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Journal articles

Forming competitively balanced teams (2015)

This article examines the problem of assigning individuals to teams to make the teams as similar as possible to each other across multiple attributes. This may be complicated by a variety of constraints, including restrictions on whether specific individuals can or should be assigned to the same team. The problem arises in multiple contexts, including youth recreation leagues and academic programs or courses with mandated project groups. A model for the problem is proposed and various solution approaches are investigated, including mixed-integer programming and several heuristics. Supplementary materials are available for this article. Go to the publisher’s online edition of IIE Transactions for datasets, additional tables, detailed proofs, etc.

Rubin, Paul A. and Lihui Bai (2015). “Forming Competitively Balanced Teams”. In: IIE Transactions 47, 6, pp. 620-633. DOI: 10.1080/0740817X.2014.953643. eprint: http://dx.doi.org/10.1080/0740817X.2014.953643 .

BibTeX

@article{Rubin2015, title = {Forming Competitively Balanced Teams}, author = {Paul A. Rubin and Lihui Bai}, journal = {IIE Transactions}, year = {2015}, month = {June}, number = {6}, pages = {620-633}, volume = {47}, doi = {10.1080/0740817X.2014.953643}, eprint = { http://dx.doi.org/10.1080/0740817X.2014.953643 }, url = { http://dx.doi.org/10.1080/0740817X.2014.953643 }, }

The problem data used in our 2015 IIE Transactions paper on team formation

Rubin, Paul A. and Lihui Bai (2015). “Forming Competitively Balanced Teams”. In: IIE Transactions 47.6, pp. 620-633. DOI: 10.1080/0740817X.2014.953643. eprint: http://dx.doi.org/10.1080/0740817X.2014.953643

is available in a zip archive.

The data set archive contains a CSV file for each of the 100 randomly generated test problems used in the paper. Not included, due to the presence of confidential information in the files, are the two real-life problems discussed in section 6.5 of the paper. Since they are quite similar to, and served as templates for, many of the random problems, their omission is not much of a loss.

Each CSV file contains multiple tables.

  • The Project Details table contains a (largely meaningless) description of the problem and all project-level settings: maximum and minimum roster counts and sizes; target values for roster counts and sizes; whether full utilization of all potential participants is mandatory/optional/not a concern; and priorities for roster size, roster count and utilization goals.

  • The Attributes table lists all attributes, their type (label, number, category or affinity), a description (empty), whether they are case-sensitive (generally false for these problems) and their priority. Labels are not used in the assignment algorithms, and in these problems the only label attribute will be the identifier for individuals. For category and affinity attributes (but not for number attributes), the remaining columns of the table will list their domains (legal values).

  • The Individuals table lists, for each individual, their identifier and attribute values.

  • The Restrictions table lists all restrictions (if any – not all problems contain restrictions). Each restriction is defined by a type (“together” or “apart”), the pair of individuals involved, whether the restriction is mandatory, and its priority (which is null for mandatory restrictions).

Click-bait: There’s a link to free software below!

Our team-building work, published as

Rubin, Paul A. and Lihui Bai (2015). “Forming Competitively Balanced Teams”. In: IIE Transactions 47.6, pp. 620-633. DOI: 10.1080/0740817X.2014.953643. eprint: http://dx.doi.org/10.1080/0740817X.2014.953643,

originated with some real-world problems, and is rather atypical of team-building research in general.

There is a significant body of research on team formation in the field of organizational behavior, most of which focuses on building a single team from a pool of subjects (not all of whom are expected to be used), with the intent of making the team as highly functional as possible. Our work, in contrast, looks at building multiple teams, using most if not all of the subjects, with the intent of making the teams as similar to each other as possible.

Uses

The work originated with a request for assistance from a volunteer in a youth recreation league. Parents sign their children up for a sport but not for a particular team. The league assigns children to teams (the task for which assistance was requested). There are various limitations on size and composition of teams, and the goal is to make the teams equally competitive. If the more athletic children wind up concentrated on a few teams, routs occur, and the experience is not a happy one for the less athletic children or the parents (or the volunteers to whom the parents vent).

A similar situation occurs when assigning students in a class to project teams. Some instructors let teams self-select. That can sometimes levae less well-networked students (particularly minority or foreign ones) to form teams of the “undrafted”. It can also engender clustering that limits the breadth of ideas expressed in the teams. One course I taught, a business cognate for non-business majors, drew students from a variety of disciplines. Left to their own devices, the building construction management majors would team with each other, the international studies majors would team with each other, and so forth, because they knew each other from other shared courses.

Free software (as promised)

I wrote some software to create balanced team, using the model and algorithms that wound up in our paper, and used it in a few of my courses to create project groups. I also released the software under an open-source license. The program’s name is ParityBuilder, and it is available from SourceForge. It should run on any platform with a recent version of the Java Runtime Environment. There is no special installation; just download it, unzip it and stick it somewhere you won’t forget. There is a built-in help system, including a tutorial on usage.

An improved formulation for the maximum coverage patrol routing problem (2015)

We present an improved formulation for the maximum coverage patrol routing problem (MCPRP). The main goal of the patrol routing problem is to maximize the coverage of critical highway stretches while ensuring the feasibility of routes and considering the availability of resources. By investigating the structural properties of the optimal solution, we formulate a new, improved mixed integer program that can solve real life instances to optimality within seconds, where methods proposed in prior literature fail to find a provably optimal solution within an hour. The improved formulation provides enhanced highway coverage for both randomly generated and real life instances. We show an average increase in coverage of nearly 20% for the randomly generated instances provided in the literature, with a best case increase over 46%. Similarly, for the real life instances, we close the optimality gap within seconds and demonstrate an additional coverage of over 13% in the best case. The improved formulation also allows for testing a number of real life scenarios related to multi-start routes, delayed starts at the beginning of the shifts, and taking a planned break during the shift. Being able to solve these scenarios in short durations help decision and policy makers to better evaluate resource allocation options while serving public.

Çapar, İbrahim, Burcu B. Keskin, and Paul A. Rubin (2015). “An improved formulation for the maximum coverage patrol routing problem”. In: Computers & Operations Research 59, pp. 1-10. DOI: http://dx.doi.org/10.1016/j.cor.2014.12.002.

