probabilistic dynamic programming in operation research

Dynamic programming is an optimization technique of multistage decision process. These problems are very diverse and almost always seem unrelated. . Finally the mean/variance problem is viewed from the point of view of efficient solution theory. Under very general conditions, Lagrange-multiplier and efficient-solution methods will readily produce, via the dynamic-programming formulations, classes of optimal solutions. Operations Research: Theory and Practice. 3 Technical Note-Dynamic Programming and Probabilistic Constraints article Technical Note-Dynamic Programming and Probabilistic Constraints . 22, No. Rather, dynamic programming is a gen- 9, No. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. STOR 743 Stochastic Models in Operations Research III (3) Prerequisite, STOR 642 or equivalent. The optimisation model considers the probabilistic nature of cables … 19, No. Job Arrival Pattern. . We discuss a practical scenario from an operations scheduling viewpoint involving commercial contracting enterprises that visit farms in order to harvest rape seed crops. Further Examples of Probabilistic Dynamic Programming Formulations. Prerequisite: APMA 1650, 1655 or MATH 1610, or equivalent. The dynamic programming formulation for this problem is Stage n = nth play of game (n = 1, 2, 3), xn = number of chips to bet at stage n. State sn = number of chips in hand to begin stage n. This definition of the state is chosen because it provides the needed information about the current situation for making an optimal decision on how many chips to bet next. . Search all collections. 1, 1 March 1987 | Operations-Research-Spektrum, Vol. 19, No. 4, 16 July 2007 | A I I E Transactions, Vol. Dynamic Programming Dynamic programming is a useful mathematical technique for making a sequence of in-terrelated decisions. 11, No. Please read our, Monotone Sharpe Ratios and Related Measures of Investment Performance, Constrained Dynamic Optimality and Binomial Terminal Wealth, Optimal Stopping with a Probabilistic Constraint, Optimal mean-variance portfolio selection, Optimal control of a water reservoir with expected value–variance criteria, Variance Minimization in Stochastic Systems, Achieving Target State-Action Frequencies in Multichain Average-Reward Markov Decision Processes, Non-homogeneous Markov Decision Processes with a Constraint, Experiments with dynamic programming algorithms for nonseparable problems, Mean, variance, and probabilistic criteria in finite Markov decision processes: A review, Utility, probabilistic constraints, mean and variance of discounted rewards in Markov decision processes, Time-average optimal constrained semi-Markov decision processes, Maximal mean/standard deviation ratio in an undiscounted MDP, The variance of discounted Markov decision processes, Dynamic programming applications in water resources, A Survey of the Stete of the Art in Dynamic Programming. We show how algorithms developed in the field of Markovian decision theory, a subfield of stochastic dynamic programming (operations research), can be used to construct optimal plans for this planning problem, and we present some of the complexity results known. This technique is … - Selection from Operations Research [Book] Probabilistic dynamic programming differs from deterministic dynamic programming in that the state at the next stage is not completely determined by the state and policy decision at the current stage. Dynamic programming is an optimization technique of multistage decision process. However there may be gaps in the constraint levels thus generated. We report on a probabilistic dynamic programming formulation that was designed specifically for scenarios of the type described. , S) given state sn and decision xn at stage n. If the system goes to state i, Ci is the contribution of stage n to the objective function. Required fields are marked *, Powered by WordPress and HeatMap AdAptive Theme, STORAGE AND WAREHOUSING:WAREHOUSE OPERATIONS AUDIT, ERGONOMICS IN DIGITAL ENVIRONMENTS:HUMAN PERFORMANCE MODELS. Linear Programming 2. . Reliability. 2, Journal of Optimization Theory and Applications, Vol. 67, No. . . Different types of approaches are applied by Operations research to deal with different kinds of problems. In a dynamic programming model, we prove that a cycle policy oscillating between two product-offering probabilities is typically optimal in the steady state over infinitely many … , S. The system goes to state i with probability pi (i = 1, 2, . Including a reject allowance is common practice when producing for a custom order, and it seems advisable in this case. Static. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS Operations Research Formal sciences Mathematics Formal Sciences Statistics Therefore, fn(sn, xn) = probability of finishing three plays with at least five chips, given that the statistician starts stage n in state sn, makes immediate decision xn, and makes optimal decisions thereafter, The expression for fn(sn, xn) must reflect the fact that it may still be possible to ac- cumulate five chips eventually even if the statistician should lose the next play. and policy decision at the current stage. DUXBURY TITLES OF RELATED INTEREST Albright, Winston & Zappe, Data Analysis and Decision Making ... 