A note on fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

In the paper “Fault-free Mutually Independent Hamiltonian Cycles in Hypercubes with Faulty Edges” (J. Comb. Optim. 13:153–162, 2007), the authors claimed that an n-dimensional hypercube can be embedded with (n−1−f)-mutually independent Hamiltonian cycles when f≤n−2 faulty edges may occur accidentally. However, there are two mistakes in their proof. In this paper, we give examples to explain why the proof is deficient. Then we present a correct proof. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-233-002.     

A note on special optimal batching structures to minimize total weighted completion time

A batch is a subset of jobs which must be processed jointly in either serial or parallel form. The algorithmic aspects of the batching scheduling problems have been extensively studied in the literature. This paper presents necessary and sufficient conditions of the existence of optimal batch sequences for the single machine, batching, total weighted completion time scheduling problems on two batching ways: (1) all jobs form one batch; (2) each batch contains a single job. This kind of conditions can help us to recognize some special optimal schedules quickly. Research supported by NSFC (10671183).     

A note on optimal price,warranty length and production rate for free replacement policy in static demand markets

This note provides some extensions on optimal policies on price, warranty length and production rate proposed by Wu et al. (2009) [1] (Wu, C.C.; Chou, C.Y.; Huang C. Optimal price, warranty length and production rate for free replacement policy in the static demand market. Omega 2009; 37: 29–39.) More specifically, the assumptions of the original paper is extended in this note from the production rate being positive and the second derivative of the demand function with respect to price and warranty-length being negative to no restrictions on both.     

A note on accounting during inflation

The paper suggests that the extensive debate over the question of whether to use a consumer price index or a specific index (or replacement costs) to adjust historic accounts is approaching the problem in the wrong way. There are several relevant concepts of income which should be reported and the relevant index differs according to the income concept reported. It is proposed that companies should publish figures for four different concepts of income and that, where specific indices are required, these should be specific to each company.     

A note on online strip packing

In online strip packing we are asked to pack a list of rectangles one by one into a vertical strip of unit width, without any information about future rectangles. The goal is to minimize the total height of strip used. The best known algorithm is First Fit Shelf algorithm (Baker and Schwarz in SIAM J. Comput. 12(3):508–525, 1983), which has an absolute competitive ratio of 6.99 under the assumption that the height of each rectangle is bounded from above by one. We improve the shelf algorithm and show an absolute competitive ratio of without the restriction on rectangle heights. Our algorithm also beats the best known online algorithm for parallel job scheduling. Ye’s research supported by NSFC(10601048). Zhang’s research supported by NSFC(60573020).     

A note on “Khouja and Park,optimal lot sizing under continuous price decrease,Omega 31 (2003)”

Khouja and Park [1] analyze the problem of optimizing the lot size under continuous price decrease. They show that the classic EOQ formula can lead to far from optimal solutions and develop an alternative lot size formula using the software package Mathematica. This formula is more exact, but also more complicated. In this note, we study the net present value formulation of the model, and thereby gain an insight that leads to the proposal of a modified EOQ formula. The modified EOQ formula, albeit not as accurate, is a good alternative to the formula developed by Khouja and Park, especially if mathematical complexity may hamper implementation.     

On the equitable k *-laceability of hypercubes

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container C _k (u,v). A k-container is a k ^*-container if it spans all vertices of G. We define G to be a k ^*-laceable graph if there is a k ^*-container joining any two vertices from different partite sets. A k ^*-container C _k ^*(u,v)={P ₁,…,P _k } is equitable if ||V(P _i )|−|V(P _j )||≤2 for all 1≤i,j≤k. A graph is equitably k ^*-laceable if there is an equitable k ^*-container joining any two vertices in different partite sets. Let Q _n be the n-dimensional hypercube. In this paper, we prove that the hypercube Q _n is equitably k ^*-laceable for all k≤n−4 and n≥5. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. The work of H.-M. Huang was supported in part by the National Science Council of the Republic of China under NSC94-2115-M008-013.     

A comment on: A note on the EOQ-JIT relationship

In a recently published note in this journal the need for independent control of the production rate (R) and production time (L) was established in order to adapt the economic production quantity (EPQ) model in JIT settings. In this comment we aim to improve the relevance of EPQ in a JIT environment. We focus on two issues, namely, modelling the production rate as a decision variable and also deriving an expession for the optimal number of batches (n* ) for the raw materials and/or subcomponents needed.     

