Dynamic Programming Applications. Capacity Expansion

Transcription

1 Dynamic Programming Applicaions Capaciy Expansion

2 Objecives To discuss he Capaciy Expansion Problem To explain and develop recursive equaions for boh backward approach and forward approach To demonsrae he mehod using a numerical example

3 Capaciy Expansion Problem Consider a municipaliy planning o increase he capaciy of is infrasrucure (ex: waer reamen plan, waer supply sysem ec) in fuure Sequenial incremens are o be made in specified ime inervals The capaciy a he beginning of ime period be S Required capaciy a he end of ha ime period be K x Thus, be he added capaciy in each ime period Cos of expansion a each ime period can be expressed as a funcion of S and x, i.e. C S, x ) ( 3

4 Capaciy Expansion Problem cond. Opimizaion problem: To find he ime sequence of capaciy expansions which minimizes he presen value of he oal fuure coss Objecive funcion: T Minimize C ( S, x ) C : Presen value of he cos of adding an addiional ( S, x ) capaciy in he ime period Consrains: Capaciy demand requiremens a each ime period S x : Iniial capaciy = 4

5 Capaciy Expansion Problem cond. Each period s final capaciy or nex period s iniial capaciy should be equal o he sum of iniial capaciy and he added capaciy S + = S + x for =,,..., T A he end of each ime period, he required capaciy is fixed S + K for =,,..., T Consrains o he amoun of capaciy added in each ime period i.e. can ake only some feasible values. x Ω x 5

6 Capaciy Expansion Problem: Forward Recursion Sages of he model: Time periods in which capaciy expansion o be made Sae: Capaciy a he end of each ime period, S f ( S + ) : Presen capaciy before expansion S + : Minimum presen value of oal cos of capaciy expansion from presen o he ime C C C T S S Sage S Sage S + S T S T+ Sage T 6 x x x T

7 Capaciy Expansion Problem: Forward Recursion cond. For he firs sage, objecive funcion f ( S ) = min = min C ( S, x, S ) S Value of S can be beween K and K T where K is he required capaciy a he end of ime period and K T is he final capaciy required Now, for he firs wo sages ogeher, f ( S 3 ) = = min x x Ω min x x Ω C ( S [ C ( S, x ) + f ( S )] [ C ( S x, x ) + f ( S x )] 3 ) 3 7 Value of S 3 can be beween K and K T

8 Capaciy Expansion Problem: Forward Recursion cond. In general, for a ime period, f Subjeced o [ C ( S x, x ) + f ( S x )] ( S+ ) = min + + x x Ω K S + KT For he las sage, i.e. =T, f T ( S T + ) need o be solved only for S T + = K T 8

9 Capaciy Expansion Problem: Backward Recursion Sages of he model: Time periods in which capaciy expansion o be made Sae: Capaciy a he beginning of each ime period, f S ( ) : Minimum presen value of oal cos of capaciy expansion in periods hrough T For he las period T, he final capaciy should reach K T afer doing he capaciy expansions f T ( S T ) = min xt x Ω T T [ C ( S, x )] T T T S Value of S T can be beween K T- and K T 9

10 Capaciy Expansion Problem: Backward Recursion cond. In general, for a ime period, f [ C ( S, x ) + f ( S + x )] ( S ) = min + x x Ω Solved for all values of S ranging from K - and K For period, he above equaion mus be solved only for S = S 0

11 Capaciy Expansion: Numerical Example Consider a five sage capaciy expansion problem The minimum capaciy o be achieved a he end of each ime period is given in he able below D

12 Capaciy Expansion: Numerical Example cond. Expansion coss for each combinaion of expansion

13 Numerical Example: Forward Recursion Consider he firs sage, = The final capaciy for sage, S can ake values beween D o D 5 Le he sae variable can ake discree values of 5, 0, 5, 0 and 5 Objecive funcion for s subproblem wih sae variable as S can be expressed as f ( S ) = min = min C ( S C ( S, x, S ) S ) 3

14 Numerical Example: Forward Recursion cond. Compuaions for sage are given in he able below 4

15 Numerical Example: Forward Recursion cond. Now considering he s and nd sages ogeher Sae variable S 3 can ake values from D o D 5 Objecive funcion for nd subproblem is f ( S 3 ) = = min x min x x Ω x Ω [ C ( S, x ) + f ( S )] [ C ( S x, x ) + + f ( S x )] 3 The value of x should be aken in such a way ha he minimum capaciy a he end of sage should be 0, i.e. S

16 Numerical Example: Forward Recursion cond. Compuaions for sage are given in he able below 6

17 Numerical Example: Forward Recursion cond. Like his, repea his seps ill = 5 The compuaion ables are shown 7

18 Numerical Example: Forward Recursion cond. For he 5 h subproblem, sae variable S 6 = D 5 8

19 Numerical Example: Forward Recursion cond. 9 Figure showing he soluions wih he cos of each addiion along he links and he minimum oal cos a each node Opimal cos of expansion is 3 unis By doing backracking from he las sage (farhes righ node) o he iniial sage, he opimal expansion o be done a s sage = 0 unis, 3rd sage = 5 unis and res all sages = 0 unis

20 Numerical Example: Backward Recursion Capaciy a he final sage is given as S 6 = 5 Consider he las sage, =5 Iniial capaciy for sage 5, S 5 can ake values beween D 4 o D 5 Objecive funcion for s subproblem wih sae variable as S 5 can be expressed as f [ f ( S )] 5 S5 ) = min 5 5, xt x Ω ( x T T 5 0 The opimal cos of expansion can be achieved by following he same procedure o all sages

21 Numerical Example: Backward Recursion cond. Compuaions for all sages

22 Numerical Example: Backward Recursion cond.

23 Numerical Example: Backward Recursion cond. 3

24 Numerical Example: Backward Recursion cond. 4

25 Numerical Example: Backward Recursion cond. Opimal cos of expansion is obained from he node value a he firs node i.e. 3 unis Opimal expansions o be made are 0 unis a he firs sage and 5 unis a he las sage 5

26 Capaciy Expansion Problem: Uncerainy The fuure demand and he fuure cos of expansion in his problem are highly uncerain Hence, he soluion obained canno be used for making expansions ill he end period, T Bu, decisions abou he expansion o be done in he curren period can be very well done hrough his For he uncerainy on curren period decisions o be less, he final period T should be seleced far away from he curren period 6

27 Thank You 7