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Linear programming and finite Markovian control problems
276
Book Reviews
ten for politicians (their last chance! (htr)), social scientists and 'merely curious citizens'. It gives a good idea of how people may get excited about one seat more or less when they let their private 'interests prevail over the interest of the community to which they belong. The principles discussed seem very reasonable and fair (like the principle of avoiding the Alabama paradox, i.e., avoiding the possibility that a state loses a seat when the total number of seats is increased by one, and the principle of avoiding the population paradox, i.e., avoiding the possibility that a state gains a seat when its quota decreases), except for one: the principle of staying within the quota, i.e., that no state should get more than its quota rounded up nor less than its quota rounded down. What should we do with 12 states, one with quota 57.1, the eleven others all with quota 3.9, and h = 100? It seems natural to give 4 seats to each of the smaller states and 56 to the larger, but this would violate staying within the quota. Fortunately, the authors recognize that this principle is not really compatible with the very idea of proportionality, and they replace it by a weaker one, already introduced by Webster. It is the principle of staying near the quota, i.e., that it should not be possible to take a seat away from one state and give it to another and simultaneously bring both of them nearer to their true quotas. The 'natural' apportionment in the example above satisfies this principle. The book gives a carefully balanced appraisal of all the historical quarrels and squabbles concerning the apportionment problem in the US House of Representatives. It comes, therefore, as a small shock to read the passage on page 79 where the authors comment on Huntington's view against the principle of staying within the quota. I cite: E. V. Huntington expressed the contrary view that "'it is a common misconception that in a good apportionment the actual assignment shouM not differ from the exact quota by more than one whole unit", no doubt because he realized that 'his' method (Hill's) did not invariably stay within the quota.
An eminent mathematician like Huntington has not deserved such a misplaced, non-academic accusation of wishful thinking. My personal explanation for this slip of the pen is the recent strong effort by the authors to propagate apportionment methods which stay within the quota (cf. [1]). Anyhow, any effort to find 'fair' apportionment
methods is of relative importance. This may be illustrated by the reviewer's idea of the weighted vote in the House [2] which guarantees a perfect representation of the states (or parties), whatever apportionment method is used. It is the idea of giving the vote of any representative of a state (or party) the following weight: the ratio of the true quota and the actual number of seats of that state (or party). Herman J.J. te R I E L E Centrum voor Wiskunde en Informatica Amsterdam, Netherlands
R eferen ces [1] M.L. Balinski and H.P. Young, The quota method of apportionment, Amer. Math. Monthly 82 (1975) 701-730. [21 H.J.J. te Riele, The proportional representation problem in the Second Chamber: an approach via minimal distances, Statist. Neerlandica 32 (1978) 163-179.
L.C.M. K A L L E N B E R G Linear Programming and Finite Markovian Control Problems Volume 148 in: Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam, 1983, iv + 245 pages, Dfl.33.00 This is another excellent publication from the Mathematical Centre in Amsterdam. The Mathematical Centre is sponsored by the Netherlands Government and is a well-known institution for promotion of mathematics and computer science. Since the introduction of Markov decision models by Bellman [1] there has been considerable interest in this field and it is now an active area of research. The book under review has evolved from the author's thesis and presents a linear programming approach for the solution of certain Markov control problems. The book comprises seven chapters, each of which begins with an introduction and a summary which provides an overview of the chapter. The first two chapters are introductory and survey some of the basic results. Markov decision problems with expected total reward as optimality criterion are dealt with in chapter 3. Comparisons
Book Reviews
of a linear programming approach with both iterative and policy improvement approaches are given. Contracting, positive and negative dynamic programming problems have also been discussed. Average reward is the optimality criterion for the problems in chapter 4. A linear programming formulation is presented and advantages over policy improvement techniques are described. Constrained Markov decision models are also discussed in this chapter. The concept of bias optimality is introduced in chapter 5. Some equivalent statements for bias optimality are given, leading to an algorithm for the computation of a bias optimal policy. In chapter 6, the two-person zero-sum stochastic game is considered in which one player controls the transition probabilities using a total reward criterion (under a contraction assumption) and an average reward criterion. A new proof for the existence of the value of the game is presented. Finally semi-Markov decision models are discussed in chapter 7 using discounted and undiscounted reward criteria. For further reading an up-to-date list of references is included, followed by a list of algorithms (34 in number) presented in the book. In conclusion this is a clearly written, cohesive research monograph, which will be of considerable interest to specialists in the areas of linear programming and Markov decision models, and would also be useful to advanced graduate students.
S.M. Janaid ZA ID1 University of Birmingham Birmingham, United Kingdom Reference [1] R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957).
S.G. TZAFESTAS (Ed.)