BibTeX

@article{Capar2015, title = {An improved formulation for the maximum coverage patrol routing problem}, author = {{.I}brahim {\c C}apar and Burcu B. Keskin and Paul A. Rubin}, journal = {Computers & Operations Research}, year = {2015}, month = {July}, pages = {1-10}, volume = {59}, doi = {http://dx.doi.org/10.1016/j.cor.2014.12.002}, }

Combinatorial Benders cuts for the minimum tollbooth problem (2009)

We address a toll pricing problem in which the objective is to minimize the number of required toll facilities in a transportation network while inducing drivers to make the most efficient collective use of the network. We formulate the problem as a mixed-integer programming model and propose a solution method using combinatorial Benders cuts. Computational study of real networks as well as randomly generated networks indicates that our proposed method is efficient in obtaining provably optimal solutions for networks with small to medium sizes.

Bai, Lihui and Paul A. Rubin (2009). “Combinatorial Benders Cuts for the Minimum Tollbooth Problem”. In: Operations Research 57, 6, pp. 1510-1522. DOI: 10.1287/opre.1090.0694.

BibTeX

@article{Bai2009, title = {Combinatorial Benders Cuts for the Minimum Tollbooth Problem}, author = {Lihui Bai and Paul A. Rubin}, journal = {Operations Research}, year = {2009}, month = {November-December}, number = {6}, pages = {1510–1522}, volume = {57}, doi = {10.1287/opre.1090.0694}, url = {http://or.journal.informs.org/cgi/content/abstract/opre.1090.0694v1}, }

A heuristic procedure for sequence dependent scheduling with stock cutting (2008)

This paper introduces a method to increase productivity and competitiveness through a heuristic solution procedure for a formulation describing sequence-dependent production scheduling where the produced material will be cut into finished products. The heuristic solution procedure obtains good, but not necessarily optimal, solutions to a production scheduling problem common in the paper industry.

D’Itri, Michael and Paul Rubin (2008). “A Heuristic Procedure for Sequence Dependent Scheduling with Stock Cutting”. In: Competition Forum 6, 1, pp. 198-202.

BibTeX

@article{DItri2008, title = {A Heuristic Procedure for Sequence Dependent Scheduling with Stock Cutting}, author = {Michael D’Itri and Paul Rubin}, journal = {Competition Forum}, year = {2008}, number = {1}, pages = {198-202}, volume = {6}, }

Scheduling and sequencing with stock cutting considerations (2007)

This paper introduces a method to increase productivity and competitiveness through an explicit formulation for sequence dependent production scheduling where the produced material will be cut into finished products. The formulation is then applied to a production scheduling problem typical of the paper industry. Results are contrasted with those produced by a heuristic simulating a manual scheduling.

D’Itri, Michael and Paul Rubin (2007). “Scheduling and Sequencing with Stock Cutting Considerations”. In: Competition Forum 5, 1, pp. 265-273.

BibTeX

@article{DItri2007, title = {Scheduling and Sequencing with Stock Cutting Considerations}, author = {Michael D’Itri and Paul Rubin}, journal = {Competition Forum}, year = {2007}, number = {1}, pages = {265-273}, volume = {5}, }

Oscillation heuristics for the two-group classification problem (2004)

We propose a new nonparametric family of oscillation heuristics for devising linear classifiers to solve the two-group discriminant problem. The heuristics are motivated by the intuition that the classification accuracy of a separating hyperplane can be improved through small perturbations to its slope and position, accomplished by substituting training observations near the hyperplane for those used to generate it. In an extensive simulation study, using data generated from multivariate normal distributions under a variety of conditions, the oscillation heuristics consistently outperform the classical linear and logistic discriminant functions, as well as two published linear programming-based heuristics. They approach, and in 40% of experimental cases attain, the best possible accuracy on the training samples, as determined by a mixed integer programming (MIP) model, at a much smaller computational cost.

Asparouhov, Ognian K. and Paul A. Rubin (2004). “Oscillation Heuristics for the Two-group Classification Problem”. In: Journal of Classification 21, 2, pp. 255-277. DOI: 10.1007/s00357-004-0019-7.

BibTeX

@article{Asparouhov2004, title = {Oscillation Heuristics for the Two-group Classification Problem}, author = {Ognian K. Asparouhov and Paul A. Rubin}, journal = {Journal of Classification}, year = {2004}, month = {September}, number = {2}, pages = {255–277}, volume = {21}, doi = {10.1007/s00357-004-0019-7}, }

Comment on “A Nonlinear Lagrangian Dual for Integer Programming” (2004)

We present a counterexample and correction to the contention by Xu and Li that the nonlinear Lagrangian dual problem they propose [Oper. Res. Lett. 30 (2002) 401] asymptotically has no duality gap.

Rubin, Paul A. (2004). “Comment on “A Nonlinear Lagrangian Dual for Integer Programming””. In: Operations Research Letters 32, 2, pp. 197-198. DOI: 10.1016/S0167-6377(03)00096-8.

BibTeX

@article{Rubin2004, title = {Comment on “A Nonlinear Lagrangian Dual for Integer Programming”}, author = {Paul A. Rubin}, journal = {Operations Research Letters}, year = {2004}, month = {March}, number = {2}, pages = {197–198}, volume = {32}, doi = {10.1016/S0167-6377(03)00096-8}, url = {http://www.ingentaconnect.com/content/els/01676377/2004/00000032/00000002/art00096}, }

Offsetting inventory cycles of items sharing storage (2003)

The ability to determine the optimal frequencies and offsets for independent and unrestricted ordering cycles for multiple items can be very valuable for managing storage capacity constrained facilities in a supply chain. The complexity of this problem has resulted in researchers focusing on more tractable surrogate problems that are special cases of the base problem. This research has focused on developing fundamental properties of the original problem. We exploit the problem structure and present a heuristic for offsetting independent and unrestricted ordering cycles for items to minimize their joint storage requirements. Heuristics of this type may prove useful in solving the more general problem of selecting order quantities to minimize combined holding, ordering, and storage costs.

Murthy, Nagesh N, W. C. Benton, and Paul A. Rubin (2003). “Offsetting Inventory Cycles of Items Sharing Storage”. In: European Journal of Operational Research 150, 2, pp. 304-319. ISSN: 03772217. DOI: 10.1016/S0377-2217(02)00518-0.

