In particular, if σ > 0 is smaller than a certain threshold and the first and third linear operators in the linear equation constraint are injective, then all the three added semi-proximal terms can be dropped and consequently, the convergent 3-block sPADMM reduces to the directly extended 3-block ADMM with .

We show the recent sufficient global optimality condition for the quadratic constrained bivalent quadratic optimization problem is equivalent to verify the zero duality gap. Then, based on the optimal parametric Lagrangian dual model, we establish improved sufficient conditions by strengthening the dual bound.

In this study, we consider a scheduling problem with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects on a single machine. We prove that the shortest processing time (SPT) rule is optimal for the makespan minimization problem, the sum of the θth power of job completion times minimization problem, and the total lateness minimization problem, respectively. For the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, we present heuristic algorithms (the worst-case bound of these heuristic algorithms are also given) according to the corresponding single machine scheduling problems without learning considerations. It also shows that the problems of minimizing the total tardiness, the total weighted completion time and the discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.

This paper studies the online hierarchical scheduling problem on two uniform machines with rejection. Two uniform machines M₁, M₂ run at the speeds of s ∈ (0, +∞), 1 separately; and they are provided with different capabilities. Each machine has a certain GOS level 1 or 2 and every job is also associated with a hierarchy 1 or 2. The job can only be assigned to the machine whose GOS level does not exceed the job's hierarchy. Preemption is permitted but idle is not introduced. Jobs come one by one over list. When a job arrives, it can be accepted and scheduled on some machine or rejected by paying its penalty. The objective is to minimize the sum of makespan yielded by accepted jobs and total penalties of all rejected jobs. For this problem, we propose a family of several online algorithms according to the range of s and the related lower bound is also obtained. These algorithms achieve optimal competitive ratio when s ∈ (0, 1) ∪ [1.618, +∞), but have a small gap between upper bound and lower bound in interval [1, 1.618).

In this paper, we consider the online-list scheduling on a single bounded parallel-batch machine to minimize makespan. In the problem, the jobs arrive online over list. The first unassigned job in the list should be assigned to a batch before the next job is released. Each batch can accommodate up to b jobs. For b = 2, we establish a lower bound 1 + γ of competitive ratio and provide an online algorithm with a competitive ratio of , where γ is the positive root of γ(γ + 1)² = 1. For b = 3, we establish a lower bound 1 + α of competitive ratio and provide an online algorithm with a competitive ratio of 2, where α is the positive root of the equation (1 + α)(1 + α²) = 2.

In this paper, we consider parallel-machine scheduling with past-sequence-dependent (p-s-d) delivery times and deteriorating maintenance. The delivery time of a job is proportional to its waiting time in the system. Each machine has a deteriorating maintenance activity, i.e., delaying the maintenance increases the time required to perform it. We consider three versions of the problem to minimize the total absolute deviation of job completion times, the total load on all the machines, and the total completion time. We develop polynomial-time algorithms to solve them.

In this paper, we consider the online scheduling on m identical machines in which jobs arrive over time. The goal is to determine a nonpreemptive schedule that minimizes the weighted makespan, i.e., the maximum weighted completion time of jobs. When m = 1, we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3. For the case in which m ≥ 1, and all jobs have a common processing time p > 0, we obtain a best possible online algorithm with a competitive ratio of .