Content and Mathematical Model of the Problem.


Machine

CN

A 10

B 25

C 25

Type I: 30

50

45

30

Type II: 20

40

42

28

Type III: 10

-

38

25

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Content and Mathematical Model of the Problem.

How to assign workers to operate machines so that the total number of products made in an hour is the largest, knowing that type III workers cannot operate machine A.

a) Model the problem. b) Solve the problem.


Answer : b) X

10

= 0

20 0

5 15 , f


= 2280

opt

0

0 10

max

Lesson 3.8A farm needs to grow 30 hectares of rice X 1 ; 20 hectares of rice X 2and 40 hectares of rice X 3on three plots of land I, II, III with areas of 25 ha, 25 ha, 40 ha respectively. The yield of each type of rice on each plot of land (tons/ha) is given in the following table:


Land

Rice varieties

I 25

II 25

III 40

X 1 : 30

-

8

6

X 2 : 20

6

9

7

X 3 : 40

5

4

6

Plan the land allocation so that the total yield is the highest knowing the rice variety X 1Cannot be grown on grade I soil.

a) Model the problem. b) Solve the problem.


Answer : b) X

0

= 0

25 5

0 20


or X

0

= 0

5 25

20 0 f


= 585

opt

25

0 15

opt

25

0 15

max

Lesson 3.9People need to grow 4 types of flowers: Chrysanthemum, Rose, Orchid, Gladiolus on 3 different gardens I, II, III. Knowing that the existing land area corresponding to each garden is 40, 60, 80 (a). The planned planting area for each type is Chrysanthemum - 50 (a); Rose - 70 (a); Orchid - 30 (a); Gladiolus - 30 (a). In addition, due to the nature of different types of soil, roses cannot be grown on land I, and Gladiolus cannot be grown on land III. Knowing the estimated harvest of each type of flower on each type of soil is as follows: (unit: hundred thousand VND/a). Plan to grow flowers to get the most profit.

10

6

C =

8

9 12

9

12

15 10 10

Solve the problem of finding the optimal plan.





Answer : X

10

= 0

0 0

30 30

30

0


or X

0

= 0

0 10

40 20

30

0 f


= 200,000,000 dong

opt

0

40 40 0 ​​

opt

50 30 0

max

0

Lesson 3.10Solve the transportation problem given by the following table: ( f min )

Collect

Play

B 1 60

B 2 70

B 3 40

B 4 30

A 1 : 100

2

1

4

3

A2 : 80

5

3

2

6

A 3 : 20

6

2

1

5

Lesson 3.11The transportation problem is given by the table ( f min )


Collect

Play

10

30

50

25

7

6

5

10

2

1

4

45

3

5

2

a) Model the problem.

b) What would the model look like if the second receiving station had to receive the full load?

c) Solve the problem in two cases.

Lesson 3.12The Olympic Games are held simultaneously on the same day in 4 locations. Material requirements (tons) are sent from 3 locations. Data on transmission and reception requirements and distance (km) are given in the table below. Due to the characteristics of material means, time and means of transport, it is not possible to transfer more than 150 km. Find a transportation plan so that the total number of T.km is the smallest.


Collect

Play

15

10

17

18

20

160

50

100

70

30

100

200

30

60

10

50

40

30

50

Lesson 3.13A mechanical factory has 3 workers A, B, C who have to operate 3 lathes I, II, III to produce a type of machine part. The productivity of each worker for each type of machine (parts/day) is given in the following table.

Machine

Human resource

I 1

II 1

III 1

A : 1

19

21

25

B : 1

20

18

24

C : 1

17

26

18


Make a plan to assign workers to operate the machine so that the total number of parts produced per day is the largest.

a) Model the problem.

b) How would the model change if worker B could not operate machine I?

c) Solve the problem in two cases.