BibTeX

@article{Murthy2003, title = {Offsetting Inventory Cycles of Items Sharing Storage}, author = {Nagesh N. Murthy and W. C. Benton and Paul A. Rubin}, journal = {European Journal of Operational Research}, year = {2003}, month = {October}, number = {2}, pages = {304–319}, volume = {150}, doi = {10.1016/S0377-2217(02)00518-0}, url = {http://dx.doi.org/10.1016/S0377-2217(02)00518-0}, }

Feature selection for multi-class discrimination via mixed-integer linear programming (2003)

We reformulate branch-and-bound feature selection employing \(L_\infty\) or particular \(L_p\) metrics, as mixed-integer linear programming (MILP) problems, affording convenience of widely available MILP solvers. These formulations offer direct influence over individual pairwise interclass margins, which is useful for feature selection in multiclass settings.

Iannarilli Jr, Frank J. and Paul A. Rubin (2003). “Feature Selection for Multi-class Discrimination via Mixed-integer Linear Programming”. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 25, 6, pp. 779-783. DOI: 10.1109/TPAMI.2003.1201827.

BibTeX

@article{IannarilliRubin, title = {Feature Selection for Multi-class Discrimination via Mixed-integer Linear Programming}, author = {Frank J. {Iannarilli Jr.} and Paul A. Rubin}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, year = {2003}, month = {June}, number = {6}, pages = {779–783}, volume = {25}, doi = {10.1109/TPAMI.2003.1201827}, }

Evaluating jointly constrained order quantity complexities for incremental discounts (2003)

In this paper we consider the purchasing decisions facing a buying firm which receives incrementally discounted price schedules for a group of items in the presence of constraints such as budgets and space limitations. The constraints take the form of upper bounds on positively weighted linear combinations of order size, dollar volume, unit price and order frequency. We adapt the solution procedures from an earlier paper [Naval Res. Logis. 40 (2) (1993) 255] to treat this case. The results show that simultaneous order quantities for multiple items, offered with separate incremental discounts and subject to joint resource constraints, can be calculated efficiently using a combination of Lagrangean relaxation and, as needed, partial enumeration. This problem proved to be no more and no less tractable than its counterpart with all units discounts.

Rubin, Paul A. and W. C. Benton (2003). “Evaluating Jointly Constrained Order Quantity Complexities for Incremental Discounts”. In: European Journal of Operational Research 149, 3, pp. 557-570.

BibTeX

@article{Rubin2003a, title = {Evaluating Jointly Constrained Order Quantity Complexities for Incremental Discounts}, author = {Paul A. Rubin and W. C. Benton}, journal = {European Journal of Operational Research}, year = {2003}, month = {September}, number = {3}, pages = {557–570}, volume = {149}, }

A generalized framework for quantity discount pricing schedules (2003)

The use of price to influence a buyer’s purchasing behavior and thus improve supply chain coordination has received considerable attention. The vendor and buyer are independent economic entities, each maximizing its own profit. We consider the case of a buyer with fixed annual demand, independent of cost. The vendor’s objective is to set a price schedule that encourages the buyer to raise its order quantity, increasing the vendor’s profits. We present a unified treatment of the problem, categorize different variations, and provide a common solution procedure for all cases.

Rubin, Paul A. and W. C. Benton (2003). “A Generalized Framework for Quantity Discount Pricing Schedules”. In: Decision Sciences 34, 1, pp. 173-188. ISSN: 0011-7315. DOI: 10.1111/1540-5915.02437.

BibTeX

@article{Rubin2003, title = {A Generalized Framework for Quantity Discount Pricing Schedules}, author = {Paul A. Rubin and W. C. Benton}, journal = {Decision Sciences}, year = {2003}, month = {February}, number = {1}, pages = {173–188}, volume = {34}, doi = {10.1111/1540-5915.02437}, url = {http://doi.wiley.com/10.1111/1540-5915.02437}, }

A comparison of four methods for minimizing total tardiness on a single processor with sequence dependent setup times (2000)

Much of the research on operations scheduling problems has either ignored setup times or assumed that setup times on each machine are independent of the job sequence. This paper considers the problem of scheduling a single machine for minimizing total tardiness in a sequence dependent setup environment. The comparative performance of branch-and-bound, genetic search, simulated annealing and random-start pairwise interchange was evaluated in this problem setting. The experimental results suggest that simulated annealing and random-start pairwise interchange are viable solution techniques that can yield good solutions to a large combinatorial problem when considering the tardiness objective with sequence dependent setup times. However, branch-and-bound may be the preferred solution technique in solving smaller problems, and it is the only solution technique tested that will confirm an optimum solution has been reached. The methods considered in this research offer promise to deal with a class of scheduling problems, which have been considered diffcult by both researchers and practitioners.

Tan, Keah-Choon, Ram Narasimhan, Paul A. Rubin and Gary L. Ragatz (2000). “A Comparison of Four Methods for Minimizing Total Tardiness on a Single Processor with Sequence Dependent Setup Times”. In: Omega 28, 3, pp. 313-326.

BibTeX

@article{Tan2000, title = {A Comparison of Four Methods for Minimizing Total Tardiness on a Single Processor with Sequence Dependent Setup Times}, author = {Keah-Choon Tan and Ram Narasimhan and Paul A. Rubin and Gary L. Ragatz}, journal = {Omega}, year = {2000}, month = {June}, number = {3}, pages = {313–326}, volume = {28}, }

Adapting the Warmack-Gonzalez algorithm to handle discrete data (1999)

Soltysik and Yarnold propose, as a method for two-group multivariate optimal discriminant analysis (MultiODA), selecting a linear discriminant function based on an algorithm by Warmack and Gonzalez. An important assumption underlying the Warmack-Gonzalez algorithm is likely to be violated when the data in the discriminant training samples are discrete, and in particular when they are nominal, causing the algorithm to fail. We offer modest changes to the algorithm that overcome this limitation.

Rubin, Paul A. (1999). “Adapting the Warmack-Gonzalez Algorithm to Handle Discrete Data”. In: European Journal of Operational Research 113, 3, pp. 632-642.

BibTeX

@article{Rubin1999, title = {Adapting the {W}armack-{G}onzalez Algorithm to Handle Discrete Data}, author = {Paul A. Rubin}, journal = {European Journal of Operational Research}, year = {1999}, month = {March}, number = {3}, pages = {632–642}, volume = {113}, }

Solving mixed integer classification problems by decomposition (1997)

Research into the accuracy of mixed integer programming models for discrimination and classification, and the efficacy of heuristics developed for them, has been hampered by the inability to solve to optimality problems with moderate to large sample sizes. We present encouraging preliminary results for a decomposition approach that allows solution of models with dimensions previously considered prohibitive.

Rubin, Paul A. (1997). “Solving Mixed Integer Classification Problems by Decomposition”. In: Annals of Operations Research 74, pp. 51-64.

