This article describes a new procedure for estimating parameters of a stochastic activity network of N arcs. The parameters include the probability that path m is the longest path, the probability that path m is the shortest path, the probability that arc i is on the longest path, and the probability that arc i is on the shortest path. The proposed procedure uses quasirandom points together with information on a cutset ℋ of the network to produce an upper bound of O[(log K)N−|ℋ|+1/K] on the absolute error of approximation, where K denotes the number of replications. This is a deterministic bound and is more favorable than the convergence rate of 1/K1/2 that one obtains from the standard error for K independent replications using random sampling. It is also shown how series reduction can improve the convergence rate by reducing the exponent on log K. The technique is illustrated using a Monte Carlo sampling experiment for a network of 16 relevant arcs with a cutset of ℋ = 7 arcs. The illustration shows the superior performance of using quasirandom points with a cutset (plan A) and the even better performance of using quasirandom points with the cutset together with series reduction (plan B) with regard to mean square error. However, it also shows that computation time considerations favor plan A when K is small and plan B when K is large.
本文描述了一种估计N个圆弧随机活动网络参数的新方法。参数包括路径m是最长路径的概率，路径m是最短路径的概率，路径i在最长路径上的概率，以及路径i在最短路径上的概率。该方法利用准随机点和网络割集h上的信息，得到近似绝对误差的上界为O[(log K)N−| h |+1/K]，其中K表示重复次数。这是一个确定性界，比从使用随机抽样的K个独立复制的标准误差中得到的1/K1/2的收敛率更有利。本文还展示了级数约简是如何通过减少log k的指数来提高收敛速度的。本文用蒙特卡罗抽样实验对一个由16个相关弧组成的网络进行了说明，该网络的割集为h = 7弧。该图显示了使用准随机点与切割集(方案a)的优越性能，以及使用准随机点与切割集以及序列约简(方案B)在均方误差方面的更好性能。然而，它也表明，当K较小时，计算时间的考虑更倾向于方案A，而当K较大时，则倾向于方案B。
{"title":"Estimating critical path and arc probabilities in stochastic activity networks","authors":"G. S. Fishman","doi":"10.1002/NAV.3800320206","DOIUrl":"https://doi.org/10.1002/NAV.3800320206","url":null,"abstract":"This article describes a new procedure for estimating parameters of a stochastic activity network of N arcs. The parameters include the probability that path m is the longest path, the probability that path m is the shortest path, the probability that arc i is on the longest path, and the probability that arc i is on the shortest path. The proposed procedure uses quasirandom points together with information on a cutset ℋ of the network to produce an upper bound of O[(log K)N−|ℋ|+1/K] on the absolute error of approximation, where K denotes the number of replications. This is a deterministic bound and is more favorable than the convergence rate of 1/K1/2 that one obtains from the standard error for K independent replications using random sampling. It is also shown how series reduction can improve the convergence rate by reducing the exponent on log K. The technique is illustrated using a Monte Carlo sampling experiment for a network of 16 relevant arcs with a cutset of ℋ = 7 arcs. The illustration shows the superior performance of using quasirandom points with a cutset (plan A) and the even better performance of using quasirandom points with the cutset together with series reduction (plan B) with regard to mean square error. However, it also shows that computation time considerations favor plan A when K is small and plan B when K is large.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130811506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article compares two types of approximating strategies for solving some continuous review inventory models noniteratively. One of these strategies is to approximate the normalized loss integral by an exponential function whereas the other strategy is to estimate the loss integral as a quadratic function of the right‐tail probability. It is found that the latter method is significantly more accurate and versatile than the former method. Theoretical arguments are given to emphasize that both the right‐tail probability and the loss integral are key functions involved in those models. Therefore, a good strategy should be not only to estimate these two functions, but also to retain the interrelationships between them. The quadratic method is better than the exponential method primarily because of the latter property.
{"title":"On the solution of some approximate continuous review inventory models","authors":"C. Das","doi":"10.1002/NAV.3800320210","DOIUrl":"https://doi.org/10.1002/NAV.3800320210","url":null,"abstract":"This article compares two types of approximating strategies for solving some continuous review inventory models noniteratively. One of these strategies is to approximate the normalized loss integral by an exponential function whereas the other strategy is to estimate the loss integral as a quadratic function of the right‐tail probability. It is found that the latter method is significantly more accurate and versatile than the former method. Theoretical arguments are given to emphasize that both the right‐tail probability and the loss integral are key functions involved in those models. Therefore, a good strategy should be not only to estimate these two functions, but also to retain the interrelationships between them. The quadratic method is better than the exponential method primarily because of the latter property.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133222011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On-site verification of ICBMs in the context of an arms control agreement might involve a situation where an inspector would choose one or more of a number of identical areas to inspect and would have confidence that the other areas had the same characteristics. This article considers optimal attack and defense of missiles deceptively based in a number of identical areas. The attacker may allocate warheads across areas as he desires and uniformly within areas. The defender may allocate interceptors across areas as he desires and either uniformly or preferentially within areas. The effect of restricting the defender to uniform allocation across areas is explored for various assumptions. Robustness of surviving missiles with respect to the number of attacking warheads is studied. Results are presented for a wide range of cases.
