ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR Summary Report DEVELOPMENT OF GENERALIZED MATHEMATICAL PROCEDURES FOR OPTIMUM ASSEMBLY OF POTENTIALLY EFFECTIVE COMBAT CREWS March 1, 1954, to June 30, 1955 Project 2226 U. S. AIR FORCE AIR RESEARCH AND DEVELOPMENT COMMAND CONTRACT NO. AF 18(600)-1050 June 1955

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ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Contract No.: AF 18(600)-1050 Budget Project No.: 670-193 Contract Title: Development of Generalized Mathematical Procedures for Optimum Assembly of Potentially Effective Combat Crews Issuing Office: The Air Research and Development Command Contractor: The Regents of the University of Michigan Monitoring Agency: Director, Detachment 4 (Crew Research Laboratory), Air Force Personnel and Training Research Center, Randolph Field, Texas Principal Investigator: Dr. Paul S. Dwyer Period: March 1, 1954, to June 30, 1955 ii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN ABSTRACT This report provides summary information about the work on the project. It presents: (1) an outline of the contents of the extended research report, previously submitted, which gives the detailed results of the study, (2) a statement of the basic conclusions and recommendations resulting from the study, (3) the names, duties, extent of service, reimbursement, and work accomplished for the various personnel engaged on the project, and (4) a summary accounting for the use of contract funds. OBJECTIVE The general objective of this contract is the development of generalized mathematical procedures for optimum assembly of potentially effective combat crews. More detailed objectives call for (a) a study of the general mathematical theory underlying the group assembly problem, (b) the determination of suitable methods of predicting crew scores from individual scores, (c) the development of a method and-tedhnique for finding the maximum assembly sum, (d) the practical adaptation of this technique to a high-speed digital computer, (e) the development of methods for obtaining approximate solutions, and (f) the determination of suitable measures of the adequacy of an approximation. iii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 1. OUTLINE OF THE CONTENTS OF THE REPORTS PREVIOUSLY SUBMITTED A copy of a research report, containing 14 chapters with references, appendix materials, and extensive illustrations, which may be said to give a detailed statement of the work on and results of the project, has been placed in the hands of Dr. Roby. Copies of an abbreviated report of about 80 pages, which may serve as the basis of a technical report in the AFPTRC series, have also been presented. The following outline of topics in the extended research report gives some indication of the various topics studied and the detail of the presentation. The outline of topics in the abbreviated report is similar. 2. OlTLTLE OF THE EXTENDED REPORT Chapter Contents I The general group assembly problem 1. Introduction 2. Group scores 3. Assembly scores 4. Mathematical statement of the problem 5. Groupings 6. Relation to personnel classification problem and similar problems 7. Use of permutation sets 8. Restatement of the problem using permutation sets II Transformations 1. Introduction 2. Subtraction of a constant 3. Deviate transformations 4. Approximate deviate transformations 5. Large deviate transformations 6. Extreme transformations III The distribution of all possible assembly sums 1. Introduction 2. The mean and variance of the distribution of all possible assembly sums for any k.1 —.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Chapter Contents 3. The third central moment of the distribution of all possible assembly sums for k = 2, 3, and 4 4. The fourth central moment when k = 2 IV Application of analysis of variance and determination of a mathematical model appropriate to empirical data 1. Introduction 2. Analysis of variance when k = 2 3. Analysis of variance when k = 3 4. Analysis of variance when k = 4 Analysis of variance for higher values of k 6. Determination of a mathematical model appropriate to empirical data V Mathematical models for group scores 1. Introduction 2. The observed score matrix for individuals 3. The rating matrix for individuals 4. The rating matrix for subgroups 5. The observed score matrices for subgroups 6. A mathematical model for group scores 7. Special cases of the general model 8. The simplification of the mathematical model by ignoring the main effects 9. Determination of a mathematical model from empirical group scores without the necessity of ratings for individuals 10. Conclusion VI Condensation of group scores 1. Introduction 2. Groupings of observed scores for individuals 3. Groupings by ratings for individuals 4. Disregard of ratings for individuals 5. Groupings of ratings for subgroups obtained from scores or from ratings for individuals or classes 6. Ratings based on scores for subgroups only 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Chapter Contents 7. Groupings of observed scores for subgroups 8. Groupings of ratings for subgroups 9. Effective reduction of k 10. Precise functional models 11. Conclusion VII The group assembly problem as a problem in linear programming 1. Linear programming problems 2. The two-dimensional problem 3. The general assembly problem 4. Methods of solution when k = 2 5. The method of reduced matrices VIII The two-dimensional assembly problem 1. Introduction 2. Conditions of solution 3. Use of extreme transformations 4. Method of bounding sets 5. Marginal zero transformations 6. Determination of a completely reduced matrix 7. Determination of an optimal solution from a completely reduced matrix 8. Solution with the method of reduced matrices 9. Solution of the quota problem with the detailed method of optimal regions IX Successive applications of two-dimensional techniques 1. Introduction 2. A succession of two-dimensional problems 3. Approximate solution of the general problem, using totals of subclasses 4. Use of deviate scores in determining suitable subclasses Use of approximate deviate scores in determining suitable subclasses 6. Use of results of analysis of variance in determining suitable subclasses 7. Conclusion 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Chapter Content s X The three-dimensional assembly problem 1. Introduction 2. The use of extreme transformations 3. The use of marginal zero transformations 4. The reduction of the Gijh matrix 5. The determination of the solution from the reduced matrix 6. The three-dimensional problem with frequencies 7. The reduced grouped matrix 8. Transformations leading to reduced grouped matrices 9. Reduced grouped matrix transformations 10. The determination of the solution from the reduced grouped matrix 11. Conclusion XI The general assembly problem 1. Introduction 2. Solution of the general problem with k = 4 and N = 2 3. Solution of the general problem with k = 4 and N = 3 4. Solution of the general problem with k = 4 and N = 3, using the method of reduced matrices 5. The solution of a frequency problem with k = 4 and n = 3 6. The condensed solution of the frequency problem of section 11.5 7. The solution of a frequency problem with k = 5 and n = 3 8. The generality of the method of reduced matrice s XII Approximate solutions 1. Introduction 2. Approximate solutions using deviates 3. Approximate solutions using approximate deviates 4. Approximate solutions using large deviates Approximate solutions using large row deviates 6. Approximate solutions of problems with 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Chapter Contents k > 2, using a succession of two-dimensional techniques 7. Approximate solutions using reduced matrices 8. Approximate solutions using reduced matrices and successive interchanges 9. Measures of the adequacy of an approximation 10. Conclusion XIII Punched-card and machine methods 1. Use of marginal punched cards 2. Use of IBM punched cards and machines 3. Use of electronic digital computers XIV Concluding remarks 1. Summary 2. Recommendations for further research 3. BASIC CONCLUSIONS AND RECOMMENDATIONS A. BASIC CONCLUS.IONSi The following conclusions parallel the objectives of the contract as stated above. 1. A study of the general mathematical theory of the group assembly problem has shown it to be more than a generalization of the transportation problem; many methods used in solving the transportation problem are not adequate for handling the group assembly problem. This study led to the method of reduced matrices, which is directly applicable to the transportation problem as well as to the general group assembly problem. 2. The study of group scores, and particularly the study of the variation of the group assembly sum by analysis of variance techniques, has led to a suitable mathematical model for determining the group scores. The analysis leads to the conclusion that the interaction terms are the important ones. 3. The study leads to the conclusion that the method of reduced matrices is a recommended method both for the transportation problems and for the general assembly sum problem. 4. The conclusion is drawn that the method of reduced matrices is adapted to high-speed digital computers. This was demonstrated by 5