BibTeX

@article{Rubin1997, title = {Solving Mixed Integer Classification Problems by Decomposition}, author = {Paul A. Rubin}, journal = {Annals of Operations Research}, year = {1997}, month = {November}, pages = {51–64}, volume = {74}, }

Scheduling in a sequence dependent setup environment with genetic search (1995)

Work on scheduling in the presence of squence dependent setup times has generally focused on minimizing the total setup time (or cost) or minimizing the makespan of a set of jobs. We explore the problem of sequencing to minimize the total tardiness of a set of jobs in a single-stage process where setup times are sequence dependent. In particular, we examine the efficacy of using genetic search to develop near optimal schedules in this environment.

Rubin, Paul A. and Gary L. Ragatz (1995). “Scheduling in a Sequence Dependent Setup Environment with Genetic Search”. In: Computers and Operations Research 22, 1, pp. 85-99. DOI: 10.1016/0305-0548(93)E0021-K.

BibTeX

@article{RubinRagatz95, title = {Scheduling in a Sequence Dependent Setup Environment with Genetic Search}, author = {Paul A. Rubin and Gary L. Ragatz}, journal = {Computers and Operations Research}, year = {1995}, month = {January}, number = {1}, pages = {85–99}, volume = {22}, doi = {10.1016/0305-0548(93)E0021-K}, }

The source code used in our 1995 Computers & Operations Research paper on minimum tardiness scheduling

Rubin, Paul A. and Gary L. Ragatz (1995). “Scheduling in a Sequence Dependent Setup Environment with Genetic Search”. In: Computers and Operations Research 22.1, pp. 85-99. DOI: 10.1016/0305-0548(93)E0021-K

is available in a zip archive. This being somewhat antiquated research, the programs are written in FORTRAN. They use a couple of libraries from a Watcom compiler that no longer exists, so your best bet may be to mimic the code in a more current language/system.

The source code archive contains a file (FILES.TXT) listing the contents of the archive. It is reproduced here for your convenience:

The files in this archive contain the FORTRAN source code used to produce the results cited in the Rubin and Ragatz 1995 Computers & Operations Research paper and in a subsequent (unpublished) work by the same authors presented in 1995 at the Fall INFORMS meeting. The programs are in reasonably generic ANSI ‘77 FORTRAN, but may need a little tweaking to compile on specific platforms. (The authors used the Watcom FORTRAN 77 compiler under MS-DOS on a Pentium PC.) The code references two library routines specific to the Watcom compiler: URAND, which takes as argument a seed and generates a single pseudorandom number; and GETTIM, which returns the current time of day as reported by the system clock.

Program MUTATE.FOR is the random mutation heuristic used as a benchmark in both works. It generates a random starting schedule, performs all pairwise interchanges that improve its tardiness, compare it to the incumbent schedule and then generates a new starting schedule.

Program GENSRCH1.FOR is the genetic search routine used in the Computers and Operations Research paper. The coding scheme makes each allele a sequence of job ordinals.

Program GENSRCH2.FOR was used in the follow-up work presented at the INFORMS meeting. Its coding scheme makes an allele a sequence of job assignment rules, where the k-th rule is applied to the remaining unscheduled jobs after the first k-1 slots in the schedule have been filled. There are twelve specific rules available (see the comments in the source code for details), plus a thirteenth rule (“rule 0”) which allows for arbitrary selection of a specific job (again, see the comments for details).

Program GENSRCH5.FOR is a slightly modified version of GENSRCH2.FOR, in which the fitness function is based on the rank of the tardiness of each schedule rather than on the tardiness itself.

The problem data used in our 1995 Computers & Operations Research paper on minimum tardiness scheduling

Rubin, Paul A. and Gary L. Ragatz (1995). “Scheduling in a Sequence Dependent Setup Environment with Genetic Search”. In: Computers and Operations Research 22.1, pp. 85-99. DOI: 10.1016/0305-0548(93)E0021-K

is available in a zip archive.

The data set archive contains a file (FORMAT.TXT) that describes the contents of the archive. It is reproduced here for your convenience:

Files PROB401.TXT through PROB708.TXT contain the data for the 32 test problems used in table 2 of the Rubin and Ragatz 1995 Computers & Operations Research paper. The first numeral in the file name signifies the problem size (4 -> 15 jobs, …, 7 -> 45 jobs). The files are plain ASCII text, with columns separated by spaces and/or end-of-line codes.

The first record of each file (one line) contains the number of jobs (n) and the branch-and-bound solution reported in table 2. The next line, or group of lines, contains the n processing times. The following line, or group of lines, contains the n due dates. The due dates are a monotone sequence (jobs were indexed in earliest due date order); a negative due date signifies a job already overdue.

The next group of lines gives the setup times as an n+1 x n matrix, the row indexed by the current state, the column indexed by the next state. The first row, row 0, contains setup times coming out of the initial state of the processor.

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A comment regarding polynomial discriminant functions (1994)

Rubin, Paul A. (1994). “A Comment Regarding Polynomial Discriminant Functions”. In: European Journal of Operational Research 72, 1, pp. 29-31. ISSN: 0377-2217.

BibTeX

@article{Rubin1994a, title = {A Comment Regarding Polynomial Discriminant Functions}, author = {Paul A. Rubin}, journal = {European Journal of Operational Research}, year = {1994}, month = {January}, number = {1}, pages = {29–31}, volume = {72}, publisher = {Elsevier}, url = {http://econpapers.repec.org/RePEc:eee:ejores✌️72:y:1994:i:1:p:29-31}, }

Jointly constrained order quantities with all-units discounts (1993)

In this article we investigate situations where the buyer is offered discounted price schedules from alternative vendors. Given various discount schedules, the buyer must make the best buying decision under a variety of constraints, such as limited storage space and restricted inventory budgets. Solutions to this problem can be utilized by the buyer to improve profitability. EOQ models for multiple products with all-units discounts are readily solvable in the absence of constraints spanning the products. However, constrained discounted EOQ models lack convenient mathematical properties. Relaxing the product-spanning constraints produces a dual problem that is separable, but lack of convexity and smoothness opens the door for duality gaps. In this research we present a set of algorithms that collectively find the optimal order vector. Finally, we present numerical examples using actual data. to illustrate the application of the algorithms.

Rubin, Paul A. and W. C. Benton (1993). “Jointly Constrained Order Quantities with All-Units Discounts”. In: Naval Research Logistics 40, 2, pp. 255-278. DOI: 10.1002/1520-6750(199303)40:2255::AID-NAV32204002093.0.CO;2-G.

