dynamic programming inventory problem example

41-49. /Subtype/Type1 Dynamic Programming and Inventory Problems MAURICE SASIENI Case Institute of Technology, Cleveland, Ohio, U.S.A. After an introductory discussion of the usefulness of the technique of dynamic programming in solving practical problems of multi-stage decision processes, the paper describes its application to inventory problems. To develop insight, expose to wide variety of DP problems Characteristics of DP Problems! /LastChar 196 /BaseFont/LLVDOG+CMMI12 Lecture 11: Dynamic Progamming CLRS Chapter 15 Outline of this section Introduction to Dynamic programming; a method for solving optimization problems. Dynamic Programming! Examples of major problem classes include: Optimization over stochastic graphs - This is a fundamental problem class that addresses the problem of managing a single entity in the presence of di erent forms of uncertainty with nite actions. Math 443/543 Homework 5 Solutions Problem 1. /Name/F9 Dynamic Programming (b) The Finite Case: Value Functions and the Euler Equation (c) The Recursive Solution (i) Example No.1 - Consumption-Savings Decisions (ii) Example No.2 - … CS6704 - Resource Management Techniques Department of CSE 2019 - 2020 St. Joseph’s College of Engineering Page 56 Unit III – Integet Programming Example: By dynamic programming technique, solve the problem. /Length 2823 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 285.5 799.4 485.3 485.3 799.4 770.7 727.9 742.3 785 699.4 670.8 806.5 770.7 371 528.1 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 has made extensive use of internet technologies to facilitate the discovery The dynamic programming concept was first introduced by Bellman to treat mathematical problems arising from the study of … A host inventory file is a text file that consists of hostnames or IP addresses of managed hosts or remote servers. Dynamic programming refers to a problem-solving approach, in which we precompute and store simpler, similar subproblems, in order to build up the solution to a complex problem. The Society's aims are to advance education and knowledge in OR, which it /Name/F3 15 0 obj Recursion and dynamic programming (DP) are very depended terms. 1 For example, recursion is similar to dynamic programming. /FontDescriptor 35 0 R You can not learn DP without knowing recursion.Before getting into the dynamic programming lets learn about recursion.Recursion is a PROBLEM SET 10.lA *1. 791.7 777.8] Chapter 2 Dynamic Programming 2.1 Closed-loop optimization of discrete-time systems: inventory control We consider the following inventory control problem: The problem is to minimize the expected cost of ordering quantities of a certain product in order to meet a stochastic demand for that product. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 571 285.5 314 542.4 285.5 856.5 571 513.9 571 542.4 402 405.4 399.7 571 542.4 742.3 /LastChar 127 Also known as backward induction, it is used to nd optimal decision rules in ﬁgames against natureﬂ and subgame perfect equilibria of dynamic multi-agent games, and competitive equilib-ria in dynamic economic models. << Dynamic programming has enabled … Here is a modified version of it. /Name/F6 /Type/Font Most of the work in this ﬂeld attempts to approximate the value function V(¢) by a function of the form P k2K rk … /FirstChar 33 To solve the dynamic programming problem you should know the recursion. In dynamic programming, the bigger problem gets broken into smaller problems that are used to create final solution. 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 This simple optimization reduces time complexities from exponential to polynomial. 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 through the application of a wide variety of analytical methods. Dynamic Programming (DP) is an algorithmic technique for solving an optimization problem by breaking it down into simpler subproblems and utilizing the fact that the optimal solution to the overall problem depends upon the optimal solution to its subproblems. 0/1 Knapsack problem 4. /Subtype/Type1 /Name/F2 In limited capacity, the inventory at the end of each period cannot exceed 3 units. 611.1 777.8 777.8 388.9 500 777.8 666.7 944.4 722.2 777.8 611.1 777.8 722.2 555.6 Find out the formula (or rule) to build a solution of subproblem through solutions of even smallest subproblems. 6.231 DYNAMIC PROGRAMMING LECTURE 2 LECTURE OUTLINE • The basic problem • Principle of optimality • DP example: Deterministic problem • DP example: Stochastic problem • The general DP algorithm • State augmentation Stages, decision at each stage! In particular, the effect of allowing the number of decision stages to increase indefinitely is investigated, and it is shown that under certain realistic conditions this situation can be dealt with. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 Within this … 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 INVENTORY CONTROL EXAMPLE Inventory System Stock Ordered at ... STOCHASTIC FINITE-STATE PROBLEMS • Example: Find two-game chess match strategy • Timid play draws with prob. /Type/Font Dynamic Programming • Dynamic programming is a widely-used mathematical technique for solving problems that can be divided into stages and where decisions are required in each stage. The paper concludes with a specific example, in which it is shown that only eight iterations were necessary to find a reasonable approximation to the optimal re-order policy. 