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APPLICATIONS OF LINEAR PROGRAMMING TECHNIQUES TO SOr-IE PROBLEMS OF PRODUCTION PLANNING OVER TIME by R M Ziki and RB Lo Anderson Institute of Statistios Mimeograph Series No o 2.88 AprU~ 1961 iv TABLE OF CONTENTS 1.0 LIST OF TABLES • • • • • • 2.0 INTRODUCTION • • • • • • • • • • • • • • • · • • • • • • • · · • • • 2.1 Decision Problems • • 2.2 The Problem of Fluctuations Production Over Time • 2.3 Mathematical Formulation of 3.0 A SIMPLIFIED MODEL • • • • • in a Firm's • • • • • • • • • • a Basic Model • • • • • • • • • • • • • • • •• • • • • Page vi 1 3.1 A Procedure for Obtaining ¢ = Max Z • • • • 11 1.2 Allocation to a S:i.ngle Time Period 11 3.3 Allocation to Two ~ime Periods 3.4 Two Useful Lemmas • • • • • 12 Allocation to Three Time Periods • • • • • 13 3.6 Allocati on Theorems • • • • • • • • • • • 18 3.7 Su.mrnary • • • • • • • • • • • • • • • • • • • • • • • • 27 4.0 LINEAR PROGRAMMING • 4.1 4.2 4.3 4.4 4.5 5.0 • • • • • • · 29 29 The Problem of Line ar Programming • • • • Brief Mathematical Background • • • • • Properties of a Solution to the Linear Programming Problem • • • • • • • • • • • • • • The Simplex Computational Procedure • • • • • The ll1al Problem of Linear Programming USE OF THE LINEAR PROGRAMMING PROCEDURES FOR THE PRODUCTION STABILIZATION PROBLEM • • • • • • • • 30 32 35 40 A Linear Programming Formulation of the Production Stabilization Problem • • A Line ar Programming Formulati on of the Simplified Model • • • , • • • • • • • • •• • ••• • • • • A Solution for the Simplified Problem by the Simplex Method • • • • • • • • • • • • A Useful Initial Basis to Start the Iterative Procedure for Solving the General Production Stabilization Problem • • • 0 A Criterion for Selecting the Vector to be Introduced into the Basis • • Procedure for Finding the Optimal Solutions (One or More) to the General Production Stabilization Problem for Each Value of c in the Interval c ~O 41 43 45 60 68 74 v TABlE OF CONTENTS (continued) Page 6.0 77 SUMMARY AND LIMITATIONS AND ElTENSIONS OF THE STUDIED MODEL 601 6~2 Sunnnary 0- • • • e e e e • • • Limitations and Extensions of the Studied Model • LIST OF REFERENCES & • & • • • 0 o & • e 0' • e • • • • • • • • 77 83 • 87 • vi 1.0 LIST OF TAB LBS Page 3,,1 Optimal allocation plans for the simplified model, (3.01) 28 5.1• 5.2 5.3, Simplex tableau for B(l) 48 Simplex tableau for B(.f,) " • • • • • • • • • •• 52 Simplex tableau for B(T) • • • • • • • • • • • ~7 Simplex tableau for B(T+l) " 61 5.5 Matrix formulation of linear constraints for the general stabilization problem ••• 62 e e " W W" and Farr$ Do 195Jo Linear programming and profit preference scheduling for a manufacturing firm Journal of Operations Research Society of America, Vol Charnes" Ao , Cooper, W W and Henderson, A 1953 Introduction to Linear Programming" John Wiley and Sons, Inc., New York Dantzig.ll G B o 1951 Maximization of a linear function of variables subject to linear inequalities Chapter XXI in Activity Analysis of Production and Allocati on o Edited by T Koopmans Cowles Commission Monograph 13$1 John Wiley and Sons, InG New York n: Dantzig:.> G B"$! Orden$) A" and Wolfe.9 P~ 1954" Generalized Simplex Method for Minimizing a Linear Form Under Linear Inequality Restraints RAND Report RM-1264" The RAND Corporation, Santa Monica,\) California Dorfman$) Ro 1951 Application of Linear Programming to Theory of the Firm University of California Press, Berkeley, California Gass, S" 10 1958 Linear Programming)! Methods and Applications McGraw;=Hill,Book COo$l Inc New York o Hoffman$) A Jo 1955" How to solve a linear programming problem ppo 397-424 in Directorate of Management Analysis: Proceedings of the SeconaSymposium in Linear Programming Edited by H Antosiewicz" DeS/Comptroller)! Headquarters, UoS Air Force, Washington$) D Co Karlin, So 