BibTeX

@article{Rubin1993, title = {Jointly Constrained Order Quantities with All-Units Discounts}, author = {Paul A. Rubin and W. C. Benton}, journal = {Naval Research Logistics}, year = {1993}, month = {March}, number = {2}, pages = {255–278}, volume = {40}, doi = {10.1002/1520-6750(199303)40:2255::AID-NAV32204002093.0.CO;2-G}, }

Separation failure in linear programming discriminant models (1991)

Linear programming discriminant analysis (LPDA) models are designed around a variety of objective functions, each representing a different measure of separation of the training samples by the resulting discriminant function. A separation failure is defined to be the selection of an “optimal” discriminant function which incompletely separates a pair of completely separable training samples. Occurrence of a separation failure suggests that the chosen discriminant function may have an unnecessarily low classification accuracy on the actual populations involved. In this paper, a number of the LPDA models proposed for the two-group case are examined to learn which are subject to separation failure. It appears that separation failure in any model can be avoided by applying the model twice, reversing group designations.

Rubin, Paul A. (1991). “Separation Failure in Linear Programming Discriminant Models”. In: Decision Sciences 22, 3, pp. 519-535. DOI: 10.1111/j.1540-5915.1991.tb01279.x.

BibTeX

@article{Rubin1991, title = {Separation Failure in Linear Programming Discriminant Models}, author = {Paul A. Rubin}, journal = {Decision Sciences}, year = {1991}, month = {July}, number = {3}, pages = {519–535}, volume = {22}, doi = {10.1111/j.1540-5915.1991.tb01279.x}, }

Joint optimality in buyer-supplier negotiations (1990)

The relationship between supplier and customer in the United States is gradually changing from adversarial to cooperative. The potential benefits of collaborative negotiation typically have been developed only in highly specialized or unique operating situations. This article demonstrates the general superiority of cooperative negotiation, from the standpoint of overall cost reduction. It develops a generalized model and discusses some possible pitfalls in its implementation.

Rubin, Paul A. and Joseph R. Carter (1990). “Joint Optimality in Buyer-Supplier Negotiations”. In: Journal of Purchasing and Materials Management 26, 2, pp. 20-26.

BibTeX

@article{Rubin1990a, title = {Joint Optimality in Buyer-Supplier Negotiations}, author = {Paul A. Rubin and Joseph R. Carter}, journal = {Journal of Purchasing and Materials Management}, year = {1990}, month = {April}, number = {2}, pages = {20–26}, volume = {26}, }

Heuristic solution procedures for a mixed-integer programming discriminant model (1990)

Mixed-integer programming models for the two-group discriminant problem appear to be more promising, in terms of accuracy, than are linear programming models, but at a substantial computational cost. This paper poses a particular mixed-integer model and suggests heuristics, based on linear programming, for obtaining suboptimal but ‘good’ solutions to it. The heuristics are compared to the mixed-integer model using Monte Carlo simulation with Gaussian data.

Rubin, Paul A. (1990). “Heuristic Solution Procedures for a Mixed-Integer Programming Discriminant Model”. In: Managerial and Decision Economics 11, 4, pp. 255-266.

BibTeX

@article{Rubin1990, title = {Heuristic Solution Procedures for a Mixed-Integer Programming Discriminant Model}, author = {Paul A. Rubin}, journal = {Managerial and Decision Economics}, year = {1990}, month = {October}, number = {4}, pages = {255–266}, volume = {11}, }

A comparison of linear programming and parametric approaches to the two-group discriminant problem (1990)

Recent simulation-based studies of linear programming models for discriminant analysis have used the Fisher linear discriminant function as the benchmark for parametric methods. This article reports experimental evidence which suggests that, while some linear programming models may match or even exceed the Fisher approach in classification accuracy, none of the fifteen models tested is as accurate on normally distributed data as the Smith quadratic discriminant function. At the minimum, further testing is warranted with an emphasis on data sets that arise from significantly non-Gaussian populations.

Rubin, Paul A. (1990). “A Comparison of Linear Programming and Parametric Approaches to the Two-Group Discriminant Problem”. In: Decision Sciences 21, 2, pp. 373-386. ISSN: 0011-7315. DOI: 10.1111/j.1540-5915.1990.tb01691.x.

BibTeX

@article{Rubin1990c, title = {A Comparison of Linear Programming and Parametric Approaches to the Two-Group Discriminant Problem}, author = {Paul A. Rubin}, journal = {Decision Sciences}, year = {1990}, month = {June}, number = {2}, pages = {373–386}, volume = {21}, doi = {10.1111/j.1540-5915.1990.tb01691.x}, url = {http://doi.wiley.com/10.1111/j.1540-5915.1990.tb01691.x}, }

Evaluating the maximize minimum distance formulation of the linear discriminant problem (1989)

The “maximize minimum distance” (MMD) linear programming model for the two group discriminant problem has been noted to produce occasionally a trivial (identically zero) discriminant function, one which classifies all observations into a single category. In tests against other methods, both parametric and nonparametric, MMD has fared poorly. In this paper, we attribute the propensity of the MMD model to produce trivial solutions to a specific aspect of its formulation; this same facet may also cause unnessarily high misclassification rates even when a nontrivial function is found. We note a simple revision of a model which ensures an acceptable solution in those instancesin which the calibration samples can be classified with 100% accuracy by a single function. This rises the question of whether the inferior performance of MMD in previous studies was due to inherent limitations in MMD, or to the particular formulation used.

Rubin, Paul A. (1989). “Evaluating the Maximize Minimum Distance Formulation of the Linear Discriminant Problem”. In: European Journal of Operational Research 41, 2, pp. 240-248. ISSN: 03772217. DOI: 10.1016/0377-2217(89)90390-1.

BibTeX

@article{Rubin1989a, title = {Evaluating the Maximize Minimum Distance Formulation of the Linear Discriminant Problem}, author = {Paul A. Rubin}, journal = {European Journal of Operational Research}, year = {1989}, month = {July}, number = {2}, pages = {240–248}, volume = {41}, doi = {10.1016/0377-2217(89)90390-1}, url = {http://dx.doi.org/10.1016/0377-2217(89)90390-1}, }

Incentive payment and nonmanagerial productivity: an interrupted time series analysis of magnitude and trend (1988)

Using an interrupted time series design, this paper examined long-term changes in the magnitude and trend of productivity following the introduction of nonmanagerial incentive payment. Analyses of three 114-month data series from a unionized iron foundry revealed a strong asymptotic increase in productivity without corresponding increases in labor costs or employee grievances. These findings substantiated a power-curve trend in productivity, implying that wage incentives might stimulate employees to learn efficacious task behaviors, and suggesting that incentive payment’s long-term productive efficacy might be greater than many of the short-term gains reported in prior research.