A note on anti-coordination and social interactions

This note confirms a conjecture of (Bramoullé in Games Econ Behav 58:30–49, 2007). The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than n ^1?? , where n is the number of nodes, and ? arbitrarily small, unless $\mathrm{P} = \mathrm{NP}$ . For the rather special case where each node has a degree of at most four, the problem is still APX-hard.     

A note on hierarchical scheduling on two uniform machines

This paper studies online hierarchical scheduling on two uniform machines with the objective to minimize makespan. Machines are provided with different capability, i.e., the one with speed s can schedule all jobs, while the other one with speed 1 can only process partial jobs. Optimal algorithms for any 0<s<∞ are given in the paper. For 0<s<1, it has a competitive ratio of min{1+s,1+\frac1+s1+s+s²}\min\{1+s,1+\frac{1+s}{1+s+s^{2}}\} . For s>1, the competitive ratio is min{\frac1+ss,1+\frac2s1+s+s²}\min\{\frac{1+s}{s},1+\frac{2s}{1+s+s^{2}}\} .     

A note on supply risk and inventory outsourcing

To fully accommodate the correlations between semiconductor product demands and external information such as the end market trends or regional economy growth, a linear dynamic system is introduced in this paper to improve the forecasting performance in supply chain operations. In conjunction with the generic Gaussian noise assumptions, the proposed state-space model leads to an expectation-maximisation (EM) algorithm to estimate model parameters and predict production demands. When the dimension of external indicators is high, principal component analysis (PCA) is applied to reduce the model order and corresponding computational complexity without loss of substantial statistical information. Experimental study on some real electronic products demonstrates that this forecasting methodology produces more accurate predictions than other conventional approaches, which thereby helps improve the production planning and the quality of semiconductor supply chain management.     

A note on corporate valuation using imprecise cost of capital

We compare the estimator for discount rates according to Elsner and Krumholz (EK) (J Bus Econ 83:985–1014, 2013) to the approach propagated in Breuer et al. (BFM) (Eur J Financ 20:568–594, 2014). While the EK estimator is derived analytically and implies a correcting factor by which the original arithmetic mean estimator is adjusted, the BFM approach is based on a simple ad hoc modification and recommends to truncate the time horizon for cash flow projection up to about N = 30 years. The BFM approach is reasonable, as the most relevant bias problems are implied by terminal value computations. Rather surprisingly, for our main simulation analysis based on real-world capital market data for 19 countries over the period 2008–2012, the EK estimator turns out inferior to the BFM approach. However, results depend on the kind of bias measure and on the issue of whether using total historical returns or only excess returns for evaluation purposes. In any case, our findings imply that a rather straightforward rule of thumb for valuation may be of value to practitioners: ‘stick to the simple arithmetic mean estimator without terminal value computation, but consider future cash flows up to a time horizon of about 30 years’.     

A note on the deterministic EPQ with partial backordering

In a recent paper published in Omega [Pentico DW, Drake MJ, Toews C. The deterministic EPQ with partial backordering: a new approach. Omega 2009; 37(3):624–36], the authors proposed an EPQ model with partial backordering. In the EPQ model, a critical value of backordering rate was developed, below which the optimal production policy is either to lose all sales or meet all demand, above which the optimal policy is to allow stockouts with partial backordering and meet fractional demand. However, there exists a flaw in the critical value of backordering rate, which will be amended in this note.     

A note on domination and total domination in prisms

Recently, Azarija et al. (Electron J Combin:1.19, 2017) considered the prism \(G \mathop {\square }K_2\) of a graph G and showed that \(\gamma _t(G \mathop {\square }K_2) = 2\gamma (G)\) if G is bipartite, where \(\gamma _t(G)\) and \(\gamma (G)\) are the total domination number and the domination number of G. In this note, we give a simple proof and observe that there are similar results for other pairs of parameters. We also answer a question from that paper and show that for all graphs \(\gamma _t(G \mathop {\square }K_2) \ge \frac{4}{3}\gamma (G)\), and this bound is tight.