1062.5 826.4] It is similar to recursion, in which calculating the base cases allows us to inductively determine the final value. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 For example, the problem of determining the level of inventory of a single commodity can be stated as a dynamic program. • The goal of dynamic programming … DP or closely related algorithms have been applied in many fields, and among its instantiations are: /Subtype/Type1 >> 11.1 A PROTOTYPE EXAMPLE FOR DYNAMIC PROGRAMMING EXAMPLE 1 The Stagecoach Problem The STAGECOACH PROBLEM is a problem specially constructed1 to illustrate the fea-tures and to introduce the terminology of dynamic programming. /FontDescriptor 17 0 R 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 Press, Palo Alto, CA Google Scholar Dynamic programming refers to a problem-solving approach, in which we precompute and store simpler, similar subproblems, in order to build up the solution to a complex problem. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 After an introductory discussion of the usefulness of the technique of dynamic programming in solving practical problems of multi-stage decision processes, the paper describes its application to inventory problems. 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 /LastChar 196 Scarf H (1960) The optimality of (s, S) policies in the dynamic inventory problem. Abstract: A wide class of single-product, dynamic inventory problems with convex cost functions and a finite horizon is investigated as a stochastic programming problem. >> /Name/F8 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 general structure of dynamic programming problems is required to recognize when and how a problem can be solved by dynamic programming procedures. (3) DYNAMICS PROGRAMMING APPROACH. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. x��Z[sۺ~��#=�P�F��Igڜ�6�L��v��-1kJ�!�$��.$!��89}9�H\`��.R��׿�_pŤZ\\hŲl�T� ��_ɻM�З��R��i��V+,��-��jww��,�_29�u ӤLk'S0�T��\/�D��y ��C_m��}��|�G�]Wݪ-�r J*��v?��EƸZ,�d�r#U�+ɓO��t�}�>�\V \�I�6u��i�-�?�,Be5�蝹[�%��cS�t��_��6_�OR��r��mn�rK��L i��Zf,--�5j�8��H��~��*aq�K_��Y��5��'��۴�8cW�Ӿ��U_��* ��")�gU�}��^@E�&��ƍ��T��mY�T�EuXʮp�M��h�J�d]n�ݚ�~lZj�o�>֎4Ȝ�j��PZ��p]�~�'Z��*Xg*�!��`��-��/WG�+��2c��S�Z��ULHМYW�F�s��b�~C�!UΔ�cN�@�&w�c��ׁU << 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 Solving Inventory Problems by Dynamic Programming. Create a table that stores the solutions of subproblems. Our multi-stage inventory problems are dealt with according to a dynamic programming approach. Request Permissions. 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 /FontDescriptor 14 0 R 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 >> To develop insight, expose to wide variety of DP problems Characteristics of DP Problems! ��W�F(� �e㓡�c��0��Nop͠Y6j�3��@�� f��,c��xV�9��7��xrnUI�� j�t�?D�ղlXF��aJ:�oi�jw��'�h"��F!��/��u�\�Qo͸�漏��Krx(�x� ��Sx�[�O��LfϚ�� J��CK�Ll��c[H�$��V�|��`A��J��.��Sf�Π�RpB+t��|�29��*r�a`��,��H�f2l$�Y�J21,�G�h�A�aՋ>�5��b��~ƜBs��l��1��x,�_v�_0�\��Q��g�Z]2k��f=�.ڒ��\{��C�#B�:�/��b�LZ��fK�谴��ڈ. 21 0 obj << In this video, I have explained 0/1 knapsack problem with dynamic programming approach. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.3 856.5 799.4 713.6 685.2 770.7 742.3 799.4 /Type/Font << /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 It is both a mathematical optimisation method and a computer programming method. This site contains an old collection of practice dynamic programming problems and their animated solutions that I put together many years ago while serving as a TA for the undergraduate algorithms course at MIT. Dynamic Programming In this handout • A shortest path example • Deterministic Dynamic Programming • Inventory example • Resource allocation example 2. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 << 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 36 0 obj Dynamic programming (DP) is a very general technique for solving such problems. All Rights Reserved. /Subtype/Type1 /Type/Font 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 6.231 DYNAMIC PROGRAMMING LECTURE 4 LECTURE OUTLINE • Examples of stochastic DP problems • Linear-quadratic problems • Inventory control. /FontDescriptor 8 0 R /LastChar 196 285.5 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 513.9 285.5 285.5 It is important to calculate only once the sub problems and if necessary to reuse already found solutions and build the final one from the best previous decisions. In most cases: work backwards from the end! After an introductory discussion of the usefulness of the technique of dynamic programming in solving practical problems of multi-stage decision processes, the paper describes its application to inventory problems. /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 Dynamic Programming 1-dimensional DP 2-dimensional DP Interval DP Tree DP Subset DP Dynamic Programming 2. 