endobj endobj For example, recursion is similar to dynamic programming. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. /BaseFont/PLLGMW+CMMI8 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 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. >> /LastChar 196 (special interest) groups and regional groups. Wikipedia definition: “method for solving complex problems by breaking them down into simpler subproblems” This definition will make sense once we see some examples – Actually, we’ll only see problem solving examples today Dynamic Programming 3. << The range of problems that can be modeled as stochastic, dynamic optimization problems is vast. /Name/F7 Request Permissions. /FirstChar 33 /BaseFont/AMFUXE+CMSY10 It appears to be generally true that the average cost per period will converge, for an optimal policy, as the number of periods considered increases indefinitely, and that it is feasible to search for the policy which minimizes this long-term average cost. 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 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. Dynamic Programming 1. /Type/Font /LastChar 196 33 0 obj 36 0 obj This item is part of JSTOR collection 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 Theory of dividing a problem into subproblems is essential to understand. /BaseFont/AKSGHY+MSBM10 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 << 1 /Type/Font 826.4 295.1 531.3] /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 /Name/F3 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 In recent years the Society Optimisation problems seek the maximum or minimum solution. 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 /BaseFont/VYWGFQ+CMEX10 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, . >> /FontDescriptor 32 0 R INVENTORY CONTROL EXAMPLE Inventory System Stock Ordered at ... STOCHASTIC FINITE-STATE PROBLEMS • Example: Find two-game chess match strategy • Timid play draws with prob. for the single-item, multi-period stochastic inventory problem in the dynamic-programming framework. 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. /LastChar 196 I am keeping it around since it seems to have attracted a reasonable following on the web. Dynamic programming … 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 41-49. 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. /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 Practitioners of Operational Research (OR) provide advice on complex issues Then calculate the solution of subproblem according to the found formula and save to the table. 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 15 0 obj 1:09:12. 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 In 6.231 DYNAMIC PROGRAMMING LECTURE 4 LECTURE OUTLINE • Examples of stochastic DP problems • Linear-quadratic problems • Inventory control. /FirstChar 0 /LastChar 127 In many models, including models with Markov-modulated demands, correlated demand and forecast evolution (see, for example, Iida and Zipkin [10], Ozer and Gallego [23], and Zipkin [28]), the optimal policy can be shown to be a state-dependent base-stock policy. 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. 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 /FirstChar 33 /Widths[285.5 513.9 856.5 513.9 856.5 799.4 285.5 399.7 399.7 513.9 799.4 285.5 342.6 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 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. 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 Dynamic Programming 1-dimensional DP 2-dimensional DP Interval DP Tree DP Subset DP Dynamic Programming 2. Dynamic programming is related to a number of other fundamental concepts in computer science in interesting ways. 6.231 DYNAMIC PROGRAMMING LECTURE 4 LECTURE OUTLINE • Examples of stochastic DP problems • Linear-quadratic problems • Inventory control. Dynamic Programming 1 Dynamic programming algorithms are used for optimization (for example, nding the shortest path between two points, or the fastest way to multiply many matrices). does through the publication of journals, the holding of conferences and meetings, 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 Single-product inventory problems are widely studied and have been optimally solved under a variety of assumptions and settings. endobj Here is a modified version of it. For this problem, we are given a list of items that have weights and values, as well as a max allowable weight. 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 /FontDescriptor 29 0 R The dynamic programming is a linear optimization method that obtains optimum solution of a multivariable problem by decomposition of the problem into sub problems [2]. << 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. 666.7 722.2 722.2 1000 722.2 722.2 666.7 1888.9 2333.3 1888.9 2333.3 0 555.6 638.9 513.9 770.7 456.8 513.9 742.3 799.4 513.9 927.8 1042 799.4 285.5 513.9] 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 << /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 Stages, decision at each stage! In this article, I break down the problem in order to … >> To develop insight, expose to wide variety of DP problems Characteristics of DP Problems! /BaseFont/VFQUPM+CMBX12 Get a good grip on solving recursive problems. 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 /Widths[777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 Dynamic programming is breaking down a problem into smaller sub-problems, solving each sub-problem and storing the solutions to each of these sub-problems in an array (or similar data structure) so each sub-problem is only calculated once. 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 Each stage has assoc states! You can not learn DP without knowing recursion.Before getting into the dynamic programming lets learn about recursion.Recursion is a 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 2 We use the basic idea of divide and conquer. /Subtype/Type1 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /Subtype/Type1 To solve the dynamic programming problem you should know the recursion. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 Press, Palo Alto, CA Google Scholar /FirstChar 33 and exchange of information by its members. Methods in Social Sciences. 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. More so than the optimization techniques described previously, dynamic programming provides a general framework for analyzing many problem types. 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. 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 At the beginning of period 1, the firm has 1 unit of inventory. 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. When applied to the inventory allocation problem described above, both of these methods run into computational di–culties. Any inventory on hand at the end of period 3 can be sold at $2 per unit. Dynamic Programming In this handout • A shortest path example • Deterministic Dynamic Programming • Inventory example • Resource allocation example 2. It is required that all demand be met on time. Recursion and dynamic programming (DP) are very depended terms. x��Z[sۺ~��#=�P�F��Igڜ�6�L��v��-1kJ�!�$��.$!���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 /Name/F5 /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 /FontDescriptor 17 0 R limited capacity, the inventory at the end of each period cannot exceed 3 units. 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. /Type/Font /BaseFont/AAIAIO+CMR9 The second problem that we’ll look at is one of the most popular dynamic programming problems: 0-1 Knapsack Problem. 799.2 642.3 942 770.7 799.4 699.4 799.4 756.5 571 742.3 770.7 770.7 1056.2 770.7 /Name/F6 Dynamic Programming! Originally established in 1948 as the OR Club, it is the Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. 24 0 obj 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 777.8 1000 1000 1000 1000 1000 1000 777.8 777.8 555.6 722.2 666.7 722.2 722.2 666.7 MIT OpenCourseWare 149,405 views. A general approach to problem-solving! endobj Steps for … << /BaseFont/LLVDOG+CMMI12 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 Dynamic Programming is mainly an optimization over plain recursion. 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 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. 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 /LastChar 196 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 OR (1960). >> Stanford Univ. Recall the inventory considered in the class. p DP or closely related algorithms have been applied in many fields, and among its instantiations are: << /Type/Font /Subtype/Type1 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 /BaseFont/EBWUBO+CMR8 /FontDescriptor 8 0 R To solve a problem by dynamic programming, you need to do the following tasks: Find solutions of the smallest subproblems. endobj 0/1 Knapsack problem 4. >> /Subtype/Type1 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 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. 18 0 obj 1-2, pp. Dividing the problem into a number of subproblems. 1062.5 826.4] << In this Knapsack algorithm type, each package can be taken or not taken. 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 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 Dynamic Programming - Examples to Solve Linear & Integer Programming Problems Inventory Models - Deterministic Models Inventory Models - Discount Models, Constrained Inventory Problems, Lagrangean Multipliers, Conclusions To develop insight, expose to wide variety of DP problems Characteristics of DP Problems! 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