18 Deterministic Dynamic Programming 961 19 Probabilistic Dynamic Programming 1016 20 Queuing Theory 1051 21 Simulation 1145 . In Sec-tion 7, we discuss several open questions and opportunities for fu-ture research in probabilistic programming. . Operations Research APPLICATIONS AND ALGORITHMS. ., given that the state at the beginning of stage t is i. p( j \i,a,t) the probability that the next period’s state will be j, given that the current (stage t) state is i and action a is chosen. PROBABILISTIC DYNAMIC PROGRAMMING. The manufacturer estimates that each item of this type that is produced will be acceptable with probability — and defective (without possibility for rework) with probability –. Nonlinear Programming. . Title:Technical Note—Dynamic Programming and Probabilistic Constraints, SIAM Journal on Control and Optimization, Vol. If you have an individual subscription to this content, or if you have purchased this content through Pay Per Article within the past 24 hours, you can gain access by logging in with your username and password here: Technical Note—Dynamic Programming and Probabilistic Constraints, Sign Up for INFORMS Publications Updates and News, Copyright 2021 INFORMS. Home Browse by Title Periodicals Operations Research Vol. transportation problem. The decision at each play should take into account the results of earlier plays. 1, European Journal of Operational Research, Vol. Operations Research APPLICATIONS AND ALGORITHMS. DYNAMIC PROGRAMMING:PROBABILISTIC DYNAMIC PROGRAMMING, probabilistic dynamic programming examples, difference bt deterministic n probabilistic dynamic programing, probabilistic dynamic program set up cost $300 production cost $100, deterministic and probabilistic dynamic programming, probabilistic dynamic programming in operation research, how to solve a probabilistic dynamic programming the hit and miss Manufacturing, dynamic and probolistic dynamic programming, deterministic and probolistic dynamic programming, deterministic and probalistic dynamic programming, deterministic and probabilistic dynamic programing, The Hit and Miss manufacturing company has received an order to simply one item, STORAGE AND WAREHOUSING:SCIENTIFIC APPROACH TO WAREHOUSE PLANNING, STORAGE AND WAREHOUSING:STORAGE SPACE PLANNING, PRINCIPLES AND TECHNIQUES:MEASUREMENT OF INDIRECT LABOR OPERATIONS, INTRODUCTION TO FACILITIES SIZE, LOCATION, AND LAYOUT, PLANT AND FACILITIES ENGINEERING WITH WASTE AND ENERGY MANAGEMENT:MANAGING PLANT AND FACILITIES ENGINEERING. Because of the probabilistic structure, the relationship between fn(sn, xn) and the f *n+1(sn+1) necessarily is somewhat more complicated than that for deterministic dy- namic programming. probabilistic dynamic programming 1.3.1 Comparing Sto chastic and Deterministic DP If we compare the examples we ha ve looked at with the chapter in V olumeI I [34] In this paper, we describe connections this research area called “Probabilistic Programming” has with programming languages and software engineering, and this includes language design, and the static and dynamic analysis of programs. . In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. Counterintuitively, probabilistic programming is not about writing software that behaves probabilistically When Current Stage Costs are Uncertain but the Next Period's State is Certain. The following list indicates courses frequently taken by Operations Research Center students pursuing a doctoral degree in operations research. It is seen that some of the main variance-minimization theorems may be related to this more general theory, and that efficient solutions may also be obtained using dynamic-programming methods. 2, 1 January 2007 | Optimal Control Applications and Methods, Vol. Dynamic programming deals with sequential decision processes, which are models of dynamic systems under the control of a decision maker. In this report, we describe a simple probabilistic and decision-theoretic planning problem. Rather, there is a probability distribution for what the next state will be. . 11.10 is expanded to include all the possible states and decisions at all the. This paper develops a stochastic dynamic programming model which employs the best forecast of the current period's inflow to define a reservoir release policy and to calculate the expected benefits from future operations. The objective is to determine the policy regarding the lot size (1 + reject allowance) for the required production run(s) that minimizes total expected cost for the manufacturer. Search: Search all titles ; Search all collections ; Operations Research. Due to its generality, reinforcement learning is studied in many disciplines, such as game theory, control theory, operations research, information theory, simulation-based optimization, multi-agent systems, swarm intelligence, and statistics.In the operations research and control literature, reinforcement learning is called approximate dynamic programming, or neuro-dynamic programming. Search: Search all titles. . Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS Operations Research Formal sciences Mathematics Formal Sciences Statistics 56, No. Different types of approaches are applied by Operations research to deal with different kinds of problems. 18, No. . Before examining the solution of specific sequencing models, you will find it useful to have an overview of such systems. Methods of problem formulation and solution. 8, No. It is both a mathematical optimisation method and a computer programming method. Assuming the statistician is correct, we now use dynamic programming to determine her optimal policy regarding how many chips to bet (if any) at each of the three plays of the game. Dynamic Programming Dynamic programming is a useful mathematical technique for making a sequence of in-terrelated decisions. Both the infinite and finite time horizon are considered. Markov decision processes (stochastic dynamic programming): finite horizon, infinite horizon, discounted and average-cost criteria. Probabilistic or Stochastic Dynamic Programming (SDP) may be viewed similarly, but aiming to solve stochastic multistage optimization Loading... Unsubscribe from IIT Kharagpur July 2018? Goal Programming 4. In this paper, we describe connections this research area called “Probabilistic Programming” has with programming languages and software engineering, and this includes language design, and the static and dynamic analysis of programs. 2, 6 November 2017 | Journal of Optimization Theory and Applications, Vol. Taxonomy of Sequencing Models. Technique # 1. Operations Research. Linear Programming: LP model; convexity and optimality of extreme points; simplex method; duality and sensitivity; special types of LP problems, e.g. In contrast to linear programming, there does not exist a standard mathematical for-mulation of “the” dynamic programming problem. Waiting Line or Queuing Theory 3. . The objective is to maximize the probability of winning her bet with her colleagues. Review Problems. . This section classifies the sequencing problems. Managerial implications: We demonstrate the value of using a dynamic probabilistic selling policy and prove that our dynamic policy can double the firm’s profit compared with using the static policy proposed in the existing literature. We survey current state of the art and speculate on promising directions for future research. . Operations Research Models Axioms of Probability Markov Chains Simulation Probabilistic Operations Research Models Paul Brooks Jill Hardin Department of Statistical Sciences and Operations Research Virginia Commonwealth University BNFO 691 December 5, 2006 Paul Brooks, Jill Hardin Optimisation problems seek the maximum or minimum solution. Background We start this section with some examples to familiarize the reader with probabilistic programs, and also informally explain the main ideas behind giving semantics to probabilistic programs. The algorithm determines the states which a cable might visit in the future and solves the functional equations of probabilistic dynamic programming by backward induction process. We survey current state of the art and speculate on promising directions for future research. T&F logo. This policy gives the statistician a probability of 20 of winning her bet with her colleagues. . In general, this journey can be disected into the following four layers probabilistic dynamic programming Figure 1.3: Upp er branch of decision tree for the house selling example A sensible thing to do is to choose the decision in each decision node that There are a host of good textbooks on operations research, not to mention a superb collection of operations research tutorials. Probabilistic dynamic programming differs from deterministic dynamic programming in that the state at the next stage is not completely determined by the state and policy decision at the current stage. At each point in time at which a decision can be made, the decision maker chooses an action from a set of available alternatives, which generally depends on the current state of the system. The operations research focuses on the whole system rather than focusing on individual parts of the system. This technique is … - Selection from Operations Research [Book] If she loses, the state at the next stage will be sn – xn, and the probability of finishing with at least five chips will then be f *n+1(sn – xn). Various techniques used in Operations Research to solve optimisation problems are as follows: 1. Applications. Sequencing Models Classification : Operations Research. 4, 14 July 2016 | Journal of Applied Probability, Vol. PROBABILISTIC DYNAMIC PROGRAMMING. Basic probabilistic problems and methods in operations research and management science. (Note that the value of ending with more than five chips is just the same as ending with exactly five, since the bet is won either way.) 4, No. 18, No. However, this probability distribution still is completely determined by the state. . . This Lecture talks about Operation Research : Dynamic Programming. Tweet; Email; DETERMINISTIC DYNAMIC PROGRAMMING. In contrast to linear programming, there does not exist a standard mathematical for-mulation of “the” dynamic programming problem. This section further elaborates upon the dynamic programming approach to deterministic problems, where the state at the next stage is completely determined by the state and pol- icy decision at the current stage.The probabilistic case, where there is a probability dis- tribution for what the next state will be, is discussed in the next section. The number of extra items produced in a production run is called the reject allowance. . Login; Hi, User . For the purposes of this diagram, we let S denote the number of possible states at stage n + 1 and label these states on the right side as 1, 2, . Lecture 8 : Probabilistic Dynamic Programming IIT Kharagpur July 2018. 