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Game Theory
Published in Wayne Patterson, Cynthia E. Winston-Proctor, Behavioral Cybersecurity, 2020
Wayne Patterson, Cynthia E. Winston-Proctor
For any fixed value of p, the row player is sure that his or her average winnings are at least the minimum of these four functions evaluated for the chosen value of p. Thus, in this range for p, we want to find the p that achieves the maximum of this “lower envelope.” In geometry or linear programming, this lower envelope is also referred to as the convex hull. In this example, we can see that maximal value for the lower envelope occurs when p = 8/15. Because the only columns that intersect at the critical point are columns 3 and 4, we can also conclude that we can reduce the game to the second and third columns, therefore once again reducing us to the 2 × 2 game for which we already have a methodology for a solution.
Game Theory
Published in Wayne Patterson, Cynthia E. Winston-Proctor, Behavioral Cybersecurity, 2019
Wayne Patterson, Cynthia E. Winston-Proctor
For any fixed value of p, the row player is sure that his or her average winnings are at least the minimum of these four functions evaluated for the chosen value of p. Thus, in this range for p, we want to find the p that achieves the maximum of this “lower envelope.” In geometry or linear programming, this lower envelope is also referred to as the convex hull. In this example, we can see that maximal value for the lower envelope occurs when p = 8/15. Because the only columns that intersect at the critical point are columns 3 and 4, we can also conclude that we can reduce the game to the second and third columns, therefore once again reducing us to the 2 × 2 game for which we already have a methodology for a solution.
Improved algorithms for dynamic lot sizing problems with incremental discount
Published in Optimization Methods and Software, 2019
Now we consider the case where the procurement cost is stationary, i.e. Ks,j = Kj and ps,j = pj, for s = 1, … ,T. We use the line segments approach provided by Hwang and van den Heuvel [13]. Hwang and van den Heuvel [13] developed a line segments approach, a generalized version of the lines approach proposed by van Hoesel et al. [26], to solve the DLS problem with inventory bounds and the non-speculative procurement cost. The corresponding dynamic program formulation is given as where values rs and Dt are monotone in s and t, respectively, and π(s) is a subset of {T, … ,1}. The approach maintains the lower envelope of line segments. Given a set of linear functions with intercept and slope Dt for , define valid line segment for each line by the corresponding valid domain of , where the valid domains are given. The envelope is given by the minimum of the line segments . After determining envelop at each stage , the value of G(s) can be obtained by . To implement the line segments approach, one needs to carefully determine the valid domain of each line and to efficiently maintain the envelope with the dynamic programming recursion.