OR II Project Definition

Transcription

1 OR II Project Definition Prof. Mohammad Modarres Designed by Nikrooz Nasr 2 2

2 About the Project 3 3

3 About the Project You should mail your group (2 or 3) to until 1st of Aban. Then you will receive your project. For phase I, you should send your model to until 15th Aban. Then you will receive the correct model. For phase II, you should send your GAMS model and documentation until 15th Azar. The complete instruction will be attached with the project. The subject of your mails should begins with OR. 3 3

4 Sudoku!!! 4 4

5 Sudoku!!! The objective is to fill a 9x9 grid Each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9, Only ONE time each (that is, exclusively). 4 4

6 Sudoku!!! The objective is to fill a 9x9 grid Each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9, Only ONE time each (that is, exclusively) A sudoku puzzle

7 Sudoku!!! The objective is to fill a 9x9 grid Each column, each row, and each of the nine 3x3 boxes (also called blocks or regions) contains the digits from 1 to 9, Only ONE time each (that is, exclusively) and its solution numbers marked in red 4 4

8 History of Sudoku Le Siècle's rival, La France, published the puzzle (195). But did not mark the sub-squares. It cannot be considered the first Sudoku L'Echo de Paris published it for about a decade but disappeared about the time of the First World War. The modern Sudoku was most likely designed anonymously by Howard Garns. First published in 1979 by Dell Magazines as Number Place (the earliest known examples of modern Sudoku). The puzzle was introduced in Japan by Nikoli in the paper Monthly Nikolist in 194 as Suuji wa dokushin ni kagiru ( ), which can be translated as "the digits must be single" or "the digits are limited to one occurrence." 5 5

9 Sudoku in Media The world's first live TV Sudoku show, Sudoku Live, was a puzzle contest first broadcast on 2005 on Sky One. Later in 2005, the BBC launched SUDO-Q, a game show that combines Sudoku with general knowledge. 5 5

10 Sudoku is NP-complete 6 6

11 A Bad BILP Model for Sudoku xijk = 1, if the element (i,j) of n n Sudoku matrix contains the integer k 0, otherwise n i =1 n j =1 xijk = 1, j =1: n, k =1: n only one k in each column xijk = 1, i =1: n, k =1: n only one k in each row 6 6

12 A Bad BILP Model for Sudoku xijk = 1, if the element (i,j) of n n Sudoku matrix contains the integer k 0, otherwise n k =1 mq j =mq-m+1 xijk = 1, i =1: n, j =1: n every position in matrix must be filled mp i =mp-m+1 xijk = 1, k =1: n, p =1: m, q =1: m only one k in each submatrix 6 6

13 A Bad BILP Model for Sudoku xijk = 1, if the element (i,j) of n n Sudoku matrix contains the integer k 0, otherwise xijk = 1 for given elements xijk = 0 or 1 6 6

14 Sudoku Model-Continued min s.t. 0 T x n i =1 n j =1 xijk = 1, xijk = 1, mq mp j =mq-m+1 i =mp-m+1 n k =1 xijk = 1 j =1: n, k =1: n i =1: n, k =1: n xijk = 1, xijk = 1, i =1: n, j =1: n for given elements k =1: n, p =1: m, q =1: m xijk = 0 or 1 7 7

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16 Sets set_name [ text ] [/element [ text ] {,element [ text ] } /] {, set_name [ text ] [/element [ text ] {,element [ text ] } /] };

17 The asterisk * is used for sequence of elements The Alias statement is used for multiple names for a set

18 Parameters param_name [text] [/element [=] signed_num {, element [=] signed_num} /] {, param_name [text] [/element [=] signed_num {, element [=] signed_num} /] };

19 Variables var_name [text] {(domain)} {, var_name [text] {(domain)} }; [var-type] Variables var_name {,var_name};

20 Equations eqn_name [text] {(domain)} {, eqn_name [text] {(domain)} };

21 eqn_name {(domain)}.. expression eqn_type expression; eqn_type: =e= = =g= =l=

22 indexed_op ( (controlling_indices), expression) $(condition)

23 Model(s) model_name [text] [/all eqn_name {, eqn_name} /] {, model_name [text] [/all eqn_name {, eqn_name} /] }; Solve model_name using model_type maximizing minimizing var_name;

24 Display ident-ref quoted text {, ident-ref quoted text} Ident-ref means the name without domain list or driving indices of a set or parameter..l level or primal value.m marginal or dual value

25 GAMS Output ECHO PRINTS 9 9

26 GAMS Output EQUATIONS LISTINGS 9 9

27 GAMS Output MODEL STATISTICS 9 9

28 GAMS Output STATUS REPORTS 9 9

29 GAMS Output SOLUTION REPORTS 9 9

30 Any Questions? 10