Lesson 3.14

To prepare goods for sale during the Lunar New Year, the transport team must transport goods from 4 factories: A, B, C, D to 5 stores: C 1 , C 2 , C 3 , C 4 , C 5. The number of tons of goods needed at the stores, the distance between the factories and the stores (km) are given in the table:


CH

XN

C 110

C 210

C 310

C 420

C 520

A : 5

5

1

4

6

7

B : 15

3

4

2

7

8

C : 20

4

3

1

7

9

D : 30

6

5

4

9

11

Want to plan transportation so that the total T.km is the smallest.

a) Solve the problem

b) Solve the problem in the case where bridge A has just collapsed and the route from A to C 2 and from C to C 3must cross this bridge

Lesson 3.15A mechanical company signed a contract to deliver to a customer 7 products of type A1 , 8 products of type A2 , 5 products of type A3 .within 1 month .The company has 3 factories I, II, III and factory II cannot produce product A2 . The production capacity of each factory I, II, III in 1 month is 9 products, 5 products, 6 products respectively. The profit earned on each product produced by the factories is given in the following table (unit: 15,000,000 VND/product).


X.Product Industry

I 9

II 5

III 6

A 1 : 7

9

5

10

A2 : 8

5

--

6

A 3 : 5

7

4

4

Ask how to assign production to the factories to carry out the contract with the greatest profit and tell the amount of profit earned?

Lesson 3.16

Solve the transportation problem (f min) below with the condition that station B 1must collect enough goods


Collect

Play

B 1 70

B 2 80

B 3 65

B 4 85

A 1 : 90

6

7

4

5

A 2 :100

5

4

6

7

A 3 : 50

8

5

6

6.5

Lesson 3.17 Solve the transportation problem ( f max ) below with the condition that station B 2must collect enough goods

Collect

Play

B 1 80

B 2 120

B 3 100

A 1 :140

7

5.5

8

A 2 :110

5

6

5.5

Lesson 3.18A company has 3 factories located at 3 different locations to produce the same product. The capacity (products/month) and production cost (thousand VND/product) at each factory are given in the following table:


Factory

Capacity

Production cost

A 1A 2

A 3

100,000

150 000

200 000

24

22

23

The company's products are distributed to 4 general agents with consumption levels and selling prices (thousand VND/product) as follows:


General agent

Consumption capacity

Selling price

B 1 B 2 B 3 B 4

70 000

95 000

145 000

100,000

32

31

30

33

The shipping cost of a product from factories to general agents is given in the following table:

Factory General Agent

B 1

B 2

B 3

B 4

A 1

5

4

1

5

A 2

6

4

4

6

A 3

3

3

2

3


a) Make a mathematical model of the problem of determining the production and distribution plan from factories to general agents in a month so that the company's total profit is maximized.

b) Determine the company's optimal production plan and product distribution plan.

Answer : b)


13

Factory A produces 100,000 products and distributes them all to general agent B.


Factory A 2produce 110,000 products and distribute to general agent B 2,000 products, B 315,000 products.

95

Factory A 3produce 200,000 products and distribute to general agent B 1,000 products, B 330,000 products, B4100,000 products.

70

Maximum total profit: 2,305,000,000 VND


Multiple choice questions

(choose 1 of 4 sentences: A, B, C, D)

Sentence 1Which of the following statements is false?

A) The balanced transmission and reception transport problem always has PATU.

B) The dual problem of the balanced transmission and reception transport problem always has PATU.

C) The objective function of the transportation problem is always bounded.

D) The optimal solution of the transportation problem is always unique.

Sentence 2Which of the following statements is false?

A) The unbalanced transport problem always has PATU.

B) The dual problem of the unbalanced transmission and reception problem always has PATU.

C) The objective function of the unbalanced transmission and reception problem is always bounded.

D) The optimal solution of the unbalanced transmission and reception problem is always unique.

Sentence 3Which of the following statements is false?

A) The transport problem with the objective function maximizing the balance of transmission and reception always has PATU.

B) The dual problem of the transport problem with the objective function maximizing the balance of transmission and reception always has PATU.

C) The objective function of the transportation problem is always bounded.

D) The optimal solution of the transportation problem has a maximum objective function that is always unique.

Sentence 4Which of the following statements is false?

A) The transport problem with the objective function maximizing the balance of transmission and reception always has PATU.

B) The dual problem of the transport problem with the objective function maximizing the balance of transmission and reception always has PATU.