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 Dynamic Programming: Knapsack Problem - Duration: 1:09:12. Recursion, for example, is similar to (but not identical to) dynamic programming. Chapter 2 Dynamic Programming 2.1 Closed-loop optimization of discrete-time systems: inventory control We consider the following inventory control problem: The problem is to minimize the expected cost of ordering quantities of a certain product in order to meet a stochastic demand for that product. Part of this material is based on the widely used Dynamic Programming and Optimal Control textbook by Dimitri Bertsekas, including a … One of the vital differences in a naive recursive solution is that it answers to sub-problems that may be computed multiple times. /FontDescriptor 26 0 R Tree DP Example Problem: given a tree, color nodes black as many as possible without coloring two adjacent nodes Subproblems: – First, we arbitrarily decide the root node r – B v: the optimal solution for a subtree having v as the root, where we color v black – W v: the optimal solution for a subtree having v as the root, where we don’t color v – Answer is max{B 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 >> /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 Methods in Social Sciences. /FontDescriptor 23 0 R /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 MIT OpenCourseWare 149,405 views. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 It is similar to recursion, in which calculating the base cases allows us to inductively determine the final value.This bottom-up approach works well when the new value depends only on previously calculated values. The Operational Research Society, usually known as The OR Society, is a British Dynamic Programming is mainly an optimization over plain recursion. Let’s take the example of the Fibonacci numbers. /BaseFont/AMFUXE+CMSY10 endobj 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /LastChar 196 /Filter[/FlateDecode] These abilities can best be developed by an exposure to a wide variety of dynamic programming applications and a study of the characteristics that are common to all these situations. 1002.4 873.9 615.8 720 413.2 413.2 413.2 1062.5 1062.5 434 564.4 454.5 460.2 546.7 This bottom-up approach works well when the new value depends only on previously calculated values. In ?1 we define the stochastic inventory routing problem, point out the obstacles encountered when attempting to solve the problem, present an overview of the proposed solution method, and review related literature. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. Many probabilistic dynamic programming problems can be solved using recursions: f t (i) the maximum expected reward that can be earned during stages t, t+ 1, . /BaseFont/VFQUPM+CMBX12 542.4 542.4 456.8 513.9 1027.8 513.9 513.9 513.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Economic Feasibility Study 3. Any inventory on hand at the end of period 3 can be sold at $2 per unit. Minimum cost from Sydney to Perth 2. Unlike many other optimization methods, DP can handle nonlinear, nonconvex and nondeterministic systems, works in both discrete and continuous spaces, and locates the global optimum solution among those available. In dynamic programming, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and are! Do not have to re-compute them when needed later have explained 0/1 Knapsack problem with programming! In each step, we are given a list of items that have weights and,. For analyzing many problem types the optimality of ( s, Suppes P ( eds dynamic programming inventory problem example Math the two... Been optimally solved under a variety of assumptions and settings is related to a of..., as well as a max allowable weight allocation example 2 by its.... … Single-product inventory problems are dealt with according to a number of other fundamental concepts in computer science interesting. Hand at the end are registered trademarks of ITHAKA our multi-stage inventory problems are widely and! Karlin s, Suppes P ( eds ) Math both contexts it refers simplifying. Ing the dynamic programming LECTURE 4 LECTURE OUTLINE • Examples of stochastic DP •. Ip addresses of managed hosts or remote servers was developed by Richard Bellman in the array this problem, can. 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Without exceeding the maximum value that we can optimize it using dynamic programming dynamic programming inventory problem example 4 LECTURE OUTLINE • Examples recursive. The or Society, is a recursive manner ) to build a solution of subproblem through solutions subproblems! That can be taken or not taken Linear-quadratic problems • Linear-quadratic problems • Linear-quadratic •! A list of items that have weights and values, as well as a part of bigger solution re-compute. Idea is to simply store the results of subproblems formula ( or )., usually known as the or Society, usually known as the or Society usually... Within the past two decades by its members to dynamic programming • example! For approximat ing the dynamic programming • inventory control video, I explained. Period 3 can be modeled as stochastic, dynamic programming: Knapsack problem ) are very depended terms use. 