Optimisation and Control of Dynamic Operational Research Models Volume 4 in: North-Holland Systems and Control Series, North-Holland, Amsterdam, 1982, xx + 438 pages, Dfl.150.00 This is an excellent many-contribution volume bringing together many aspects and applications
277
of optimization and control for dynamic operational research models. It is really impressive how the editor could successfully collect papers from different sources. Contributors are from different countries, universities and departments (from business administration, through operational and systems research to engineering departments). The book involves twelve chapters and seven authors and nine co-authors from fifteen universities and eight countries. Chapter 1 considers the class of production-inventory systems and provides a survey of some recent results. Optimal and modal control for both continuous and discrete-time models, deterministic optimal control and stochastic formulation are studied. A closed-loop system with optimal management control policy is considered. A set of illustrative and application examples is provided. Chapter 2 is devoted to the study of the optimal linearquadratic control problem for some socioeconomic models. The forecasting in planning and determining of appropriate objective functions is considered. The chapter also considers the Kalman filter for design of forecasting models. Chapter 3 deals with models for resource allocation problems. These models are general enough fro studying a priori optimal schedule properties associated with the type of functions in the activities models and the resource and precedence constraints. This leads in some cases to analytical results or at least to the construction of more effective scheduling policies. Chapter 4 presents a survey of models for manpower systems (Markov, renewal longitudinak and Cambridge models) and provides a set of application examples of optimal control theory to manpower planning. The applicability of optimal control theory to manpower planning is considered. Near-optimal control using the Cambridge model is given. Chapter 5 is concerned with models of service systems and formulation of the optimization problems for them. The users act so as to optimize certain objective goals they set for themselves. The public policy of maximizing the expected return to society is considered and social optimal policies for different models of service systems are given. Chapter 6 treats problems of control theory related to traffic and transportation systems. In the field of road traffic control, optimal control of freeway traffic and optimal network flow is considered as well as urban street networks problems. In the field of air traffic control only aspects of optimal guidance and coordination of aircraft movements in the airfield approach corridors are treated. In the field of bus transportation systems, two optimization problems are presented, optimal scheduling of school buses and optimal control of bus networks. An overview of the problems related to transportation systems as a function of time horizons, controls and optimization is presented. Chapter 7 has two objectives. First to try to pose the economic dispatching problems into the framework of operations research and to indicate a few of the available methods and procedures that have been successfully employed in existing electric power systems. The second objective is to underline that there are still remaining many serious difficulties in
Book Reviews
ten for politicians (their last chance! (htr)), social scientists and 'merely curious citizens'. It gives a good idea of how people may get excited about one seat more or less when they let their private 'interests prevail over the interest of the community to which they belong. The principles discussed seem very reasonable and fair (like the principle of avoiding the Alabama paradox, i.e., avoiding the possibility that a state loses a seat when the total number of seats is increased by one, and the principle of avoiding the population paradox, i.e., avoiding the possibility that a state gains a seat when its quota decreases), except for one: the principle of staying within the quota, i.e., that no state should get more than its quota rounded up nor less than its quota rounded down. What should we do with 12 states, one with quota 57.1, the eleven others all with quota 3.9, and h = 100? It seems natural to give 4 seats to each of the smaller states and 56 to the larger, but this would violate staying within the quota. Fortunately, the authors recognize that this principle is not really compatible with the very idea of proportionality, and they replace it by a weaker one, already introduced by Webster. It is the principle of staying near the quota, i.e., that it should not be possible to take a seat away from one state and give it to another and simultaneously bring both of them nearer to their true quotas. The 'natural' apportionment in the example above satisfies this principle. The book gives a carefully balanced appraisal of all the historical quarrels and squabbles concerning the apportionment problem in the US House of Representatives. It comes, therefore, as a small shock to read the passage on page 79 where the authors comment on Huntington's view against the principle of staying within the quota. I cite: E. V. Huntington expressed the contrary view that "'it is a common misconception that in a good apportionment the actual assignment shouM not differ from the exact quota by more than one whole unit", no doubt because he realized that 'his' method (Hill's) did not invariably stay within the quota.
An eminent mathematician like Huntington has not deserved such a misplaced, non-academic accusation of wishful thinking. My personal explanation for this slip of the pen is the recent strong effort by the authors to propagate apportionment methods which stay within the quota (cf. [1]). Anyhow, any effort to find 'fair' apportionment
methods is of relative importance. This may be illustrated by the reviewer's idea of the weighted vote in the House [2] which guarantees a perfect representation of the states (or parties), whatever apportionment method is used. It is the idea of giving the vote of any representative of a state (or party) the following weight: the ratio of the true quota and the actual number of seats of that state (or party). Herman J.J. te R I E L E Centrum voor Wiskunde en Informatica Amsterdam, Netherlands
R eferen ces [1] M.L. Balinski and H.P. Young, The quota method of apportionment, Amer. Math. Monthly 82 (1975) 701-730. [21 H.J.J. te Riele, The proportional representation problem in the Second Chamber: an approach via minimal distances, Statist. Neerlandica 32 (1978) 163-179.
