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Linear Programming (LP)

線性規劃- George Dantzig, 1947

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[1] LP Formulation

(a) Decision Variables :

All the decision variables are non-negative.

(b) Objective Function : Min or Max

(c) Constraints

nxxx ,,, 21

21 32 xxMinimize

0,0

414

343..

21

21

21

xx

xx

xxts

s.t. : subject to

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[2] Example

A company has three plants, Plant 1, Plant 2, Plant 3. Because of declining earnings, top management has decided to revamp the company’s product line.

Product 1: It requires some of production capacity

in Plants 1 and 3.

Product 2: It needs Plants 2 and 3.

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The marketing division has concluded that the

company could sell as much as could be

produced by these plants.

However, because both products would be

competing for the same production capacity in

Plant 3, it is not clear which mix of the two

products would be most profitable.

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The data needed to be gathered:

1. Number of hours of production time available per week in each plant for these new products. (The available capacity for the new products is quite limited.)

2. Production time used in each plant for each batch to yield each new product.

3. There is a profit per batch from a new product.

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Production Timeper Batch, Hours

Production TimeAvailable

per Week, HoursPlant

Product

Profit per batch

1

2

3

4

12

18

1 2

1 0

0 2

3 2

$3,000 $5,000

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: # of batches of product 1 produced per week : # of batches of product 2 produced per week : the total profit per week

Maximizesubject to

1x2x

Z

1 2

1 2

1 2

1 2

1 2

3 5

1 0 4

0 2 12

3 2 18

0, 0

Z x x

x x

x x

x x

x x

81x0 2 4 6 8

2x

2

4

6

8

10

[3] Graphical Solution (only for 2-variable cases)

0,0 21 xx

Feasibleregion

91x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

41 x

Feasibleregion

101x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

Feasibleregion

111x0 2 4 6 8

2x

2

4

6

8

10

0,0 21 xx

122 2 x41 x

1823 21 xx

Feasibleregion

12

1x0 2 4 6 8 10

2x

2

4

6

8

21 5310 xxZ

21 5320 xxZ

Maximize:

21 5336 xxZ

)6,2(

The optimal solution

The largest value

Slope-intercept form:

21 53 xxZ

Zxx

5

1

5

312

13

1 1 2 2 n nZ c x c x c x

22222121

11212111

bxaxaxa

bxaxaxa

nn

nn

0,,0,0 21

2211

n

mnmnmm

xxx

bxaxaxa

Max

s.t.

[4] Standard Form of LP Model

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[5] Other Forms

The other LP forms are the following:

1. Minimizing the objective function:

2. Greater-than-or-equal-to constraints:

.2211 nn xcxcxcZ

1 1 2 2i i in n ia x a x a x b

Minimize

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3. Some functional constraints in equation form:

4. Deleting the nonnegativity constraints for

some decision variables:

ininii bxaxaxa 2211

jx : unrestricted in sign

where 0, 0j j j j jx x x x x

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[6] Key Terminology

(a) A feasible solution is a solution

for which all constraints are satisfied

(b) An infeasible solution is a solution

for which at least one constraint is violated

(c) A feasible region is a collection

of all feasible solutions

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(d) An optimal solution is a feasible solution

that has the most favorable value of

the objective function

(e) Multiple optimal solutions have an infinite

number of solutions with the same

optimal objective value

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,23 21 xxZ

1x

0,0

1823

21

21

xx

xx

122 2 x4

and

Maximize

Subject to

Example

Multiple optimal solutions:

19

21 2318 xxZ

1x0 2 4 6 8 10

2x

2

4

6

8

Feasibleregion

Every point on this red line

segment is optimal,

each with Z=18.

Multiple optimal solutions

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(f) An unbounded solution occurs when

the constraints do not prevent improving

the value of the objective function.

2x

1x

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[7] Case Study

The Socorro Agriculture Co. is a group of three farming communities.

Overall planning for this group is done in its Coordinating Technical Office.

This office currently is planning agricultural production for the coming year.

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GroupUsable Land

(Acres)Water Allocation

(Acre Feet)

123

400600300

600800375

The agricultural production is limited by both the amount of available land and the quantity of water allocated for irrigation.

These data are given below.

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The crops suited for this region include sugar

beets, cotton, and sorghum, and these are the three

products being considered for the upcoming

season.

These crops differ primarily in their expected net

return per acre and their consumption of water.

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CropSugar beetsCottonSorghum

Net Return($/Acre)

In addition, there is a maximum quota for the total acreage that can be devoted to each of these crops by Socorro Agriculture Co .

Water Consumption

(Acre Feet/Acre)

MaximumQuota

(Acres)600500325

321

1,000750250

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Because of the limited water available for irrigation, the Socorro Agriculture Co. will be unable to use all its land for planting crops.

To ensure equity among the three groups, it has been agreed that it will plant the same proportion of its available land.

For example, if Group 1 plants 200 of its available 400 acres, then Group 2 must plant 300 of its 600 acres, while Group 3 plants 150 acres of its 300 acres.

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The job facing the Coordinating Technical

Office is to plan how many acres to devote to

each crop while satisfying the given

restrictions.