1959 Mathematical Methods and Theory in Games, VoL 1.9 Addison-Wesley Publishing Programming and Economics Co".9 Inc Reading,\) Massachusetts" Riley, V" and Gass,\l So I 19580 Bibliography on Linear Programming and Related Techniques" Johns Hopkins Press, Baltimore, Maryland Simmonds, No W'" England 1959 Bananas Longmans Green and Coo, Ltd., London, INSTITUTE OF STATISTICS NORTH CAROLINA STATE COLLEGE (Mimeo Series available for distribution) 258 Hoeffding,Wassily On sequences of sums of independent random vectors 259 Webster, J T., A H E Grandage, R J Hader, R L Anderson dent variate in a linear estimator June, 1960 260 Chakravarti, I M A decision procedure for the inclusion of an indepen- On some methods of construction of partially balanced arrays July, 1960 261 Roy, S N and R Gnanadesikan On certain altemative hypotheses on dispersion matrices August, 1960 262 Murthy, V K On the distribution of averages over the val'ious lags of certain statistics related to the serial correlation coefficients August,} 960 263 Anderson, R L Some needed developments in multivariate analysis 264 Chapman, D G., W S Overton and A L Finkner August, 1960 Methods of estimating dove kill October, 1959 265 Eicker, FriedheIm Consistency of parameter-estimates in a linear time-series model 266 Eicker, FriedheIm 1960 A necessary and sufficient condition for consistency of the LS estimates in linear regression 267 Smith, W L October, 1960 On some general renewal theorems for nonidentically distributed variables 268 Duncan, D B 1960 October, 1960 Bayes rules for a common multiple comparisons problem and related Student-t problems 269 Bose, R C Theorems in the additive theory of numbers 270 Cooper, Dale and D D Mason 271 Eicker, FriedheIm October, November, November, 1960 Available soil moisture as a stochastic process December, 1960 Central limit theorem and consistency in linear regression December, 1960 272 Rigney, Jackson A The cooperative organization in wildlife statistics Presented at the 14th Annual Meeting, Southeastern Association of Game and Fish Commissioners, Biloxi, Mississippi, October 23-26, 1960 Published in Mimeo Series, January, 1961 273 Schutzenberger, M T On the definition of a certain class of automata January, 1961 274 Roy, S N and J N Shrizastaza January, 1961 275 Ray-Chaudhuri, D K Inference on treatment effects and design of experiments in relation to such inferences An algorithm for a minimum cover of an abstract complex February, 1961 276 Lehman, E H., Jr and R L Anderson Estimation of the scale parameter in the Weibull distribution using samples cen· sored by time and by number of failures March, 1961 277 Hotelling, Harold The behavior of some standard statistical tests under non-standard conditions February, 1961 278 Foata, Dominique 1961 On the construction of Bose-Chaudhuri matrices with help of Abelian group characters 279 Eicker, Friedheim Central limit theorem for sums over sets of random variables February, 1961 280 Bland, R P A minimum average risk solution for the problem of choosing the largest mean March, 1961 281 Williams, J S., S N Roy and C C Cockerham An evaluation of the worth of some selected indices 282 Roy, S N and R Gnanadesikan April, 1961 283 Schutzenberger, M T On the recurrence of patterns April, 1961 Investigations on factorial designs May, 1961 A quantitative analysis of the growth and regrowth of a forage crop 288 Zaki, R M and R L Anderson ning over time May, 1961 April, 1961 May, 1961 286 Bishir, J W Two problems in the theory of stochastic branching processes 287 Konsler, T R May, 1961 Equality of two dispersion matrices against altematives of intermediate specificity 284 Bose, R C and I M Chakravarti A coding problem arising in the transmission of numerical data 285 Patel, M S February, May, 1961 Applications of linear programming techniques to some problems of production plan-