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN making the reductions on M I D A C for many k = 3 problems. 5. Useful approximate solutions can be obtained by: (a) the earlier steps of the method of reduced matrices, (b) the results of the deviate transformation, and (c) the successive use of two-dimensional techniques. 6. The conclusion is drawn that suitable estimates of the adequacy of an approximation can be determined from the same partially reduced matrix which provides the value of the approximate sum. In general, the objectives of the contract seem to be completed successfully, especially when k = 3, and the accomplishment of these objectives brings the need for additional studies into focus. These are indicated by the following recommendations for future study. B. RECOMMENDATIONS 1. Additional study of the solution when the number of positions is large. 2. Extension of the techniques in which several alternative criteria of group effectiveness are to be applied simultaneously. 3. More study on the problem of grouping the group scores into classes. 4. Extended study of the use of electronic digital computers in obtaining exact and approximate solutions. 5. A study of the effects of the errors of the fallible group scores on the process of maximization. 6. An investigation leading to the identification of the group assembly problem with other problems which may be encountered by the Air Force, the Army, the Navy, or industrial organizations having contracts with the Armed Forces. 4. SUMMARY OF PERSONNEL ACTIVITY The names of the persons participating in the work on the contract, together with the duties, period of service, reimbursement, and work accomplished, are indicated in the table and footnotes below. 6