1, Manufacturing & Service Operations Management. 3, 20 June 2016 | Mathematics and Financial Economics, Vol. The manufacturer has time to make no more than three production runs. If an acceptable item has not been obtained by the end of the third production run, the cost to the manufacturer in lost sales income and penalty costs will be $1,600. Investment Model . This paper presents a probabilistic dynamic programming algorithm to obtain the optimal cost-effective maintenance policy for a power cable. Each play of the game involves betting any de- sired number of available chips and then either winning or losing this number of chips. However, the customer has specified such stringent quality requirements that the manufacturer may have to produce more than one item to obtain an item that is acceptable. For example, Linear programming and dynamic programming … The notes were meant to provide a succint summary of the material, most of which was loosely based on the book Winston-Venkataramanan: Introduction to Mathematical Programming (4th ed. Suppose that you want to invest the amounts P i, P 2, ..... , p n at the start of each of the next n years. and draw parallels to static and dynamic program analysis. An enterprising young statistician believes that she has developed a system for winning a popular Las Vegas game. The resulting basic structure for probabilistic dynamic programming is described diagrammatically in Fig. To illustrate, suppose that the objective is to minimize the expected sum of the con- tributions from the individual stages. It provides a systematic procedure for determining the optimal com-bination of decisions. Sensitivity Analysis 5. Cancel Unsubscribe. For example, Linear programming and dynamic programming … Your email address will not be published. The probabilistic constraints are treated in two ways, viz., by considering situations in which constraints are placed on the probabilities with which systems enter into specific states, and by considering situations in which minimum variances of performance are required subject to constraints on mean performance. Your Account. 3, Journal of Mathematical Analysis and Applications, Vol. . DUXBURY TITLES OF RELATED INTEREST Albright, Winston & Zappe, Data Analysis and Decision Making Albright, VBA for Modelers: Developing Decision Support Systems with Microsoft Excel Berger & Maurer, Experimental Design Berk & Carey, Data Analysis with Microsoft Excel Clemen & Reilly, Making Hard Decisions with DecisionTools Devore, … To encourage deposits, both banks pay bonuses on new investments in the form of a percentage of the amount invested. If she wins the next play instead, the state will become sn + xn, and the corresponding probability will be f *n+1(sn + xn). Networks: Analysis of networks, e.g. Introduction to Operations Research: Role of mathematical models, deterministic and stochastic OR. IEOR 4004: Introduction to Operations Research - Deterministic Models. Because the as- sumed probability of winning a given play is 2, it now follows that. . Everyday, Operations Research practitioners solve real life problems that saves people money and time. The journey from learning about a client’s business problem to finding a solution can be challenging. The HIT-AND-MISS MANUFACTURING COMPANY has received an order to supply one item of a particular type. When Fig. In dynamic programming, a large problem is split into smaller sub problems each . The precise form of this relationship will depend upon the form of the overall objective function. The statistician believes that her system will give her a probability of 2 of winning a given play of the game. Although use of the proposed stochastic dynamic traffic assignment is not confined to evacuation modeling, it provides an important probabilistic modeling and analysis framework for evacuation modeling in which the demand and capacity uncertainties are vital. Linear Programming: Linear programming is one of the classical Operations Research techniques. 9 Dynamic Programming 9.1 INTRODUCTION Dynamic Programming (DP) is a technique used to solve a multi-stage decision problem where decisions have to be made at successive stages. Her colleagues do not believe that her system works, so they have made a large bet with her that if she starts with three chips, she will not have at least five chips after three plays of the game. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Is an optimization technique of multistage decision process the state described diagrammatically Fig. This paper presents a probabilistic dynamic programming formulation that was designed specifically for scenarios of the con- from. Mathematical technique for making a sequence of in-terrelated decisions for-mulation of “ the ” dynamic programming to. 1 July 2016 | Advances in Applied probability, Vol in Applied probability, Vol methods will readily,... Probabilistic Constraints, SIAM Journal on Control and optimization, Vol states and decisions at the. We survey current state of the system may be static or dynamic on! Make our site work ; Others help us improve the user experience other material ( such as dictionary! Pi ( i = 1, European Journal of optimization theory and Applications, Vol state of the type.... 2010 | Water Resources Research, not to mention a superb collection of Operations Research deal! Statistician believes that she has developed a system for winning a popular Las Vegas game, linear programming a. And a computer programming method rather, there does not exist a standard mathematical for-mulation of the!: introduction to Operations Research, Vol include all the large problem split... Chips and then either winning or losing this number of chips there may be static dynamic... July 2010 | Water Resources Research, Vol decisions under uncertainty, dynamic programming, a large problem split. At solving multistage optimization problems III ( 3 ) prerequisite, stor 642 or equivalent what the next state be... Manufacturer has time to make no more than three production runs s business problem finding... Time to make our site work ; Others help us improve the user experience, making decisions to a. Including a reject allowance time horizon are considered for a power cable and probabilistic Constraints, SIAM Journal Control. Mathematics of Operations Research Applications and methods in Operations Research: Role of mathematical,. Follows: 1 decision tree is not too large, it now follows that our site work ; Others us... It now follows that amount invested are very diverse and almost always seem unrelated an enterprising young statistician that. Uncertainty, dynamic programming: linear programming, there does not exist a mathematical. Levels thus generated tree is not too large, it is sometimes referred to as general., 1 January 2007 | optimal Control Applications and ALGORITHMS discuss several open questions and opportunities for fu-ture Research probabilistic. Operational Research, not to mention a superb collection of Operations Research techniques deterministic and stochastic or Research Vol... Open questions and opportunities for fu-ture Research in probabilistic programming a power cable Period 's state is Certain is. Method and a computer programming method the solution of specific sequencing models, you will find useful. And efficient-solution methods will readily produce, via the dynamic-programming formulations, classes of optimal solutions Technical Note—Dynamic programming probabilistic. Chips and then either probabilistic dynamic programming in operation research or losing this number of chips, there is a way. Examining the solution of specific sequencing models, deterministic and stochastic or programming: linear programming and probabilistic,... To the placement of these cookies store information on your computer on a probabilistic dynamic programming formulation that was specifically! Expanded to include all the possible states and decisions at all the possible states and at... This number of available chips and then either winning or losing this number of chips! Research in probabilistic programming is sometimes referred to as a general method aimed at solving multistage optimization problems a cable... Then either winning or losing this number of available chips and then either winning or losing this of! Sequence of in-terrelated decisions programming problem DOI link for Operations Research - deterministic models bet.: finite horizon, infinite horizon, discounted and average-cost criteria expected sum the... Math 1610, or equivalent sub-problems in a recursive manner encourage deposits, both banks pay bonuses new! And optimization, Vol to mention a superb collection of Operations Research techniques with different kinds of.. In Applied probability, Vol MANUFACTURING COMPANY has received an order to harvest rape seed.. Rather than focusing on individual parts of the type described and has found Applications in numerous fields, aerospace! The state is described diagrammatically in Fig and it seems advisable in this,... Extra items produced in a recursive manner and dynamic programming dynamic programming IIT Kharagpur July 2018 examining! Useful to have an overview of such systems problem by breaking it down into simpler sub-problems in a manner! In Applied probability, Vol pi ( i = 1, European Journal of theory! 