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Mathematical optimization to improve cows' artificial insemination services
Abstract
Engineering tools and mathematical optimization are applied in this study to plan the work of the agents of the cow artificial insemination service (inseminator) in Israel. Time is crucial in insemination as the chances of conception decline with increasing delay between the start of estrus and insemination. About 1,090 artificial inseminations of cows are performed daily in Israel. They involve 412 farms in 283 villages, and are performed by 29 inseminators; the work plan should balance the work load among the inseminators. To this end, the working time of an inseminator in each village is required. Thus, a model to predict the working time in a village was developed. Subsequently, a mathematical optimization model was designed and solved, which aims to allocate customers to trips and to determine the itinerary of each trip to minimize total distance/time. The main benefits included a 21.4% reduction in total traveling time and a 55% reduction in the difference between the lengths of the longest and shortest working days. Moreover, the longest delay in reaching an estrous cow is reduced from 7.6 to 5.9 h (i.e., by 1.7 h), which may increase the conception ratio by some 7%. In addition, the trade-off between work balance and total traveling time was studied.
Objetivou-se identificar e quantificar indicadores-referência (benchmark) em sistemas de produção de leite da agricultura familiar no Agreste de Pernambuco. Utilizou-se uma amostragem de 30 propriedades, sendo os dados coletados mensalmente durante um período de 12 meses. A partir dos coeficientes de correlação obteve-se dez indicadores-referência, os quais foram avaliados em quatro cenários de taxa de remuneração do capital investido (4, 6, 8 e 10 % ao ano). Verificou-se que a produção média de leite, produção de leite/total do rebanho, produção de leite/vacas em lactação, produção de leite/mão de obra e produção de leite/área foram correlacionados positivamente com a taxa de remuneração do capital investido na atividade leiteira. Foi verificada correlação negativa entre os indicadores custo operacional efetivo, custo operacional total, custo total em relação ao preço do leite e gasto com concentrado em relação à renda bruta do leite. A identificação e quantificação de indicadores-referência correlacionados com a rentabilidade pode determinar os pontos frágeis da atividade leiteira e auxiliar na correção, promovendo atratividade e sustentabilidade da agricultura familiar.
This work emerged from the need to better plan the daily work of 29 travelling service agents, who provide 1090 services to 412 customers in 283 sites, on daily average. A handy and flexible tool was developed and is presented herein. A major contribution of this study is an explicit consideration of the multi-dimensional nature of the problem by the inclusion of workload balancing, which may stand in conflict to cost minimisation. Further, the geographical distribution of the demand is highly irregular. Therefore, two load measures are required and balanced. This required to fitting a proper planning scheme. The planning tool has been applied successfully by the commercial service provider. Improvements in the order of 20% and more were obtained in key performance measures. Moreover, cost reductions, service improvement and load balance were obtained simultaneously: the standard deviations of the service times and working day’s duration were reduced by 18 and 58%, respectively. This enables to reduce the number of agents with no significant harm in performance. Additional practical advantages of the proposed tool are also discussed and demonstrated, for example, the ability to cope with geographical distributions and the flexibility to respond to daily variations in demand.
The authors consider heuristic algorithms for vehicle routing, comparing techniques of G. Clarke and J. Wright, B. Gillett and L. Miller, and M. Tyagi, and presenting modifications and extensions which permit problems involving hundreds of demand points to be solved in a matter of seconds. In addition, a multidepot routing algorithm is developed. The results are illustrated with a routing study for an urban newspaper with an evening circulation exceeding 100,000.
This paper examines a class of asymmetrical multi-depot vehicle routing problems and location-routing problems, under capacity or maximum cost restrictions. By using an appropriate graph representation, and then a graph extension, the problems are transformed into equivalent constrained assignment problems. Optimal solutions are then found by means of a branch and bound tree. Problems involving up to 80 nodes can be solved without difficulty.
A heuristic approach is presented for solving the multiterminal delivery problem with constraints on the system and probabilistic demands. The algorithm is basically the same whether a commodity is being collected or delivered. This is a practical method that results in “near optimal” solutions to a difficult problem.
