or ii 04 dynamic

Operations ResearchIndustrial engineering

11/04/2023 Operations Research 2

DYNAMIC PROGRAMMING


Characteristics of dynamic programming problem1. The problem can be divided into

stages, with a policy decision required at each stage

2. Each stage has a number of states associated with it

3. The effect of the policy decision at each stage is to transform the current state into a state associated with the next stage (possibly according to a probability distribution)


Characteristics of dynamic programming problem4. The solution procedure is designed to find

an optimal policy for the overall problem, i.e., a prescription of the optimal policy decision at each stage for each of the possible states

5. Given the current state, an optimal policy for the remaining stages is independent of the policy adopted in precious stages (this is the principle of the optimality for dynamic programming)

6. The solution procedure begins by finding the optimal policy for the last stage


Characteristics of dynamic programming problem7. A recursive relationship that identifies

the optimal policy for stage n, given the optimal policy for stage (n + 1), is available


N = number of stagesn = label for current stage (n = 1, 2,

…, N)Sn = current state for stage n

Xn = decision variable for stage n

Xn* = optimal value of Xn (given Sn)

nnSXX

n XfcSfn

n

*1

* min


= contribution of stage n, n + 1, …, N to the objective function if the system starts in state Sn at stage n, the immediate decision id Xn, and optimal decisions are made thereafter

nnn XSf ,

** , nnnnn XSfSf

nnSXX

n XfcSfn

n

*1

* min


The recursive relationship will always be of the form

nnnX

nn

nnnX

nn

XSfSf

atau

XSfSf

n

n

,min

,max

*

*

nnSXX

n XfcSfn

n

*1

* min


Characteristics of dynamic programming problem8. When we use this recursive

relationship, the solution procedure moves backward stage by stage – each time finding the optimal policy for that stage – until it finds the optimal policy starting at the initial stage

X1 X2 X3S1S2S3

Xn Sn

fn(Sn, Xn)fn*(Sn) Xn*


Contoh 1Sebuah perusahaan mempunyai

usulan dari ketiga pabriknya untuk kemungkinan mengembangkan sarana produksi. Perusahaan tersebut menyediakan anggaran $5 juta untuk alokasi pada ketiga pabrik. Setiap pabrik diminta untuk menyampaikan usulannya yang memberikan jumlah biaya (c) dan jumlah pendapatan (R) untuk setiap usulan.


Contoh 1

c1 R1 c2 R2 c3 R31 0 0 0 0 0 02 1 5 2 8 1 33 2 6 3 94 4 12

Pabrik 2 Pabrik 3Usulan

Pabrik 1


Contoh 1

P

P1U3

P1U2

P1U1

P2 U4

P2 U3

P2 U2

P2 U1

P3 U2

E

P3 U1


Contoh 1: stage 1

0 1 20 0 0 01 0 5 5 12 0 5 6 6 23 0 5 6 6 24 0 5 6 6 25 0 5 6 6 2

X1 S1

f1(S1, X1)f1*(S1) X1*


Contoh 1: Stage 2

0 2 3 40 0 0 01 5 5 02 6 8 8 23 6 13 9 13 24 6 14 14 12 14 2 atau 35 6 14 15 17 17 4

X2* X2

S2 f2*(S2)f2(S2, X2)


Contoh 1: Stage 3

Dana yang tersedia $5 juta dimanfaatkan semua

Alokasi dana pabrik 1 – pabrik 2 – pabrik 3◦ 1 – 4 – 0◦ 1 – 3 – 1◦ 2 – 2 – 1

Total pendapatan = $17 juta

0 15 17 17 17 0 atau 1

X3 S3 X3*

f3(S3, X3)f3*(S3)


Contoh 1: Rekursif Mundur

0 10 0 0 01 0 3 3 12 0 3 3 13 0 3 3 14 0 3 3 15 0 3 3 1

X3 S3

f3(S3, X3)f3*(S3) X3*



0 2 3 40 0 0 01 3 3 02 3 8 8 23 3 11 9 11 24 3 11 12 12 12 3 atau 45 3 11 12 15 15 4

X2* X2

S2f2(S2, X2)

f2*(S2)



Dana yang tersedia $5 juta dimanfaatkan semua

Alokasi dana pabrik 1 – pabrik 2 – pabrik 3◦ 1 – 3 – 1◦ 1 – 4 – 0◦ 2 – 2 – 1

Total pendapatan = $17 juta

0 1 25 15 17 17 17 1 atau 2

X1 S1

f1(S1, X1)f1*(S1) X1*


Contoh 2Suatu organisasi kesehatan dunia

menyelenggarakan program peningkatan kepedulian pada kesehatan dan memberikan pendidikan kesehatan di beberapa negara terbelakang

Organisasi tersebut memiliki 5 tim medis yang siap ditugaskan di 3 negara

Satu negara paling tidak harus didatangi 1 tim medis

Performansi diukur dengan penambahan umur hidup


Contoh 2

1 2 31 45 20 502 70 45 703 90 75 804 105 110 1005 120 150 130

Number of Medical Teams

Thousands of Additional Person-Years of Life

Country


Contoh 2: Stage 3

1 2 31 45 45 12 45 70 70 23 45 70 90 90 3

X3 S3

f3(S3, X3)f3*(S3) X3*


Contoh 2: Stage 2

1 2 32 65 65 13 90 90 90 1 atau 24 110 115 120 120 3

X2 S2

f2(S2, X2)f2*(S2) X2*


Contoh 2: Stage 1

Alokasi Tim Medis◦1 – 3 – 1◦Total additional person-years of life =

170.000

1 2 35 170 160 145 170 1

X1 S1

f1(S1, X1)f1*(S1) X1*


Contoh 2: asumsi suatu negara boleh tidak dikunjungi tim medis sama sekali

0 1 2 3 4 50 0 0 01 0 45 45 12 0 45 70 70 23 0 45 70 90 90 34 0 45 70 90 105 105 45 0 45 70 90 105 120 120 5

f3*(S3) X3* X3

S3f3(S3, X3)

0 1 2 3 4 50 0 0 01 45 20 45 12 70 65 45 70 13 90 90 90 75 90 0 atau 1 atau 24 105 110 115 120 110 120 35 120 125 135 145 155 150 155 4

X2 S2

f2(S2, X2)f2*(S2) X2*


Contoh 2: asumsi suatu negara boleh tidak dikunjungi tim medis sama sekali

Alokasi tim medis◦1 – 3 – 1◦Total additional person-years of life =

170.000

0 1 2 3 4 55 155 170 160 150 145 130 170 1

X1 S1

f1(S1, X1)f1*(S1) X1*


Soal 1A college student has 7 days remaining before final

examinations begin in her four courses, and she wants to allocate this study time as effectively as possible. She needs at least 1 day on each course, and she likes to concentrate on just one course each day, so she wants to allocate 1, 2, 3, or 4 days to each course. Having recently taken an operations research course, she decides to use dynamic programming to make these allocations to maximize the total grade points to be obtained from four courses. She estimates that the alternative allocations for each course would yield the number of grade points shown in the table. Solve this problem by dynamic programming.


Soal 1

1 2 3 41 3 5 2 62 5 5 4 73 6 6 7 94 7 9 8 9

number of study days

estimated grade pointscourses

or ii 04 dynamic

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