Wagner III, John A, Paul A. Rubin, and Thomas J. Callahan (1988). “Incentive Payment and Nonmanagerial Productivity: An Interrupted Time Series Analysis of Magnitude and Trend”. In: Organizational Behavior and Human Decision Processes 42, 1, pp. 47-74. DOI: 10.1016/0749-5978(88)90019-2.

BibTeX

@article{WagnerIII1988, title = {Incentive Payment and Nonmanagerial Productivity: An Interrupted Time Series Analysis of Magnitude and Trend}, author = {John A {Wagner III} and Paul A. Rubin and Thomas J. Callahan}, journal = {Organizational Behavior and Human Decision Processes}, year = {1988}, month = {August}, number = {1}, pages = {47–74}, volume = {42}, doi = {10.1016/0749-5978(88)90019-2}, }

Examining short-rotation hybrid poplar investments by using stochastic simulation (1986)

We examined and compared short-rotation hybrid poplar investments using standard discounted cash flow and stochastic simulation. With stochastic simulation, triangular probability density functions were used to describe the values for three important uncertain factors: product price, product yield, and harvest and transport costs. We found that the net present value per acre could range from a minus $1010 with a mean value of about $140, using a 4% discount rate. Based on the assumptions we used, product price uncertainty was found to be the major cause of uncertainty surrounding the financial returns from a short-rotation system. Because information on future prices is limited, decisions about short-rotation systems should be made carefully. Analyzing uncertainty by using imprecise input data cannot produce both accurate and precise results, but thinking through the uncertainties and collecting information should help investors make better choices.

Lothner, David C, Howard M. Hoganson, and Paul A. Rubin (1986). “Examining Short-Rotation Hybrid Poplar Investments by Using Stochastic Simulation”. In: Canadian Journal of Forest Research 16, 6, pp. 127-1213. DOI: 10.1139/x86-215.

BibTeX

@article{Lothner1986, title = {Examining Short-Rotation Hybrid Poplar Investments by Using Stochastic Simulation}, author = {David C. Lothner and Howard M. Hoganson and Paul A. Rubin}, journal = {Canadian Journal of Forest Research}, year = {1986}, month = {December}, number = {6}, pages = {127–1213}, volume = {16}, doi = {10.1139/x86-215}, }

Short-run characteristics of samples drawn by random walks (1985)

An alternative to using acceptance-rejection methods to generate a sample of points distributed uniformly over a region is to employ a random walk over that region. The sequence of points generated by a random walk has been shown, under certain easily satisfied conditions, to be a realization of a vector-valued discrete parameter Markov process, and to have the uniform distribution as its limiting distribution. The purpose of this paper is to point out that even if the marginal distribution of each point is actually uniform, rather than merely being asymptotically uniform, small samples may exhibit nonuniform characteristics due to serial autocorrelation within the sample.

Rubin, Paul A. (1985). “Short-Run Characteristics of Samples Drawn by Random Walks”. In: Communications in Statistics – Simulation and Computation 14, 2, pp. 473-490. ISSN: 0361-0918. DOI: 10.1080/03610918508812451.

BibTeX

@article{Rubin1985, title = {Short-Run Characteristics of Samples Drawn by Random Walks}, author = {Paul A. Rubin}, journal = {Communications in Statistics – Simulation and Computation}, year = {1985}, month = {April}, number = {2}, pages = {473–490}, volume = {14}, doi = {10.1080/03610918508812451}, publisher = {Taylor & Francis}, url = {http://www.informaworld.com/smpp/content~{}db=all~{}content=a780081848}, }

Generating random points in a polytope (1984)

An algorithm is presented for generating pseudorandom variates distributed uniformly over an arbitrary convex polytope in a Euclidean space of arbitrary dimension. Many commonly used methods for generating uniform variates require that the polytope be expressed as the solution set of a system of linear inequalities; the algorithm presented here requires instead that the polytope be presented in terms of a finite generating set, typically the set of its vertices. Included in the algorithm are procedures for identifying all faces of the polytope and for decomposing the polytope into simplices.

Rubin, Paul A. (1984). “Generating Random Points in a Polytope”. In: Communications in Statistics – Simulation and Computation 13, 3, pp. 375-396. ISSN: 0361-0918. DOI: 10.1080/03610918408812382.

BibTeX

@article{Rubin1984a, title = {Generating Random Points in a Polytope}, author = {Paul A. Rubin}, journal = {Communications in Statistics – Simulation and Computation}, year = {1984}, month = {May}, number = {3}, pages = {375–396}, volume = {13}, doi = {10.1080/03610918408812382}, publisher = {Taylor & Francis}, url = {http://www.informaworld.com/smpp/content~{}db=all~{}content=a780075515}, }

Fuzzy goal programming with nested priorities (1984)

This paper proposes a new approach to formulating fuzzy priorities in a goal programming problem. The proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works [7, 10]. The principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker’s intentions than in other noninteractive approaches [8, 9, 10, 14], in some of which a fixed hierarchy of goals is assumed.

Rubin, Paul A. and Ram Narasimhan (1984). “Fuzzy Goal Programming with Nested Priorities”. In: Fuzzy Sets and Systems 14, 2, pp. 115-129. ISSN: 01650114. DOI: 10.1016/0165-0114(84)90095-2.

BibTeX

@article{Narasimhan1984, title = {Fuzzy Goal Programming with Nested Priorities}, author = {Paul A. Rubin and Ram Narasimhan}, journal = {Fuzzy Sets and Systems}, year = {1984}, month = {November}, number = {2}, pages = {115–129}, volume = {14}, doi = {10.1016/0165-0114(84)90095-2}, url = {http://dx.doi.org/10.1016/0165-0114(84)90095-2}, }

Implementation of a subgradient projection algorithm (1983)

This paper discusses the implementation of an algorithm due to Sreedharan [8] for the minimization, subject to linear constraints, of an objective function composed of the sum of a piecewise-affine, convex function with a smooth, strictly convex function. Successful techniques for two subproblems arising in the algorithm, a projection problem and a line search problem, are described in detail. Computational experience with the algorithm on several test problems is presented.