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Portion of Name Title Time Devoted to Contract Work Dwyer, Paul S. Principal Investigator Up to 40 pours (Professor of Mathematics, per month Consultant in Statistical Research Laboratory) Graves, Glenn Assistant in Research Varied2 (Graduate Student) Hubbell, Charles Assistant in Research Full time3 (Graduate Student) Lott, Fred Assistant in Research Half time4 (Graduate Student) Rider, Leonard Assistant in Research Half time5 (Graduate Student) Taylor, Patricia Assistant in Research Half time6 (Graduate Student) Bassett, Karen Typist Half time7 Parker, Kathryn Secretary Varied8 (Student) 1During the summer months of 1954, June 13 to September 13, Dr. Dwyer worked full time on the project. Throughout the other 13 months of the contract, be worked full time on his University duties, and his work on the project was limited to 40 hours per month. He had general charge of the work on the project and the preparation of the reports and worked a total of 982-1/2 hours at the rate of $8.40 per hour. Mr. Graves began working on the project January 27, 1955, and terminated June 30, 1955. His particular job was to assist in translating some..of the methods to routines which could be performed on M I D A C (Michigan Digital Automatic Computer), which he did successfully. He worked a total of 100 hours at the rate of $3.00 per hour. 3Mr. Hubbell began work on the project June 7, 1954, and terminated September 13, 1955. He assisted in the determination of the general solution when k = 3, and applied the methods to IBM machines and 7..

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN marginal punched cards. He worked a total of 417-1/2 hours at the rate of $2.00 per hour. 4Mr. Lott began work on the project June 7, 1954, and terminated September 13, 1955. He studied the problem of moments of the assembly sum and derived and illustrated the important formulas of Chapter III of the extended report. He worked a total of 310-1/2 hours at the rate of $2.00 per hour. 5Mr. Rider started work on the project on September 18, 1954, and terminated January 6, 1955. He assisted in perfecting the k = 2 and k = 3 techniques and in showing how the methods could be applied to k = 4 and k = 5 problems. He worked a total of 274-.1/2 hours at $2.00 per hour. 6Miss Taylor started work on the project September 23, 1954, and terminated June 30,.1955. She contributed greatly to the development of the method of reduced matrices, particularly in formalizing the reduced grouped matrix transformations. She acted as a coauthor with the principal investigator in writing all reports prepared during 1955. She worked a total of 741-1/2 hours at $2.00 per hour. 7Mrs. Bassett began work on January 10, 1955, and terminated May 27, 1955. From April 1.1 until she terminated she worked full time. She typed the reports and numerical illustrations which describe the results of the various aspects of the contract work. She worked a total of 517-1/2 hours at $1.20 per hour. 8Miss Parker worked half time on the project from August 24, 1954, to September 20, 1954 typing the project reports. She worked a total of 60 hours at $1.40 per hour. 5. SUMMARY ACCOUNTING OF UTILIZATION OF CONTRACT FUNDS This summary is made as of June 25, 1955. The exact final figures cannot at this date be determined since, for example, the cost of this summary report is not known though an estimate is available, It should be noted that the figures below apply to the 16-month contract as renegotiated rather that to the original 13-month contract. 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN USE OF CONTRACT FUNDS AS CALCULATED JUNE 25, 1955 Services of principal investigator $8,253.00 Services of assistants in research 3,788.00 Clerical (for typing reports) 705.00 Total personnel $12,746. 0 fc5%s on personnel during 1954~ 4 589.77 Service charge 537% on personnel during 1955J Reproducing Reports (includes $60.00 estimate for this summary report) 323 95 Travel (trips to San Antonio and New York) 354.13 Tabulating and M I D A C 237.90 Supplies 41.78 Remaining funds as of June 25, 1955 876.47 Total Funds $19,170.00

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