Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among n discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location i takes ti time units and detects the hider—if hidden there—independently with probability [Formula: see text] for [Formula: see text]. The hider aims to maximize the expected time until detection, whereas the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, any optimal mixed hiding strategy hides in each location with a nonzero probability, and there exists an optimal mixed search strategy that can be constructed with up to n simple search sequences.

A retailer places orders periodically for items that are shipped by a wholesaler. Items that are not sold perish randomly and independently of one another, with the perish probability depending on the age class. We consider a first-in-first-out policy for depleting items. We model this problem as a Markov decision process with stochastic demand, unit holding, outdating and ordering costs, plus unit penalty costs for lost sales. We prove convexity for the penultimate period and show convexity may not hold any earlier. A dynamic program can be solved optimally for small instances. We introduce both a one-stage-lookahead heuristic and a heuristic which is a combination of two existing standard approaches, the newsvendor and periodic review models. For simulated data, we compare these heuristics to the optimal solution for small problem instances and to further lookahead policies for larger problem instances. We show that the two new heuristics achieve results close to optimal. Our numerical study, which includes real data from a large European retail chain, highlights that products perishing independently from each other strongly affect model behavior compared to existing approaches from the literature.

An attacker with homogeneous weapons aims to destroy a target via sequential engagements over a finite planning horizon. Each weapon, with an associated cost, has a nonzero probability of destroying the target. At each decision epoch, the attacker can allocate a salvo of weapons to increase its chances, however this comes at the increasing linear cost of allocating additional weapons. We assume complete information in that the target status (dead or alive) is known. The attacker aims to maximize its chances of destroying the target while also minimizing the allocation cost. We show that the optimal salvo size, which is a function of time and inventory levels, is monotonic nondecreasing in both variables. In particular, we show that the salvo size either stays the same or decreases by one when the inventory level drops by one. The optimal allocation can be computed by solving a nonlinear stochastic dynamic program. Given the computational burden typically associated with solving Bellman recursions, we provide a scalable linear recursion to compute the optimal salvo size and numerical results to support the main ideas.

Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among n discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location i takes t i time units and detects the hider-if hidden there-independently with probability q i , for i = 1, . . . , n. The hider aims to maximize the expected time until detection, while the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, the hider's optimal mixed strategy hides in each location with a nonzero probability, and the searcher's optimal mixed strategy can be constructed with up to n simple search sequences. We develop an algorithm to compute an optimal strategy for each player, and compare the optimal hiding strategy with the simple hiding strategy which gives the searcher no location preference at the outset.