The objective is to maximize the total net

return as a whole.

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The quantities to be decided are the number of acres to devote to each of the three crops at each of the three groups.

The decision variables represent these nine quantities.

)9,,2,1( jx j

1x 2x 3xCropSugar beetsCottonSorghum

4x 5x 6x7x 8x 9x

Allocation(Acres)Group

1 2 3

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Maximize Z=

),(250)(750)(000,1 987654321 xxxxxxxxx

The measure of effectiveness Z is the total net return and the resulting linear programming model for this problem is

1. Usable land for each group:

300

600

400

963

852

741

xxx

xxx

xxx

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3. Total land use for each crop:

325

500

600

987

654

321

xxx

xxx

xxx

2. Water allocation for each group:

375xx2x3

800xx2x3

600xx2x3

963

852

741

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4. Equal proportion of land planted:

400

xxx

300

xxx300

xxx

600

xxx600

xxx

400

xxx

741963

963852

852741

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5. Nonnegativity:

,0jx ( j =1, 2, …, 9)

0)(3)(4

0)(2)(

0)(2)(3

741963

963852

852741

xxxxxx

xxxxxx

xxxxxx

The final form is

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An optimal solution is

,0,0,0,150,250,100,25,100,3

1133

)x,x,x,x,x,x,x,x,x( 987654321

The resulting optimal value of the objective

function is $633,333.33.

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[8] Case Study - Personal Scheduling

UNION AIRWAYS needs to hire additional customer service agents.

Management recognizes the need for cost control while also consistently providing a satisfactory level of service to customers.

Based on the new schedule of flights, an analysis has been made of the minimum number of customer service agents that need to be on duty at different times of the day to provide a satisfactory level of service.

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** ** ** * * * * * * * * * * * *

ShiftTime Period Covered Minimum #

of Agents needed

Time Period

6:00 am to 8:00 am8:00 am to10:00 am10:00 am to noon Noon to 2:00 pm2:00 pm to 4:00 pm4:00 pm to 6:00 pm6:00 pm to 8:00 pm8:00 pm to 10:00 pm10:00 pm to midnightMidnight to 6:00 am

1 2 3 4 548796587647382435215

170 160 175 180 195Daily cost per agent

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The problem is to determine how many agents should be assigned to the respective shifts each day to minimize the total personnel cost for agents, while meeting (or surpassing) the service requirements.

Activities correspond to shifts, where the level of each activity is the number of agents assigned to that shift.

This problem involves finding the best mix of shift sizes.

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1x2x3x

4x5x

: # of agents for shift 1 (6AM - 2PM)

: # of agents for shift 2 (8AM - 4PM)

: # of agents for shift 3 (Noon - 8PM)

: # of agents for shift 4 (4PM - Midnight)

: # of agents for shift 5 (10PM - 6AM)

The objective is to minimize the total cost of the agents assigned to the five shifts.

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Min

s.t.54321 195180175160170 xxxxx

0ix )5~1( iall 15

52

43

82

73

64

87

65

79

48

5

54

4

43

43

32

321

21

21

1

x

xx

x

xx

xx

xx

xxx

xx

xx

x

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15

52

43

82

64

87

79

48

5

54

4

43

32

321

21

1

x

xx

x

xx

xx

xxx

xx

x

)15,43,39,31,48(),,,,( 54321 xxxxx

Total Personal Cost = $30,610

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Basic assumptions for LP models:

1. Additivity: c1x1+ c2x2 ； ai1x1+ ai2x2

2. Proportionality: cixi ； aijxj

3. Divisibility: xi can be any real number

4. Certainty: all parameters are known with certainty

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Other Examples

• Galaxy manufactures two toy doll models:– Space Ray. – Zapper.

• Resources are limited to– 1000 pounds of special plastic.– 40 hours of production time per week.

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• Marketing requirement– Total production cannot exceed 700 dozens.

– Number of dozens of Space Rays cannot exceed

number of dozens of Zappers by more than 350.

• Technological input– Space Rays requires 2 pounds of plastic and

3 minutes of labor per dozen.– Zappers requires 1 pound of plastic and

4 minutes of labor per dozen.

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• The current production plan calls for: – Producing as much as possible of the more profitable

product, Space Ray ($8 profit per dozen).– Use resources left over to produce Zappers ($5 profit

per dozen), while remaining within the marketing guidelines.

• The current production plan consists of:

Space Rays = 450 dozenZapper = 100 dozenProfit = $4100 per week

8(450) + 5(100)

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Management is seeking a production schedule that will increase the company’s profit.

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A linear programming model can provide an insight and an intelligent solution to this problem.

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• Decisions variables::

– X1 = Weekly production level of Space Rays (in dozens)

– X2 = Weekly production level of Zappers (in dozens).

• Objective Function:

– Weekly profit, to be maximized

The Galaxy Linear Programming Model

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Max 8X1 + 5X2 (Weekly profit)

subject to

2X1 + 1X2 1000 (Plastic)

3X1 + 4X2 2400 (Production Time)

X1 + X2 700 (Total production)

X1 - X2 350 (Mix)

Xj> = 0, j = 1,2 (Nonnegativity)

The Galaxy Linear Programming Model