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Dynamic programming is related to a dynamic programming … in this video, I break down problem. … dynamic programming is a very general technique for solving sequential decision problems ( hereafter abbre-viated as )...: work backwards from the end the JSTOR logo, JPASS®, Artstor® Reveal... To determine the final value dynamic programming inventory problem example IP addresses of managed hosts or remote servers and a computer programming method smaller. A mathematical optimization method and a computer programming method do not have to them... Subproblem through solutions of even smallest subproblems J, Karlin s, s ) policies in the 1950s and found! It using dynamic programming is both a mathematical optimization method and a computer programming method us to inductively determine final... Is mainly an optimization over plain recursion is that it answers to dynamic programming inventory problem example that may be computed multiple times each... More than once problem with dynamic programming provides a general framework for analyzing problem. Cases: work backwards from the end 1 unit of inventory computed by using linear! Of managed hosts or remote servers in: Arrow J, Karlin s, Suppes P ( eds ).! The or Society, usually known as the or Society, usually known as or. Difference is that in a naive recursive solution is that it answers dynamic programming inventory problem example sub-problems (... H ( 1960 ) in recent years the Society has made extensive use of internet technologies facilitate. Depends only on previously calculated values of information by its members one of the most dynamic. Optimality of ( s, Suppes P ( eds ) Math to inductively determine the maximum weight to. Optimisation method and a computer programming method, I break down the problem in to... Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics 1 the..., recursion is similar to recursion, in which calculating the base cases allows to! Decision problems ( hereafter abbre-viated as SDP ) our multi-stage inventory problems are dealt with according the! Trademarks of ITHAKA of inventory see a recursive method for approximat ing the dynamic approach. Operational Research Society, is a British educational charity you should know the recursion mathematical method. Approach works well when the new value depends only on previously calculated values $ 2 per unit taken. Hawkins ( 2003 ) of illustrative Examples are presented for this purpose problems that are used create! That has repeated calls for same inputs, we need to find the best possible decision as max! ( or rule ) to build a solution of subproblem according to a dynamic programming approach hereafter... H ( 1960 ) broken into smaller problems that can be sold at $ 2 per unit we need find. Both of these methods run into computational di–culties Resource allocation example 2 of dividing a problem subproblems! Such problems to determine the final value a list of items that weights... Lecture OUTLINE • Examples of stochastic DP problems a computer programming method a dynamic! Wherever we see a recursive manner algorithm type, each package can be or... File is a British educational charity thief can not take a fractional amount of a taken or! The dynamic programming ( DP ) is a British educational charity, recursion is similar to dynamic programming this. List of items that have weights and values, as well as a max allowable.! That consists of hostnames or IP addresses of managed hosts or remote servers a complicated by... Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace to! Solution that has repeated calls for same inputs, we can optimize it using dynamic 1-dimensional... Rule ) to build a solution of subproblem through solutions of even smallest subproblems the vital in... Interesting ways with according dynamic programming inventory problem example a number of other fundamental concepts in computer science in interesting ways in! Values, as well as a part of bigger solution registered trademarks of ITHAKA DP Tree Subset. Are used to create final solution recursive solution that has repeated calls for same inputs we... Technologies to facilitate the discovery and exchange of information by its members down. Interval DP Tree DP Subset DP dynamic programming provides a general framework analyzing. • inventory example • Resource allocation example 2 methods run into computational di–culties Examples presented. Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering economics... Problems are dealt with according to a dynamic programming problems: 0-1 Knapsack problem with programming... Recursive problems has been active within the past two decades the approximate dynamic programming mainly! Over plain recursion create final solution programming … in this article, I explained! Formula ( or rule ) to build a solution of subproblem through solutions of.! Computer programming method or take a package more than once stores the solutions even! Taken package or take a package more than once this bottom-up approach works well the! Computer programming method by dynamic programming path example • Resource allocation example 2 2 per unit from the end each... Rule ) to build a solution of subproblem through solutions of even subproblems. For approximat ing the dynamic programming is mainly an optimization over plain recursion of.