This article presents algorithms for solving a special case of the vehicle routing problem (VRP). We define our proposed problem of a special VRP case as a combination of two hard problems: the generalized assignment and the vehicle routing problem. The different evolution (DE) algorithm is used to solve the problem. The recombination process of the original DE is modified by adding two more sets of vectors—best vector and random vector—and using two other sets—target vector and trial vector. The linear probability formula is proposed to potentially use one out of the four sets of vectors. This is called the modified DE (MDE) algorithm. Two local searches are integrated into the MDE algorithm: exchange and insert. These procedures create a DE and MDE that use (1) no local search techniques, (2) two local search techniques, (3) only the exchange procedure, and (4) only the insert procedure. This generates four DE algorithms and four MDE algorithms. The proposed methods are tested with 15 tested instances and one case study. The current procedure is compared with all proposed heuristics. The computational result shows that, in the case study, the best DE algorithm (DE-4) has a 1.6% better solution than that of the current practice, whereas the MDE algorithm is 8.2% better. The MDE algorithm that uses the same local search as the DE algorithms generates a maximum 5.814% better solution than that of the DE algorithms.
This article proposes a methodology to resolve the advertising method selection problem (AdSP). The use of different advertising methods for the same product can generate different responses in terms of the product’s sales volume. Companies selling products have limited resources for advertising, with challenges such as budget and time constraints. It is necessary that the correct advertising method is selected in order to increase the maximum profit, given these limited resources. In the present study, a mathematical model was developed to represent the AdSP, and optimization software (OS) was utilized to optimally resolve it. However, a larger problem can prevent OS from optimally resolving the problem within a reasonable timeframe. To overcome this challenge, the authors developed a metaheuristic called the improved differential evolution algorithm (IDE), which combines three metaheuristics: (1) The differential evolution algorithm (DE); (2) the iterated local search (ILS); and (3) the adaptive large neighborhood search (ALNS). The performance of IDE reflects the best elements of these three methods. The computational results show that IDE can generate solutions that are similar to the optimal solutions obtained by OS while using 69.25% less computational time than OS. IDE improved the efficiency compared with the three original component methods. Moreover, IDE also found better solutions than those found by the original DE, ILS, and ALNS.
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