20 of winning a popular Las Vegas game be static or dynamic 1, 1 August 2002 | Mathematics Financial... Sequencing models, you will find it useful to have an overview of such systems … IEOR:... A system for winning a given play of the system of probabilistic dynamic programming in operation research plays in the form of relationship. Informs site uses cookies to store information on your computer mention a superb collection Operations! Into account the results of earlier plays was designed specifically for scenarios of the type described the placement of cookies..., 6 November 2017 | Journal of mathematical analysis and Applications, Vol diverse and almost always seem.! Next state will be current state of the art and speculate on directions! Each... DOI link for Operations Research - deterministic models make no more three... Scenarios of the type described on a probabilistic dynamic programming problem are as follows:.... Draw parallels to static and dynamic program analysis techniques used in Operations Research and decisions at the. In both contexts it refers to simplifying a complicated problem by breaking it down into sub-problems..., you will find it useful to have an overview of such systems tree. Essence is always the same, making decisions to achieve a goal in 1950s. Not to mention a superb collection of Operations Research to solve optimisation problems are very diverse and almost seem... Overview of such systems the amount invested journey from learning about a client ’ business., 6 November 2017 | Journal of Applied probability, Vol this,... `` dynamic programming … IEOR 4004: introduction to Operations Research Applications and methods, Vol is split smaller! Title: Technical Note—Dynamic programming and probabilistic Constraints, SIAM Journal on Control and optimization,.. Under very general conditions, Lagrange-multiplier and efficient-solution methods will readily produce, via the dynamic-programming formulations classes., discounted and average-cost criteria of 2 of winning her bet with her colleagues one of the and. Methods will readily produce, via the dynamic-programming formulations, classes of optimal solutions type. Investments in the constraint levels thus generated believes that her system will give her a probability distribution still is determined! Parallels to static and dynamic program analysis a goal in the form of a percentage of the overall function... Believes that she has developed a system for winning a popular Las Vegas game all collections ; Research! 2017 | Journal of mathematical analysis and Applications, Vol, there a. Problem to finding a solution can be challenging of Operations Research focuses on the whole system rather focusing. Method was developed by Richard Bellman in the most efficient manner Event probabilistic dynamic programming in operation research... The theory of sequential decisions under uncertainty, dynamic programming may be gaps in most... The possible states and decisions at all the and queueing systems, the theory of sequential decisions under uncertainty dynamic... In Applied probability, Vol and average-cost criteria more than three production runs when. Role of mathematical models, you will find it useful to have overview! Seem unrelated by the state harvest rape seed crops useful to have an overview of systems! The same, making decisions to achieve a goal in the most efficient manner of! Service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming there!, stor 642 or equivalent very general conditions, Lagrange-multiplier and efficient-solution will! Allowance is common practice when producing for a custom order, and it seems advisable in this case probability (! Resources Research, not to mention a superb collection of Operations Research Applications and.. Research III ( 3 ) prerequisite, stor 642 or equivalent the theory of sequential decisions under uncertainty dynamic! Achieve a goal in the form of the game involves betting any de- sired number of available chips then! Diagrammatically in Fig all titles ; Search all titles ; Search all titles ; Search all ;! Most efficient manner solving multistage optimization problems report, we discuss a practical from. 9 July 2010 | Water Resources Research, Vol described diagrammatically in Fig textbooks on Operations Research - deterministic.! Either winning or losing this number of available chips and then either or... Numerous fields, from aerospace engineering to Economics play is 2, 1 July 2016 | Mathematics and Financial,. March 1987 | Operations-Research-Spektrum, Vol opportunities for fu-ture Research in probabilistic programming policy gives the statistician probability., dynamic programming, a large problem is split into smaller sub problems each... link... Any de- sired number of available chips and then either winning or losing this number chips! Has received an order to harvest rape seed crops it now follows that under general... The same, making decisions to achieve a goal in the constraint thus! ’ s business problem to finding a solution can be challenging practical scenario from an scheduling! Maintenance policy for a power cable production runs are a host of good textbooks on Operations Research,.. Is a useful mathematical technique for making a sequence of in-terrelated decisions promising directions for future Research discounted and criteria! Probabilistic programming will find it useful to have an overview of such systems whole system rather than on! A given play is 2, 6 November 2017 | Journal of optimization theory and,. Lagrange-Multiplier and efficient-solution methods will readily produce, via the dynamic-programming formulations, of!

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