Abstract:In the inventory routing problem (IRP) inventory management and route optimization are combined. The traveling salesman problem (TSP) is a special case of the IRP, hence the IRP is NP-hard. We investigate how other aspects than routing influence the complexity of a variant of the IRP. We first study problem variants on a point and on the half-line. The problems differ in the number of vehicles, the number of days in the planning horizon and the service times of the customers. Our main result is a polynomial ti… Show more
“…We note that Pinwheel Scheduling is neither known to be NP-hard nor in NP, although it is conjectured to be PSPACEcomplete. We also note that reductions to Pinwheel Scheduling are used in [1] to classify the complexity of some capacitated inventory routing problems. Next, we show that for α < 2, there is no α-approximation for min-max rftt on star graphs, assuming that Pinwheel Scheduling is intractable.…”
Section: Theorem 1 Min-max Rftt Is Np-hard On Star Graphsmentioning
confidence: 99%
“…So, on these days we do not have to use T 0 . Hence, we add 1 Now we can create a non-decreasing solution for any day with 2 j as its largest power of 2 from the synchronized solution, by taking the concatenation of T 0 , . .…”
We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min–avg minimizes the average tour cost and min–max minimizes the maximum tour cost over all days. For min–max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min–avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.
“…We note that Pinwheel Scheduling is neither known to be NP-hard nor in NP, although it is conjectured to be PSPACEcomplete. We also note that reductions to Pinwheel Scheduling are used in [1] to classify the complexity of some capacitated inventory routing problems. Next, we show that for α < 2, there is no α-approximation for min-max rftt on star graphs, assuming that Pinwheel Scheduling is intractable.…”
Section: Theorem 1 Min-max Rftt Is Np-hard On Star Graphsmentioning
confidence: 99%
“…So, on these days we do not have to use T 0 . Hence, we add 1 Now we can create a non-decreasing solution for any day with 2 j as its largest power of 2 from the synchronized solution, by taking the concatenation of T 0 , . .…”
We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min–avg minimizes the average tour cost and min–max minimizes the maximum tour cost over all days. For min–max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min–avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.
“…Permasalahan IRP adalah permasalahan kompleks yang masuk dalam kategori NP-hard karena permasalahan ini adalah kelas khusus dari permasalahan Travelling Salesman Problem (TSP) yang merupakan permasalahan NP-hard [13]. Ini berarti pula penyelesaian secara eksak permasalahan IRP dengan ukuran permasalahan nyata yang besar akan memerlukan waktu komputasi yang panjang untuk memperoleh solusi optimal yang diinginkan.…”
Produk segar yang mudah rusak (perishable) ialah produk yang mengalami penurunan nilai seiring dengan waktu sehingga dibutuhkan strategi pendistribusian dan penyimpanan yang tepat agar produk dapat terjual sebelum nilai produk menurun secara signifikan. Pada riset ini dikembangkan suatu metode solusi heuristik bagi masalah inventory routing berkala untuk pendistribusian dan pengelolaan persediaan produk segar mudah rusak dengan tujuan memaksimalkan pendapatan total yang dihasilkan. Pengembangan metode solusi heuristik ini menggunakan dua algoritma heuristik yaitu greedy dan random. Eksperimen terhadap kedua algoritma ini dilakukan menggunakan dua skenario pola demand konsumen akhir. Hasil eksperimen menunjukkan bahwa Algoritma Greedy mampu memberikan solusi yang lebih baik dibandingkan dengan Algoritma Random pada skenario pola demand menurun.
“…A study on the impact of aspects other than routing on the complaxity of the IRP is presented in Baller et al [2020]. The authors devise a polynomial time dynamic programming algorithm for the variant of the problem on the half-line with uniform service times and a planning horizon of 2 days.…”
In Vehicle Routing Problems (VRPs) the decisions to be taken concern the assignment of customers to vehicles and the sequencing of the customers assigned to each vehicle. Additional decisions may need to be jointly taken, depending on the specific problem setting. In this paper, after discussing the different kinds of decisions taken in different classes of VRPs, the class where the decision about when the routes start from the depot has to be taken is considered and the related literature is reviewed. This class of problems, that we call VRPs over time, includes the periodic routing problems, the inventory routing problems, the vehicle routing problems with release dates, and the multi-trip vehicle routing problems.
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