0 a toy production problem how many units to produce from each product type in order to maximize...

1

A Toy Production Problem

How many units to produce from each product type in order to maximize the profit?

Product Man-Power Machine Profit

Type A 3 h 1 h 25 SR

Type B 2 h 4 h 15 SR

Availability 70 h 110 h

2

A Toy Production Problem

xA : number of units of product type AxB : number of units of product type B Total Profit: 25 xA + 15 xB

 Man-Power availability: 3 xA + 2 xB 70 Machine availability: xA + 4 xB 110

Product Man-Power Machine Profit

Type A 3 h 1 h 25 SR

Type B 2 h 4 h 15 SR

Availability 70 h 110 h

3

A Toy Production Problem

The corresponding linear program (LP) is:

Max Z = 25 xA + 15 xB

Subject to

3 xA + 2 xB 70

xA + 4 xB 110

xA , xB IN

Objective function

Constraints

Decision Variables

4

A Toy Production Problem

The optimal solution of the LP is:

x*A = 22 , x*B = 2 and Z* = 580 SR

• The optimal solution of the toy production problem is to produce 22 units of toy A and 2 units of toy B.

• The optimal profit is 580 SR.

5

A Banking Problem

• A bank is in the process to allocate 12 million SR to different types of loans.• Competition with other financial institutions suggests allocating at least 40% of the funds to farm & commercial loans.• To assist in the housing industry, the home loans must be at least 50% of the personal, car and home loans. • The bank management requires that the overall bad ratio debt not to exceed 4%.• The bank wants to determine the best loan policy that will maximize its profit rate.

Type of loan Profit rate Bad debt ratio1 Personal 0.14 0.102 Car 0.13 0.073 Home 0.12 0.034 Farm 0.125 0.055 Commercial 0.10 0.03

6

A Banking Problem

Decision variables:x1: amount of personal loans (in million SR)

x2: amount of car loans (in million SR)x3: amount of home loans (in million SR)x4: amount of farm loans (in million SR)

x5: amount of commercial loans (in million SR)

Type of loan Profit rate Bad debt ratio1 Personal 0.14 0.102 Car 0.13 0.073 Home 0.12 0.034 Farm 0.125 0.055 Commercial 0.10 0.03

7

A Banking Problem

Objective function:

Max Z= 0.14(0.90)x1+ 0.13(0.93)x2+ 0.12(0.97)x3+0.125(0.95)x4+0.1 (0.98)x5 - 0.1x1- 0.07x2- 0.03x3- 0.05x4-0.02x5

Type of loan Profit rate Bad debt ratio1 Personal 0.14 0.102 Car 0.13 0.073 Home 0.12 0.034 Farm 0.125 0.055 Commercial 0.10 0.02

8

A Banking Problem

Constraints:

• The total funds shall not exceed 12 million SR: x1+x2+ x3+ x4+x5 12

• Farm and commercial loan constraint: x4+x5 0.4(x1+x2+ x3+ x4+x5)

• Home loans constraint: x3 0.5(x1+x2+ x3)

Type of loan Profit rate Bad debt ratio1 Personal 0.14 0.102 Car 0.13 0.073 Home 0.12 0.034 Farm 0.125 0.055 Commercial 0.10 0.03

9

A Banking Problem

Constraints:

• Limit on bad debts rule: (0.1x1+ 0.07x2+ 0.03x3+ 0.05x4+0.02x5)/( x1+x2+ x3+ x4+x5)

0.04

(NONLINEAR)or equivalently,

0.06x1+ 0.03x2- 0.01x3+ 0.01x4- 0.02x5 0

Type of loan Profit rate Bad debt ratio1 Personal 0.14 0.102 Car 0.13 0.073 Home 0.12 0.034 Farm 0.125 0.055 Commercial 0.10 0.03

10

A Banking Problem

The optimal solution of the LP is:

x*1 = x*2 = x*4 = 0 , x*3 = 7.2, x*5 = 4.8 and Z* = 0.99648

• The optimal solution consists in allocating 7.2 M to the home type and 4.8 M to commercial type.

• The optimal profit is 0.996 M SR.

11

A Telecommunication Problem

• We have to place transmitters on the sites A,B,…,G.• Each transmitter covers the two adjacent zones (e.g. if a transmitter is placed on site D, then it will cover both zones 3 and 4).• Each Zone must be covered with at least one transmitter.• Zone 4 must be covered by at least 2 transmitters.• What is the minimum number of transmitters to be placed? • Where should they be placed?

12

A Telecommunication Problem

Decision variablesxi = 1 if a transmitter is to be placed on site i (i =A,B,…,G) 0 otherwise Objective functionMin Z = xA + xB + xC + xD + xE + xF + xG

13

A Telecommunication Problem

Constraints• xA + xB + xC 1 (Zone 1)

• xA + xE + xF 1 (Zone 2)

• xB + xD 1 (Zone 3)

• xC + xD + xE + xG 2 (Zone 4)

• xF + xG 1 (Zone 5)

• xi {0,1} for all i=A,B,…,G

14

A Telecommunication Problem

• The optimal solution of the LP is:x*

A = x*D = x*

G = 1x*

B = x*C = x*

E = x*F = 0

Z * = 3• The optimal solution of the telecommunication problem is to place one transmitter at each of the sites A, D, and G.

15

A Transportation Problem

• All the demands must be satisfied.• All the supplies must be delivered.• How many units to transport from each source to each destination in order to minimize the total transportation cost?

D1

D2

C

B

A

7

2

4

6

3

5

Demand4

5

8

Supply

8

9

Sources Destinations

Unit transportation cost

16

A Transportation Problem

• xij : number of units transported from Source i to Destination j i=1,2 j=A,B,C

• Objective function Min Z = 5 x1A + 3 x1B + 6 x1C + 4 x2A + 7 x2B + 2 x2C

D1

D2

C

B

A

7

2

4

6

3

5

Demand4

5

8

Supply

8

9

Sources Destinations

Unit transportation cost

17

A Transportation Problem

Constraints• x1A + x1B + x1C = 8 (Supply of D1)

• x2A + x2B + x2C = 9 (Supply of D2)

• x1A + x2A = 4 (Demand of A)

• x1B + x2B = 5 (Demand of B)

• x1C + x2C = 8 (Demand of C)

• x1A ,x1B , x1C , x2A , x2B , x2C  IN

D1

D2

C

B

A

7

2

4

6

3

5

Demand4

5

8

Supply

8

9

Sources Destinations

18

A Transportation Problem

Optimal solution

x*1A = 3, x*1B = 5, x*1C = 0, x*2A = 1, x*2B = 0, x*2C = 8

Z* = 50

D1

D2

C

B

A

7

2

4

6

3

5

Demand4

5

8

Supply

8

9

Sources Destinations