This paper considers the design and analysis of algorithms for the multi-depot vehicle routing problem with time windows (MDVRPTW). Given the intrinsic difficulty of this problem class, approximation methods of the type cluster first, route second (two-step approaches) seem to offer the most promise for practical size problems. After describing six heuristics for the cluster part (assignment of customers to depots) an initial computational study of their performance is conducted. Finding, as expected, that the heuristics with the best results (in terms of the routing results) are those with the largest computational efforts.Journal of the Operational Research Society (2002) 53, 977984. doi:10.1057/palgrave.jors.2601426
The relationship between profitability and efficiency in a production line is discussed at length for the continuous production model. We prove that a solution which maximizes efficiency will not necessarily maximize profit. Several useful relationships between profitability and efficiency are developed which can greatly enhance computational efficiency. A solution procedure is developed which will determine the optimal number of work stations by maximizing an assumed profit function. An example is provided to illustrate these relationships and the entire solution procedure.
The complexity of a class of vehicle routing and scheduling problems is investigated. We review known NP-hardness results and compile the results on the worst-case performance of approximation algorithms. Some directions for future research are suggested. The presentation is based on two discussion sessions during the Workshop to Investigate Future Directions in Routing and Scheduling of Vehicles and Crews, held at the University of Maryland at College Park, June 4–6, 1979.
Heuristic programming algorithms frequently address large problems and require manipulation and operation on massive data sets. The algorithms can be improved by using efficient data structures. With this in mind, we consider heuristic algorithms for vehicle routing, comparing techniques of Clarke and Wright, Gillett and Miller, and Tyagi, and presenting modifications and extensions which permit problems involving hundreds of demand points to be solved in a matter of seconds. In addition, a multi-depot routing algorithm is developed. The results are illustrated with a routing study for an urban newspaper with an evening circulation exceeding 100,000.
A genetic algorithm (GA) with an asexual reproduction plan through a generalized mutation for an evolutionary operator is developed that can be directly applied to a permutation of n numbers for an approximate global optimal solution of a traveling salesman problem (TSP). Schema analysis of the algorithm shows that a sexual reproduction with the generalized mutation operator preserves the global convergence property of a genetic algorithm thus establishing the fundamental theorem of the GA for the algorithm. Avoiding an intermediate step of encoding through random keys to preserve crossover or permuting n and using “fixing” states for legal crossover are the chief benefits of the innovations reported in this paper. The algorithm has been applied to a number of natural and artificial problems and the results are encouraging.
The paper presents a heuristic solution technique for an extended vehicle scheduling problem. The problem is decomposed into five main decision areas: the route assignment, the depot assignment, the vehicle assignment, the delivery period, and the detailed route design. Each part is solved separately and then connected by an iterative procedure. The modular structure makes it possible to work with stochastic demands with the additional constraints of fixed depot and fixed vehicle assignment for each customer. In addition, the method includes the aspects of multiple use of vehicles and the possibility of postponing some of the deliveries. The method should, therefore, be able to handle many different long-term, as well as short-term, distribution planning problems.
This article describes a tabu search algorithm for the multi-depot vehicle routing problem with capacity and route length restrictions. The algorithm is tested on a set of 23 benchmark instances. It is shown to outperform existing heuristics.
In this paper, some of the main known algorithms for the traveling salesman problem are surveyed. The paper is organized as follows: 1) definition; 2) applications; 3) complexity analysis; 4) exact algorithms; 5) heuristic algorithms; 6) conclusion.
In this paper, some of the main known results relative to the Vehicle Routing Problem are surveyed. The paper is organized as follows: (1) definition; (2) exact algorithms; (3) heuristic algorithms; (4) conclusion.
Several polynomial time algorithms finding "good," but not necessarily optimal, tours for the traveling salesman problem are considered. For the nearest neighbor method, the worst case ratio of the obtained tour to the optimal tour is shown to increase logarithmically with the number of cities. For another method, which we call the nearest insertion method, the worst case ratio is shown to approach 2 as the number of cities increases. It is also shown that for any n ≥ 8, there are traveling salesman problems with n cities having tours which are k-optimal for all k ≤ n/4, and for which the ratio with respect to the optimal is 2(1 -1/n).