C) The transportation problem with restricted cells always has a unique optimal solution.

D) The dual problem of the synchronous production problem always has PATU.


Chapter 4

SYNCHRONIZED PRODUCTION PROBLEM

§ 1 CONTENT AND NATURE OF SYNCHRONIZED PRODUCTION ESSAY

1.1. Content and mathematical model of the problem.

Suppose a product is assembled by n different parts C 1 , C 2 , ..., C nand each product needs exactly 1 part of each type. Participating in the production process of these parts are m types of machines M 1 , M 2 , ...., M mand each type has exactly 1. Call c ijis the capacity of machine M iwhen used to produce C j parts(calculated by time unit)

can be hours, days, shifts, weeks, months,...). For convenience of presentation, we use the convention of productivity.

m n

calculated by shift. Matrix C = c ijcalled the productivity matrix (c i j 0). Arrange your time.

Working time for machines so that the total output obtained in one shift is the largest.

Problem analysis: Let x ijis the time period (in units of shifts) for machine M i

C j detail production(i = 1, m

; j = 1, n

) and z is the total number of assembled products.

The total number of products produced is at most: f(z,x) = zmax

n

Machines used up the entire shift: x ij

j 1

= 1 , i = 1, m

The number of assembled products does not exceed the number of parts of each type that can be produced.

m

in song:

z c ij x ij0 , j = 1, n

i 1

We have a mathematical model to find (z,x) = (z,x i j ) so that: (we call this problem (P) )

(1) f(z,x) = z max

(2)

n

j 1

x ij

1,


i 1, m

z

m

c

i 1

ij x ij

0,


j 1, n


(3) z 0 ; x ij 0 (i = 1, m , j = 1, n )

Each set (z,x ij) satisfying (2) and (3) is called a solution. The solution satisfying (1) is called PATU.

1.2. Put the problem in table form.

The synchronous production problem is a QHTT problem. But because it has a special form, people do not solve it by the simplex method but by another method. To facilitate the presentation of this algorithm, we represent the problem in a table form.

m n

with the yield matrix C =c ij

. All the concepts of cells, rings,... are similar.

transportation problem


Machine Details

C 1 1

C 2 1


C j 1


C n 1

M 1 : 1

c 11

x 11

c 12

x 12

c 1j

x 1j

c 1n

x 1n

M 2 : 1

c 21

x 21

c 22

x 22

c 2j

x 2j

c2n

x 2n





M i : 1

c i1

x i1

c i2

x i2

c ij

x ij

print

Ask for






M : 1

c m1

x m1

c m2

x m2

c m j

x mj

c mn

x mn


1.3. Properties of synchronous production problem.

Property 1 : Synchronous production problems always have PATU.

Prove


0,

The SXDB problem has the following solution: z = 0 , x ij= 1 ,

j 1

j 1


The SXDB problem has a bounded objective function on: f(z,x)m.maxc ij

So the SXDB problem has PATU .

Consider the synchronous production problem (P):

(1) f(z,x) = z max

(2)

n

j 1

x ij

1,


i 1, m

z

m

c

i 1

ij x ij

0,


j 1, n

(3) z 0 ; x ij0 (i = 1, m , j = 1, n ) The dual problem of this problem is (D)ø:


m

(1) g(u, v) = u i min

i 1

(2)

n

v j

j 1

1,

i 1, m

u i c ij v j 0 i 1, m; j 1, n

(3) and0 (j = 1, n ), u iarbitrary (i = 1, m ). The dual constraint pairs are:

n

z 0 v j 1

j 1

u i c ij v j 0 x ij 0 (i = 1, m

; j = 1, n )

z c i j x ij0 v j0

(j = 1, n )


Based on the weak complement theorem we have the following properties.

Property 2: The necessary and sufficient conditions for solution (z,x i j ) of problem (P) and solution (u i ,v j ) of problem (D) to be optimal are:

n

z v j 1 0

(1)

(*)

j 1

0


(2)

x ij u i c ij v j

m

v j z c ij x ij 0

(3)

i 1


This property follows easily from the weak complement theorem.

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