{"title":"Attack and defense of ICBMs deceptively based in a number of identical areas","authors":"J. Bracken, P. Brooks","doi":"10.1002/NAV.3800320202","DOIUrl":"https://doi.org/10.1002/NAV.3800320202","url":null,"abstract":"On-site verification of ICBMs in the context of an arms control agreement might involve a situation where an inspector would choose one or more of a number of identical areas to inspect and would have confidence that the other areas had the same characteristics. This article considers optimal attack and defense of missiles deceptively based in a number of identical areas. The attacker may allocate warheads across areas as he desires and uniformly within areas. The defender may allocate interceptors across areas as he desires and either uniformly or preferentially within areas. The effect of restricting the defender to uniform allocation across areas is explored for various assumptions. Robustness of surviving missiles with respect to the number of attacking warheads is studied. Results are presented for a wide range of cases.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114906509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A service center to which customers bring failed items for repair is considered. The items are exchangeable in the sense that a customer is ready to take in return for the failed item he brought to the center any good item of the same kind. This exchangeability feature makes it possible for the service center to possess spares. The focus of the article is on customer delay in the system—the time that elapses since the arrival of a customer with a failed item and his departure with a good one—when repaired items are given to waiting customers on a FIFO basis. An algorithm is developed for the computation of the delay distribution when the item repair system operates as an M/M/c queue.
{"title":"Customer delays in M/M/c repair systems with spares","authors":"M. Berg, M. Posner","doi":"10.1002/NAV.3800320209","DOIUrl":"https://doi.org/10.1002/NAV.3800320209","url":null,"abstract":"A service center to which customers bring failed items for repair is considered. The items are exchangeable in the sense that a customer is ready to take in return for the failed item he brought to the center any good item of the same kind. This exchangeability feature makes it possible for the service center to possess spares. The focus of the article is on customer delay in the system—the time that elapses since the arrival of a customer with a failed item and his departure with a good one—when repaired items are given to waiting customers on a FIFO basis. An algorithm is developed for the computation of the delay distribution when the item repair system operates as an M/M/c queue.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"27 10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123246857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note points out the omission of a simple but vital constraint in the recent articles on partial backlogging. Also, a simple intuitive interpretation of the “backorder” inequality of [2], [3], and [4] is provided.
{"title":"Recent articles on partial backorders: Comment","authors":"T. M. Whitin","doi":"10.1002/NAV.3800320215","DOIUrl":"https://doi.org/10.1002/NAV.3800320215","url":null,"abstract":"This note points out the omission of a simple but vital constraint in the recent articles on partial backlogging. Also, a simple intuitive interpretation of the “backorder” inequality of [2], [3], and [4] is provided.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"2006 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127656496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The operating characteristics of (s,S) inventory systems are often difficult to compute, making systems design and sensitivity analysis tedious and expensive undertakings. This article presents a methodology for simplified sensitivity analysis, and derives approximate expressions for operating characteristics of a simple (s,S) inventory system. The operating characteristics under consideration are the expected values of total cost per period, holding cost per period, replenishment cost per period, backlog cost per period, and backlog frequency. The approximations are obtained by using least-squares regression to fit simple functions to the operating characteristics of a large number of inventory items with diverse parameter settings. Accuracy to within a few percent of actual values is typical for most approximations. Potential uses of the approximations are illustrated for several idealized design problems, including consolidating demand from several locations, and tradeoffs for increasing service or reducing replenishment delivery lead time.
{"title":"Easily computed approximations for (s,S) inventory system operating characteristics","authors":"R. Ehrhardt","doi":"10.1002/NAV.3800320214","DOIUrl":"https://doi.org/10.1002/NAV.3800320214","url":null,"abstract":"The operating characteristics of (s,S) inventory systems are often difficult to compute, making systems design and sensitivity analysis tedious and expensive undertakings. This article presents a methodology for simplified sensitivity analysis, and derives approximate expressions for operating characteristics of a simple (s,S) inventory system. The operating characteristics under consideration are the expected values of total cost per period, holding cost per period, replenishment cost per period, backlog cost per period, and backlog frequency. The approximations are obtained by using least-squares regression to fit simple functions to the operating characteristics of a large number of inventory items with diverse parameter settings. Accuracy to within a few percent of actual values is typical for most approximations. Potential uses of the approximations are illustrated for several idealized design problems, including consolidating demand from several locations, and tradeoffs for increasing service or reducing replenishment delivery lead time.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"614 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116210041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most of the results are purely qualitative in character showing that many properties of such systems are independent of the actual numerical values of the coefficients. Finally, we discuss in some detail an example of such a system.