We propose a tabu search heuristic capable of solving three well-known routing problems: the periodic vehicle routing problem, the periodic traveling salesman problem, and the multi-depot vehicle routing problem. Computational experiments carried out on instances taken from the literature indicate that the proposed method outperforms existing heuristics for all three problems. © 1997 John Wiley & Sons, Inc. Networks 30: 105–119, 1997
The optimum routing of a fleet of trucks of varying capacities from a central depot to a number of delivery points may require a selection from a very large number of possible routes, if the number of delivery points is also large. This paper, after considering certain theoretical aspects of the problem, develops an iterative procedure that enables the rapid selection of an optimum or near-optimum route. It has been programmed for a digital computer but is also suitable for hand computation.
This paper introduces the Multi-Terminal Sweep Algorithm, a heuristic algorithm for obtaining an approximate solution to the multiple terminal vehicle-dispatch problem. The procedure determines a set of routes by which vehicles from two or more terminals can service a collection of demand points so that the total distance traveled is kept near to the minimum. This solution also satisfies constraints on the vehicle load and on the length of each route. Application of the algorithm to eleven multiple terminal vehicle-dispatch problems shows that near-optimal solutions to large-scale problems can be found in a reasonable amount of computer time.
Dynamic programming to solve the Markov decision process problem of optimal insemination and replacement decisions was adapted to address large dairy herd management decision problems in the US. Expected net present values of cow states (151,200) were used to determine the optimal policy. States were specified by class of parity (n = 12), production level (n = 15), month of calving (n = 12), month of lactation (n = 16), and days open (n = 7). Methodology optimized decisions based on net present value of an individual cow and all replacements over a 20-yr decision horizon. Length of decision horizon was chosen to ensure that optimal policies were determined for an infinite planning horizon. Optimization took 286 s of central processing unit time. The final probability transition matrix was determined, in part, by the optimal policy. It was estimated iteratively to determine post-optimization steady state herd structure, milk production, replacement, feed inputs and costs, and resulting cash flow on a calendar month and annual basis if optimal policies were implemented. Implementation of the model included seasonal effects on lactation curve shapes, estrus detection rates, pregnancy rates, milk prices, replacement costs, cull prices, and genetic progress. Other inputs included calf values, values of dietary TDN and CP per kilogram, and discount rate. Stochastic elements included conception (and, thus, subsequent freshening), cow milk production level within herd, and survival. Validation of optimized solutions was by separate simulation model, which implemented policies on a simulated herd and also described herd dynamics during transition to optimized structure.
An experiment was designed to estimate the optimal interval from the beginning of estrus to artificial insemination (AI). The data were analyzed by means of a mathematical model. The analysis was based on pedometer readings and results of rectal palpation at 42 to 49 d post-AI of 171 breedings in 121 cows. The chance of conception was highest between 6 and 17 h after increased pedometer activity; the estimated optimum was at 11.8 h. In this data file, the effects of disease, inseminator, time of AI (a.m. or p.m.), and bull did not contribute to the improvement of the model. The effects of disease were not significant because of the low incidence of any specific disease. Activity measurements can be used as a tool for AI strategy to improve conception in groups of healthy cows and heifers already showing visual signs of estrus.
This paper introduces multi-depot vehicle routing problem with fixed distribution of vehicles (MDVRPFD) which is one important and useful variant of the traditional multi-depot vehicle routing problem (MDVRP) in the supply chain management and transportation studies. After modeling the MDVRPFD as a binary programming problem, we propose two solution methodologies: two-stage and one-stage approaches. The two-stage approach decomposes the MDVRPFD into two independent subproblems, assignment and routing, and solves them separately. In contrast, the one-stage approach integrates the assignment with the routing where there are two kinds of routing methods-draft routing and detail routing. Experimental results show that our new one-stage algorithm outperforms the published methods. Note to Practitioners-This work is based on several consultancy work that we have done for transportation companies in Hong Kong. The multi-depot vehicle routing problem (MDVRP) is one of the core optimization problems in transportation, logistics, and supply chain management, which minimizes the total travel distance (the major factor of total transportation cost) among a number of given depots. However, in real practice, the MDVRP is not reliable because of the assumption that there have unlimited number of vehicles available in each depot. In this paper, we propose a new useful variant of the MDVRP, namely multi-depot vehicle routing problem with fixed distribution of vehicles (MDVRPFD), to model the practicable cases in applications. Two-stage and one-stage solution algorithms are also proposed. The industry participators can apply our new one-stage algorithm to solve the MDVRPFD directly and efficiently. Moreover, our one-stage solution framework allows users to smoothly add new specified constraints or variants.
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