Rubin, Paul A. (1983). “Implementation of a Subgradient Projection Algorithm”. In: International Journal of Computer Mathematics 12, 3-4, pp. 321-328. ISSN: 0020-7160. DOI: 10.1080/00207168208803345.

BibTeX

@article{Rubin1983a, title = {Implementation of a Subgradient Projection Algorithm}, author = {Paul A. Rubin}, journal = {International Journal of Computer Mathematics}, year = {1983}, month = {January}, number = {3-4}, pages = {321–328}, volume = {12}, doi = {10.1080/00207168208803345}, publisher = {Taylor & Francis}, url = {http://www.informaworld.com/smpp/content~{}db=all~{}content=a771010048}, }

Economic order quantities with quantity discounts: grandma does it best (1983)

We examine a new algorithm developed by Kuzdrall and Britney [5] for locating the optimal order quantity in the presence of quantity discounts. Their algorithm, based on a model for the supplier’s formulation of the price schedule, involves a regression analysis to identify the supplier’s variable cost per unit and the fixed cost that the supplier seeks to recover, followed by an iterative search for the optimum. The authors describe this method as a “convenient alternative to the aimless searching of traditional approaches” [5, p. 101]. We examine the allegation of superiority of their total setup lot-sizing model over the classical method and dispute their claim of superiority.

Rubin, Paul A., David M. Dilts, and Beth A. Barron (1983). “Economic Order Quantities with Quantity Discounts: Grandma Does It Best”. In: Decision Sciences 14, 2, pp. 270-281. DOI: 10.1111/j.1540-5915.1983.tb00185.x.

BibTeX

@article{Rubin1983, title = {Economic Order Quantities with Quantity Discounts: Grandma Does It Best}, author = {Paul A. Rubin and David M. Dilts and Beth A. Barron}, journal = {Decision Sciences}, year = {1983}, month = {April}, number = {2}, pages = {270–281}, volume = {14}, doi = {10.1111/j.1540-5915.1983.tb00185.x}, }

A note on the geometry of reciprocal fuzzy relations (1982)

A proposed characterization of the set of reciprocal fuzzy relation matrices is shown to be incorrect, and the correct dimension of the set is computed.

Rubin, Paul A. (1982). “A Note on the Geometry of Reciprocal Fuzzy Relations”. In: Fuzzy Sets and Systems 7, 3, pp. 307-309. ISSN: 01650114. DOI: 10.1016/0165-0114(82)90058-6.

BibTeX

@article{Rubin1982, title = {A Note on the Geometry of Reciprocal Fuzzy Relations}, author = {Paul A. Rubin}, journal = {Fuzzy Sets and Systems}, year = {1982}, month = {May}, number = {3}, pages = {307–309}, volume = {7}, doi = {10.1016/0165-0114(82)90058-6}, url = {http://dx.doi.org/10.1016/0165-0114(82)90058-6}, }

Conference proceedings

A novel 3D underwater WSN deployment strategy for full-coverage and connectivity in rivers (2016)

In this paper, we propose a novel 3D Underwater Wireless Sensor Network Deployment scheme for solid detection in rivers. Our objective is to minimize the number of deployed underwater sensors within a target field installation while ensuring i) the required Quality of Monitoring (QoM) (i.e., differentiated probabilistic detection) and ii) wireless network connectivity. To generate the best topology, we propose a novel deployment heuristic, named 3D-UWSN-Deploy, based on a subcube tessellation of the monitored field installation and a mixed integer linear program optimization. To gauge the effectiveness of 3D-UWSN-Deploy, we compare it with the most prominent related strategies. Simulation results show that our proposal is scalable and obtains the best performance in terms of cost deployment, quality of monitoring and connectivity.

Khalfallah, Z.; Fajjari, I.; Aitsaadi, N.; Rubin, P. & Pujolle, G. A Novel 3D Underwater WSN Deployment Strategy for Full-Coverage and Connectivity in Rivers. 2016 IEEE International Conference on Communications (ICC), 2016, 1-7

​Offsetting inventory cycles in a storage space constrained environment (2000)

Murthy, N. N.; Benton, W. C. & Rubin, P. A. Offsetting inventory cycles in a storage space constrained environment. Proceedings of the 2000 Annual Meeting of the Decision Sciences Institute, 2000.

Minimizing tardiness with sequence dependent setup times: A comparative study of branch-and-bound, genetic search and simulated annealing (1996)

Tan, K.-C.; Narasimhan, R.; Rubin, P. A. & Ragatz, G. L. Minimizing tardiness with sequence dependent setup times: A comparative study of branch-and-bound, genetic search and simulated annealing. Proceedings of the 1996 Annual Meeting of the Decision Sciences Institute, 1996.

Jointly constrained order quantities with incremental discounts (1995)

Rubin, P. A. & Benton, W. C. Jointly constrained order quantities with incremental discounts. Proceedings of the 1995 Annual Meeting of the Decision Sciences Institute, 1995.

Jointly constrained order quantities and the natural inventory cycle (1994)

Murthy, N. N.; Benton, W. C. & Rubin, P. A. Jointly constrained order quantities and the natural inventory cycle. Proceedings of the 1994 Annual Meeting of the Decision Sciences Institute, 1994.

A partial enumeration algorithm for static capacitated location problems (1994)

Rubin, P. A. & Benton, W. C. A partial enumeration algorithm for static capacitated location problems. Proceedings of the 1994 Annual Meeting of the Decision Sciences Institute, 1994.

A heuristic solution procedure for scheduling cutting stock in a multi-period, multi-process facility with sequence dependent setups (1993)

D’Itri, M. & Rubin, P. A. A heuristic solution procedure for scheduling cutting stock in a multi-period, multi-process facility with sequence dependent setups. Proceedings of the 1993 Annual Meeting of the Decision Sciences Institute, 1993.

A heuristic for the single echelon dynamic capacitated facility location problem (1993)

Benton, W. C. & Rubin, P. A. A heuristic for the single echelon dynamic capacitated facility location problem. Proceedings of the 1993 Annual Meeting of the Decision Sciences Institute, 1993.

Scheduling in a sequence dependent setup environment with genetic search (1992)

Rubin, P. A. & Ragatz, G. L. Scheduling in a sequence dependent setup environment with genetic search. Proceedings of the 1992 Annual Meeting of the Decision Sciences Institute, 1992.

A formulation for scheduling cutting stock in a multi-period, multi-process facility with sequence dependent setups (1992)

A formulation for scheduling cutting stock in a multi-period, multi-process facility with sequence dependent setups. Proceedings of the 1992 Annual Meeting of the Decision Sciences Institute, 1992.

