Discrete Optimization
The nuclear medicine production and delivery problem

https://doi.org/10.1016/j.ejor.2013.12.024Get rights and content

Highlights

  • A nuclear medicine production and delivery problem is defined and modeled.

  • A mixed integer programming model is developed.

  • A large neighborhood algorithm with improvement algorithms is proposed.

  • A benchmark problem set is developed.

  • Computational results including a case study show that the approach performs well.

Abstract

Half-life is a unique characteristic of radioactive substances used in a variety of medical treatments. Radioisotope F-18 used for diagnosing and monitoring many types of cancers has a half-life of 110 minutes. As such, it requires careful coordination of production and delivery by manufacturers and medical end-users. To model this critical production and delivery problem, we develop a mixed integer program and propose a variant of a large neighborhood search algorithm with various improvement algorithms. We conduct several computational experiments to demonstrate the effectiveness of the proposed approach. The method when applied in a case study shows that improvement in terms of both time and cost is possible in the production and delivery of F-18.

Introduction

To our knowledge, no previous work has considered the nuclear medicine production and delivery problems (NMPDP) posed by F-18, an isotope that requires scrupulous determination of manufacturing levels as well as accounting for its rapid deterioration. In this paper we develop a mixed integer programming (MIP) model and propose a variant of a large neighborhood search (LNS) algorithm with various improvement algorithms. We conduct several computational experiments to demonstrate the effectiveness of the proposed approach. We conclude that the approach will help manufacturers and the medical community produce and deliver F-18 considering both time and cost.

The remainder of this paper is organized as follows. The detailed problem is described in Section 2. A literature review is presented in Section 3. The MIP model and our proposed approach are presented in Sections 4 Mathematical model, 5 Solution method. Our experimental results are shown in Section 6. A case study is presented in Section 7 and our concluding remarks are presented in Section 8.

Section snippets

Problem description

The description below is based on a manufacturer that produces only radioisotope F-18 that we observed. Physicians order nuclear medicine, such as F-18, based on a specified level of radiation according to their injection plans for the medicine. The manufacture gathers a set of customer orders a day in advance, each of which corresponds to a hospital and perhaps multiple patients, and schedules delivery vehicles to visit each hospital. Each customer has a requested quantity of F-18 and time

Literature review

The nuclear medicine delivery problem resembles some perishable food and newspaper delivery problems. This section briefly reviews these studies and research efforts.

Tarantilis and Kiranoudis (2001) consider fresh milk distribution for a diary company in Athens, Greece, and propose a modified version of the threshold-acceptance algorithm of Dueck and Scheuer (1990), which is a deterministic version of the simulated annealing algorithm. Tarantilis and Kiranoudis (2001) treat the problem as a

Mathematical model

This section introduces an MIP model for scheduling the production and delivery of radioisotope F-18 to closely match demand. This paper uses the following notation:

Indices
i, jcustomer stop index, 1, 2,  , n; 0 = plant index (start location); n + 1 = plant index (final location)
vvehicle index, 1, 2,  , V
hvehicle tours index, 1, 2,  , H
kmachine index, 1, 2,  , K
pproduction run index, 1, 2,  , P

Parameters
dicustomer i demanded quantity
[ei, li]time window of customer i; li = the exact time the medicine is used
ckproduction

Solution method

LNS, a state-of-the-art meta heuristics, has been shown to perform well on various combinatorial optimization problems including vehicle routing problems (Kim et al., 2012, Kim et al., 2012, Pisinger and Ropke, 2010, Prescott-Gagnon et al., 2009, Ropke and Pisinger, 2006, Shaw, 1977, Wy and Kim, 2013). While local search heuristics attempt to improve solutions via incremental adjustments to current solutions, LNS perturbs a current solution significantly within a large neighborhood. Pisinger

Experimental results

We use the C++ language to implement the algorithms and an Intel Core(TM) i7 CPU 920 with 2.67 giga hertz, 6 gigabytes RAM running Windows 7 Enterprise for our eight scenarios. Because the previous research has not considered the NMPDP and there is no publicly available benchmark data, we develop 29 benchmark instances by extending the benchmark problem instances for the VRPTW (Solomon, 1987); we use the VRPTW benchmarks because VRPTW is an embedded sub-problem of the NMPDP and the optimal

Case study

This problem is motivated by a company producing and delivering radioisotope F-18.2 The problem instance has 277 customer stops and a total demand of 3672.1 mCi. The lower bound on the production quantity calculated by the method stated in the previous Section is 42242.9 mCi, which is more than 11.5 times the total demand. This large ratio is partially due to the long distances between the plant and the customers. While the current practice

Conclusions

This paper described a nuclear medicine production and delivery problem (NMPDP) in which the production quantity of the medicine, F-18, for a customer varied depending on when it was produced due to a half-life of 110 minutes, and the production time determines the vehicle’s earliest departure time from the plant to the customer. F-18’s half-life, a unique characteristic, makes this NMPDP different from other perishable product delivery, newspaper delivery, and integrated production and

Acknowledgments

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (No. 2012R1A1A2005243).

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