{"title":"The theory of combined-arms lanchester-type models of warfare","authors":"J. Maybee","doi":"10.1002/NAV.3800320204","DOIUrl":"https://doi.org/10.1002/NAV.3800320204","url":null,"abstract":"The mathematical theory necessary to solve combined arms models of military combat is presented here. We show how to apply the theory of positive operators to such models. Most of the results are purely qualitative in character showing that many properties of such systems are independent of the actual numerical values of the coefficients. Finally, we discuss in some detail an example of such a system.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128728901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we consider obtaining probability distributions for total downtime incurred for n units when we have only partial moment information about the downtime distribution per individual failure. Extremal distributions based upon the Markov‐Krein Theorem for Chebychev systems of functions and maximum entropy of distributions are presented.
{"title":"Probability bounds on downtimes","authors":"P. Brockett, M. Hinich","doi":"10.1002/NAV.3800320212","DOIUrl":"https://doi.org/10.1002/NAV.3800320212","url":null,"abstract":"In this article we consider obtaining probability distributions for total downtime incurred for n units when we have only partial moment information about the downtime distribution per individual failure. Extremal distributions based upon the Markov‐Krein Theorem for Chebychev systems of functions and maximum entropy of distributions are presented.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"166 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115202310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem treated here involves a mixed fleet of vehicles comprising two types of vehicles: K1 tanker‐type vehicles capable of refueling themselves and other vehicles, and K2 nontanker vehicles incapable of refueling. The two groups of vehicles have different fuel capacities as well as different fuel consumption rates. The problem is to find the tanker refueling sequence that maximizes the range attainable for the K2 nontankers. A tanker refueling sequence is a partition of the tankers into m subsets (2 ≤ m ≤ K1). A given sequence of the partition provides a realization of the number of tankers participating in each successive refueling operation. The problem is first formulated as a nonlinear mixed‐integer program and reduced to a linear program for a fixed sequence which may be solved by a simple recursive procedure. It is proven that a “unit refueling sequence” composed of one tanker refueling at each of K1 refueling operations is optimal. In addition, the problem of designing the “minimum fleet” (minimum number of tankers) required for a given set of K2 nontankers to attain maximal range is resolved. Also studied are extensions to the problem with a constraint on the number of refueling operations, different nontanker recovery base geometry, and refueling on the return trip.
{"title":"Optimal refueling strategies for a mixed-vehicle fleet","authors":"A. Mehrez, H. Stern","doi":"10.1002/NAV.3800320211","DOIUrl":"https://doi.org/10.1002/NAV.3800320211","url":null,"abstract":"The problem treated here involves a mixed fleet of vehicles comprising two types of vehicles: K1 tanker‐type vehicles capable of refueling themselves and other vehicles, and K2 nontanker vehicles incapable of refueling. The two groups of vehicles have different fuel capacities as well as different fuel consumption rates. The problem is to find the tanker refueling sequence that maximizes the range attainable for the K2 nontankers. A tanker refueling sequence is a partition of the tankers into m subsets (2 ≤ m ≤ K1). A given sequence of the partition provides a realization of the number of tankers participating in each successive refueling operation. The problem is first formulated as a nonlinear mixed‐integer program and reduced to a linear program for a fixed sequence which may be solved by a simple recursive procedure. It is proven that a “unit refueling sequence” composed of one tanker refueling at each of K1 refueling operations is optimal. In addition, the problem of designing the “minimum fleet” (minimum number of tankers) required for a given set of K2 nontankers to attain maximal range is resolved. Also studied are extensions to the problem with a constraint on the number of refueling operations, different nontanker recovery base geometry, and refueling on the return trip.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115745685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The literature on multicomponent systems, though extensive, has dealt mainly with formulations where the failure of a component has no effect or influence on the operation or failure of other components in the system. In many real-life multicomponent systems, the failure of a component has some effect on one or more of the remaining components. The paper considers a two-component system with a specific type failure interaction and obtains expressions for the expected cost of operating the system for both finite and infinite time.
{"title":"Study of two‐component system with failure interaction","authors":"D. Murthy, D. G. Nguyen","doi":"10.1002/NAV.3800320205","DOIUrl":"https://doi.org/10.1002/NAV.3800320205","url":null,"abstract":"The literature on multicomponent systems, though extensive, has dealt mainly with formulations where the failure of a component has no effect or influence on the operation or failure of other components in the system. In many real-life multicomponent systems, the failure of a component has some effect on one or more of the remaining components. The paper considers a two-component system with a specific type failure interaction and obtains expressions for the expected cost of operating the system for both finite and infinite time.","PeriodicalId":431817,"journal":{"name":"Naval Research Logistics Quarterly","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1985-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121920604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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