L.C.M. K A L L E N B E R G Linear Programming and Finite Markovian Control Problems Volume 148 in: Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam, 1983, iv + 245 pages, Dfl.33.00 This is another excellent publication from the Mathematical Centre in Amsterdam. The Mathematical Centre is sponsored by the Netherlands Government and is a well-known institution for promotion of mathematics and computer science. Since the introduction of Markov decision models by Bellman [1] there has been considerable interest in this field and it is now an active area of research. The book under review has evolved from the author's thesis and presents a linear programming approach for the solution of certain Markov control problems. The book comprises seven chapters, each of which begins with an introduction and a summary which provides an overview of the chapter. The first two chapters are introductory and survey some of the basic results. Markov decision problems with expected total reward as optimality criterion are dealt with in chapter 3. Comparisons
Book Reviews
of a linear programming approach with both iterative and policy improvement approaches are given. Contracting, positive and negative dynamic programming problems have also been discussed. Average reward is the optimality criterion for the problems in chapter 4. A linear programming formulation is presented and advantages over policy improvement techniques are described. Constrained Markov decision models are also discussed in this chapter. The concept of bias optimality is introduced in chapter 5. Some equivalent statements for bias optimality are given, leading to an algorithm for the computation of a bias optimal policy. In chapter 6, the two-person zero-sum stochastic game is considered in which one player controls the transition probabilities using a total reward criterion (under a contraction assumption) and an average reward criterion. A new proof for the existence of the value of the game is presented. Finally semi-Markov decision models are discussed in chapter 7 using discounted and undiscounted reward criteria. For further reading an up-to-date list of references is included, followed by a list of algorithms (34 in number) presented in the book. In conclusion this is a clearly written, cohesive research monograph, which will be of considerable interest to specialists in the areas of linear programming and Markov decision models, and would also be useful to advanced graduate students.
S.M. Janaid ZA ID1 University of Birmingham Birmingham, United Kingdom Reference [1] R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957).
S.G. TZAFESTAS (Ed.)
Optimisation and Control of Dynamic Operational Research Models Volume 4 in: North-Holland Systems and Control Series, North-Holland, Amsterdam, 1982, xx + 438 pages, Dfl.150.00 This is an excellent many-contribution volume bringing together many aspects and applications
277
of optimization and control for dynamic operational research models. It is really impressive how the editor could successfully collect papers from different sources. Contributors are from different countries, universities and departments (from business administration, through operational and systems research to engineering departments). The book involves twelve chapters and seven authors and nine co-authors from fifteen universities and eight countries. Chapter 1 considers the class of production-inventory systems and provides a survey of some recent results. Optimal and modal control for both continuous and discrete-time models, deterministic optimal control and stochastic formulation are studied. A closed-loop system with optimal management control policy is considered. A set of illustrative and application examples is provided. Chapter 2 is devoted to the study of the optimal linearquadratic control problem for some socioeconomic models. The forecasting in planning and determining of appropriate objective functions is considered. The chapter also considers the Kalman filter for design of forecasting models. Chapter 3 deals with models for resource allocation problems. These models are general enough fro studying a priori optimal schedule properties associated with the type of functions in the activities models and the resource and precedence constraints. This leads in some cases to analytical results or at least to the construction of more effective scheduling policies. Chapter 4 presents a survey of models for manpower systems (Markov, renewal longitudinak and Cambridge models) and provides a set of application examples of optimal control theory to manpower planning. The applicability of optimal control theory to manpower planning is considered. Near-optimal control using the Cambridge model is given. Chapter 5 is concerned with models of service systems and formulation of the optimization problems for them. The users act so as to optimize certain objective goals they set for themselves. The public policy of maximizing the expected return to society is considered and social optimal policies for different models of service systems are given. Chapter 6 treats problems of control theory related to traffic and transportation systems. In the field of road traffic control, optimal control of freeway traffic and optimal network flow is considered as well as urban street networks problems. In the field of air traffic control only aspects of optimal guidance and coordination of aircraft movements in the airfield approach corridors are treated. In the field of bus transportation systems, two optimization problems are presented, optimal scheduling of school buses and optimal control of bus networks. An overview of the problems related to transportation systems as a function of time horizons, controls and optimization is presented. Chapter 7 has two objectives. First to try to pose the economic dispatching problems into the framework of operations research and to indicate a few of the available methods and procedures that have been successfully employed in existing electric power systems. The second objective is to underline that there are still remaining many serious difficulties in