A fundamental theorem in discrete convex analysis states that a set function is M^♮-concave if and only if its conjugate function is submodular. This paper gives an alternative direct proof to this fact.

The fuzzy linear programming problem with triangular fuzzy numbers in its objective functions or constraints has been discussed by many scholars based on using Zadeh's decomposition theorem of fuzzy numbers and transforming it into some crisp linear programming problems. However, the existing methods and the results will be limited when the objective functions (or the constraint functions) of a fuzzy linear programming contain generalized fuzzy numbers. In this paper, we first investigate the approximate representation of the fully fuzzy constraints and the transformation theorem of the fully fuzzy linear programming problem by means of the definition of the extended LR-fuzzy numbers. At the same time, the fully fuzzy linear programming problem is solved by transforming it into a multi-objective linear programming problem under a new ordering of GLR-fuzzy numbers proposed in this paper. Finally, the results obtained are compared with the existing work, and some numerical examples are given.

While the algorithmic complexity is in general worse than the one of Tardos' original algorithms, the authors, motivated by the practicality of such methods, recently proposed a simplex-based variant that is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate. In this paper, we introduce a slight modification that circumvents the determination of the largest sub-determinant while keeping the same theoretical performance. Assuming that the coefficient matrix is integer-valued and the auxiliary problems are non-degenerate, the proposed algorithm is polynomial in the dimension of the input data and the largest absolute value of a sub-determinant of the coefficient matrix.

Deploying caches on a network is an effective way to reduce the amount of data transmitted in a network. Recently, in an academic backbone network such as SINET (the Science Information Network) in Japan, the amount of transmitted data has significantly increased. It is desired to design an efficient mechanism to allocate caches in an optimal way. In this paper, we begin by formulating a discrete optimization model to find a cache allocation that minimizes the total transmission cost. We then design two efficient algorithms to solve our proposed model. The first one makes use of the fact that a backbone network has small treewidth. The algorithm runs in polynomial time when the number of items is fixed and a graph has a bounded treewidth. The other one reduces the problem to the minimum-cost flow problem under the practical assumption that each item has at most one copy. This yields a polynomial-time combinatorial algorithm. Our numerical experiments on the real SINET network show that our algorithms can solve the cache placement problem efficiently in practice.

In this paper, we consider the inventory problem of a firm facing disruption probability in its supply chain which consists of multiple suppliers and multiple demand nodes. The firm wishes to minimize its total expected cost in a finite time horizon setting. In order to manage the supply chain disruption, we introduce flexibility into the supply chain network of our inventory management problem. The problem is formulated as a Markov decision process, and a state-dependent optimal threshold policy is derived. We show that the expected cost function is monotonic in the convex ordering of the demand distribution and that the optimal policy can be characterized with the ratio of the ordering cost and the disruption probability of supply. We also numerically demonstrate that the flexibility of the supply chain network reduces the total expected cost.