Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.     

Economic application in a finite capacity multi-channel queue with second optional channel

We consider a finite capacity M/M/R queue with second optional channel. The interarrival times of arriving customers follow an exponential distribution. The service times of the first essential channel and the second optional channel are assumed to follow an exponential distribution. As soon as the first essential service of a customer is completed, a customer may leave the system with probability (1 − θ) or may opt for the second optional service with probability θ (0 ? θ ? 1). Using the matrix-geometric method, we obtain the steady-state probability distributions and various system performance measures. A cost model is established to determine the optimal solutions at the minimum cost. Finally, numerical results are provided to illustrate how the direct search method and the tabu search can be applied to obtain the optimal solutions. Sensitivity analysis is also investigated.     

Vacation policies for machine repair problem with two type spares

This paper studies the machine repair problem consisting of M operating machines with two types of spare machines (S = S₁ + S₂), and R servers (repairmen) who leave for a vacation of random length when there are no failed machines queuing up for repair in the repair facility. At the end of the vacation the servers return and operate two vacation policies. First, the servers take vacations repeatedly until they find the repair facility has at least one waiting failed machine in the queue. Second, the servers do not take a vacation again and remain idle until the first arriving failed machine arrives, which starts a busy period in the repair facility. For both policies, the servers have two service rates for repair-slow and fast. The matrix geometric theory is used to find the steady-state probabilities of the number of failed machines in the system as well as the performance measures. Some special cases are given. A direct search algorithm is used to simultaneously determine the optimal values of the number of two types of spares and the number of servers while maintaining a minimum specified level of system availability.     

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Two-dimensional cargo overbooking models

This paper introduces two-dimensional (weight and volume) overbooking problems arising mainly in the cargo revenue management, and compares them with one-dimensional problems. It considers capacity spoilage and cargo offloading costs, and minimizes their sum. For one-dimensional problems, it shows that the optimal overbooking limit does not change with the magnitude of the booking requests. In two-dimensional problems, the overbooking limit is replaced by a curve. The curve, along with the volume and weight axes, encircles the acceptance region. The booking requests are accepted if they fall within this region. We present Curve (Cab) and Rectangle (Rab) models. The boundary of the acceptance region in the Cab (resp. Rab) model is a curve (resp. rectangle). The optimal curve for the Cab model is shown to be unique and continuous. Moreover, it can be obtained by solving a series of simple equations. Finding the optimal rectangle for the Rab model is more challenging, so we propose an approximate rectangle. The approximate rectangle is a limiting solution in the sense that it converges to the optimal rectangle as the booking requests increase. The approximate rectangle is numerically shown to yield costs that are very close to the optimal costs.     

节能情形下制造企业动态批量生产策略与节能方式选择

随着诸多严厉的环保政策的实行,节能管理已成为我国高耗能制造企业运营管理中的一个重要内容。通过构建自我节能、节能效益分享和节能量保证三种节能方式下制造企业的动态批量生产决策模型,首先分析了给定节能方式下制造企业的最优动态批量生产策略,其次探讨了制造企业最优节能方式选择。结果表明,最优动态批量生产策略满足“零库存生产”规则。进一步的数值实验分析结果表明:面临前两种节能方式时,当节能服务公司与制造企业的节能投资成本系数比率和采用合同能源管理的预计节能率较小时,制造企业应选择后者,反之应选择前者;面临后两种节能方式时,当采用合同能源管理的预计节能率较小时,制造企业选择两者无差异,反之应选择前者。     

Hölder and Minkowski type inequalities for pseudo-integral

There are proven generalizations of the Hölder’s and Minkowski’s inequalities for the pseudo-integral. There are considered two cases of the real semiring with pseudo-operations: one, when pseudo-operations are defined by monotone and continuous function g, the second semiring ([a, b], sup, ⊙), where ⊙ is generated and the third semiring where both pseudo-operations are idempotent, i.e., ⊕ = sup and ⊙ = inf.     

Quasi-steady flight to quasi-steady flight transition in a windshear: Trajectory optimization and guidance

This paper is concerned with the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The take-off problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered. It is assumed that the power setting is held at the maximum value and that the aircraft is controlled through the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the absolute path inclination and a reference value, assumed constant. Two types of optimal trajectories are studied: type 1 is concerned with gamma recovery (recovery of the initial value of the relative path inclination); and type 2 is concerned with quasisteady flight recovery (recovery of the initial values of the relative velocity, the relative path inclination, and the relative angle of attack). The numerical results show that the type 1 trajectory and the type 2 trajectory are nearly the same in the shear portion, while they diverge to a considerable degree in the aftershear portion of the optimal trajectory.Next, trajectory guidance is considered. A guidance scheme is developed so as to achieve near-optimum quasi-steady flight recovery in a windshear. The guidance scheme for quasi-steady flight recovery includes three parts in sequence. The first part refers to the shear portion of the trajectory and is based on the result that this portion of the trajectory depends only mildly on the boundary conditions; therefore, any of the guidance schemes already developed for type 1 trajectories can be employed (for instance, variable gamma guidance). The second part (constant gamma guidance) refers to the initial aftershear portion of the trajectory and is designed to achieve almost velocity recovery. The third part (constant rate of climb guidance) refers to the final aftershear portion of the trajectory and is designed to achieve almost complete restoration of the initial quasi-steady state.While the shear guidance and the initial aftershear guidance employ constant gain coefficients, the final aftershear guidance employs a variable gain coefficient. This is done in order to obtain accuracy and prompt response, while avoiding oscillations and overshoots. The numerical results show that the guidance scheme for quasi-steady flight recovery yields a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, ensures the restoration of the initial quasi-steady state, and has good stability properties.This paper is based on Refs. 1 and 2.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Aircraft Company. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, for helpful discussions.     

Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems

Extended one-step schemes of exponential type are introduced for solving singularly perturbed Volterra integro-differential problems. These schemes are of order (m + 1), m = 0, 1, 2, …, when the perturbation parameter, ε, is fixed. These schemes have the property that if ε is of order h they reduced to first order of accuracy and optimal when ε → 0. Stability analysis of these schemes are presented. Numerical results and comparisons with other schemes are presented.     

Preemptive multiprocessor order scheduling to minimize total weighted flowtime

Consider m identical machines in parallel, each of which can produce k different product types. There is no setup cost when the machines switch from producing one product type to another. There are n orders each of which requests various quantities of the different product types. All orders are available for processing at time t = 0, and preemption is allowed. Order i has a weight w_i and its completion time is the time when its last requested product type finishes. Our goal is to find a preemptive schedule such that the total weighted completion time ∑_wiCi$\sum w_i{C}_i$ is minimized. We show that this problem is NP-hard even when all jobs have identical weights and there are only two machines. Motivated by the computational complexity of the problem, we propose a simple heuristic and show that it obeys a worst-case bound of 2 − 1/m. Finally, empirical studies show that our heuristic performs very well when compared with a lower bound of the optimal cost.