Pathologies in linear programming models for the two group discriminant problem (1990)

Pathologies in linear programming models for the two group discriminant problem. Proceedings of the 1990 Annual Meeting of the Decision Sciences Institute, 1990.

Efficient heuristics for identification of outliers in discriminant analysis (1989)

Rubin, P. A. Efficient heuristics for identification of outliers in discriminant analysis. Proceedings of the 1989 Annual Meeting of the Decision Sciences Institute, 1989.

Economic benefits of cooperation between purchaser and vendor (1988)

Rubin, P. A. Economic benefits of cooperation between purchaser and vendor. Proceedings of the Nineteenth Annual Meeting of the Midwest Decision Sciences Institute, 1988.

The maximize minimum distance formulation of the linear discriminant problem (1987)

Rubin, P. A. The maximize minimum distance formulation of the linear discriminant problem. Proceedings of the 1987 Annual Meeting of the Decision Sciences Institute, 1987.

Forcing convexity in quadratic regression models (1986)

Rubin, P. A. Forcing convexity in quadratic regression models. Proceedings of the Seventeenth Annual Meeting of the Midwest Decision Sciences Institute, 1986.

Exponential smoothing of series with geometric trend: an experimental analysis (1986)

Rubin, P. A. Exponential smoothing of series with geometric trend: an experimental analysis. Proceedings of the 1986 Annual Meeting of the Decision Sciences Institute, 1986.

Analyzing simulation output using time series intervention analysis (1985)

Narasimhan, R.; Rubin, P. A. & Melynk, S. A. Analyzing simulation output using time series intervention analysis. Proceedings of the Seventeenth Annual Meeting of the American Institute for Decision Sciences, 1985.

An intervention analysis of the productive efficacy of an equitable group incentive plan (1985)

Wagner III, J. A; Rubin, P. A. & Callahan, T. J. An intervention analysis of the productive efficacy of an equitable group incentive plan. Proceedings of the Seventeenth Annual Meeting of the American Institute for Decision Sciences, 1985.

A test of purchase decision rules for the case of stochastic prices (1984)

Rubin, P. A. A test of purchase decision rules for the case of stochastic prices. Proceedings of the Sixteenth Annual Meeting of the American Institute for Decision Sciences, 1984.

A comparison of three approaches to linear programming with multiple objectives (1981)

Rubin, P. A. A comparison of three approaches to linear programming with multiple objectives. Proceedings of the Midwest American Institute for Decision Sciences Conference, 1981.

Book chapters

Mixed Integer Classification Problems (2009)

Rubin, Paul A. (2009). “Mixed Integer Classification Problems”. In: Encyclopedia of Optimization. Ed. by Christodoulos A. Floudas and Panos M. Pardalos. 2nd. Springer, pp. 2210-2214. ISBN: 978-0-387-74758-3.

BibTeX

@incollection{Rubin2009, title = {Mixed Integer Classification Problems}, author = {Paul A. Rubin}, booktitle = {Encyclopedia of Optimization}, publisher = {Springer}, year = {2009}, edition = {2nd}, editor = {Christodoulos A. Floudas and Panos M. Pardalos}, pages = {2210–2214}, }

Linear Programming Models for Classification (2009)

Rubin, Paul A. (2009). “Linear Programming Models for Classification”. In: Encyclopedia of Optimization. Ed. by Christodoulos A. Floudas and Panos M. Pardalos. 2nd. Springer. ISBN: 978-0-387-74758-3.

BibTeX

@incollection{PaulA.Rubin2009, title = {Linear Programming Models for Classification}, author = {Paul A. Rubin}, booktitle = {Encyclopedia of Optimization}, publisher = {Springer}, year = {2009}, edition = {2nd}, editor = {Christodoulos A. Floudas and Panos M. Pardalos}, }

Optimizing with Piecewise Smooth Functions (1996)

Rubin, Paul A. (1996). “Optimizing with Piecewise Smooth Functions”. In: Computational Economics and Finance. Ed. by Hal R. Varian. Springer-Verlag. ISBN: 0387945180.

BibTeX

@incollection{PaulA.Rubin1996, title = {Optimizing with Piecewise Smooth Functions}, author = {Paul A. Rubin}, booktitle = {Computational Economics and Finance}, publisher = {Springer-Verlag}, year = {1996}, editor = {Hal R. Varian}, url = {http://books.google.com/books?hl=en&lr=&id=iXFkN278_rQC&pgis=1}, }

A Simulation-Based Time-Series Policy-Capturing Methodology for Studying Recurring Decision Making in Organizations (1988)

Moch, Michael, Aaron Buchko, and Paul A. Rubin (1988). “A Simulation-Based Time-Series Policy-Capturing Methodology for Studying Recurring Decision Making in Organizations”. In: Advances in Information Processing in Organizations. Ed. by Robert L. Cardy, Sheila M. Puffer and Jerry M. Newman. JAI Press. ISBN: 978-0892326891.

BibTeX

@incollection{Moch1988, title = {A Simulation-Based Time-Series Policy-Capturing Methodology for Studying Recurring Decision Making in Organizations}, author = {Michael Moch and Aaron Buchko and Paul A. Rubin}, booktitle = {Advances in Information Processing in Organizations}, publisher = {JAI Press}, year = {1988}, editor = {Robert L. Cardy and Sheila M. Puffer and Jerry M. Newman}, }

Future market uncertainty: a case for short-rotation systems (1987)

Hoganson, Howard M, David C. Lothner, and Paul A. Rubin (1987). “Future market uncertainty: a case for short-rotation systems”. In: Achieving Wood Energy Potential: Evidence in Northeastern Minnesota. Ed. by Dennis P. Bradley and David C. Lothner. Vol. General Technical Report NC-114. Forest Service, North Central Forest Experiment Station, St. Paul MN: U. S. Department of Agriculture.

BibTeX

@incollection{Hoganson1987, title = {Future market uncertainty: a case for short-rotation systems}, author = {Howard M. Hoganson and David C. Lothner and Paul A. Rubin}, booktitle = {Achieving Wood Energy Potential: Evidence in Northeastern Minnesota}, publisher = {U. S. Department of Agriculture}, year = {1987}, address = {Forest Service, North Central Forest Experiment Station, St. Paul MN}, editor = {Dennis P. Bradley and David C. Lothner}, volume = {General Technical Report NC-114}, }