?????? Linear Programming Models

1
??????Linear Programming Models
Chapter 3
2
?????? Introduction to Linear Programming

• ??????(Linear Programming model)????????????(a
  set of linear constraints)??,?????(maximize)????(m
  inimize)?????????(objective function)
• ???????????????
• ?????? (A set of decision variables)
• ?????????(An objective function)
• ?????????? (A set of constraints)

3
?????? Introduction to Linear Programming

• ???????
• ?????????????????
• ????????????
• ??????????????
• Manufacturing
• Marketing
• Finance (investment)
• Advertising
• Agriculture

4
?????? Introduction to Linear Programming

• ???????
• ????????????????????????? what if ?????

5
????????? Assumptions for Linear Programming

• ?????????(certainty)
• ?????????????????????(constant returns to scale)
• ?????????????????? ,??????????????????
• ????? (Continuity) ?????????????????

500????45002000,35001,500Hrs??
1 ????4, 3Hrs??
6
???? The Galaxy Industries Production Problem

• Galaxy ????????
• ???Space Ray.
• ???Zapper.
• ????(Resources)
• 1000 ???????? (special plastic)
• ??40 ??????(40 hrs of production time per week)

7
???? The Galaxy Industries Production Problem

• ????(Marketing requirement)
• ??????? 700 ?
• Space Rays??????Zappers 350???

• ???? (Technological inputs) (Table 2.2)
• Space Rays ???? 2 pounds ??? 3??????
• Zappers???? 1pound ??? 4??????

8
???? The Galaxy Industries Production Problem

• ????
• Space Ray????(profit) 8,Zappers????(profit) 5
• ?????Space Ray,???????Zapper

• ??????
• Space Rays 450 dozen
• Zapper 100 dozen
• Profit 4100 per week

9

• ??????????????????????
• Management is seeking a production schedule
  that will increase the companys profit.

10
????????????? ????????????? A linear
programming model can provide an insight and an
intelligent solution to this problem.
11
??????????The Galaxy Linear Programming Model

• ????(Decisions variables)
• X1 ????? Space Rays ??
• X2 ????? Zappers ??
• ????(Objective Function)
• ????????

12
??????????The Galaxy Linear Programming Model

• Max 8X1 5X2 (?????)
• subject to
• 2X1 1X2 1000 (????,Plastic)
• 3X1 4X2 2400 (????,Production Time)
• X1 X2 700 (????,Total production)
• X1 - X2 350 (??)
• Xjgt 0, j 1,2 (???,Nonnegativity)

13
?????????? Graphical Analysis of
Linear Programming
????????????????? The set of all points
that satisfy all the constraints of the model is
called a
???? FEASIBLE REGION
14

• ?????(graphical presentation)
• ?????(all the constraints)
• ????(objective function)
• ???(three types of feasible points)

15
???? ????Graphical Analysis the Feasible
Region
The non-negativity constraints (?????)
X2
X1
16
???? ????Graphical Analysis the Feasible
Region
X2
1000
700
Total production ??? X1X2 700 (??)
500
Infeasible
Feasible
Production Time ??? 3X14X2 2400
X1
500
700
17
???? ???? (p. 6768)Graphical Analysis the
Feasible Region
X2
1000
Plastic??? 2X1X2 1000
700
Total production ??? X1X2 700 (??)
500
Infeasible
Mix??? X1-X2 350
Feasible
Production Time ??? 3X14X2 2400
X1
500
700
???Interior points.
??? Boundary points.
??Extreme points.

• ???(feasible points)???

18
?????????????Solving Graphically for an Optimal
Solution
19
????????? (p.71)The search for an optimal
solution
???? profit??, say profit 1,250.
X2
????????? increase the profit, if possible...
1000
????????????? continue until it becomes
infeasible
700
500
Optimal Profit 4360
???? Profit 1250
X1
500
20
??? (p.69) Summary of the optimal solution

• Space Rays X1 320 dozen
• Zappers X2 360 dozen
• Profit Z 4360
• ??????????????(plastic)????? (production hours).
• 2X1 1X2 1000 (????,Plastic)
• 3X1 4X2 2400
  (????,Production Time)

Excel???
?????(Binding Constraints)?????????
21
??? (p.7071) Summary of the optimal solution

• ???(Total production) 680 ? (not 700?)
• Space Rays ????? Zappers 40?

X1 X2 680 lt 700 (???) X1 - X2 -40 lt
350 (????)
????700-68020??? ?????350-(-40) 390???
??????(Non-Binding Constraints)???????????? ??(Sl
ack)???????????,?????????
22
?????? (p.72)Extreme points and optimal solutions

• ???????????????,????????????
  (??????????,True/False)
• ???????????????????

3X14X2 2400 X1 0 ??
(0,600)
2X1 X2 1000 3X14X2 2400 ??
(320,360)
?????????
2X1 X2 1000 X1-X2 350 ??
(450,100)
23
?????Multiple optimal solutions

• ????????,????????????????

• ????????????????????

XaX1(1-a)X2 , a?0,1 ?????
X2(0,600) ???2
X1(350,0) ???1
???? Z
24
??????????? The Role of Sensitivity
Analysis of the Optimal Solution (p.75)

• ???????????????????
• ??????????
• ?????????????????
• ???????????,??????????
• ??..?(What-if)???????????????.

25
(1) ????????????Sensitivity Analysis of
Objective Function Coefficients.

• ????(Range of Optimality) (p.76)
• ??????????,???????????,??????????????
• (p.77)????????,?
• ?????????????
• ?????????
• ???????????????,????????.

26
????????????Sensitivity Analysis of Objective
Function Coefficients.
X2
1000
??C1???8?3.75
Max 4X1 5X2
?????(320,360)
Max 3.75X1 5X2
Max 8X1 5X2
(0,600)
600
C1??2,????(0,600) ?(320,360)??????
(320,360)
Max 2X1 5X2
X1
500
800
27
????????????Sensitivity Analysis of Objective
Function Coefficients.
X2
1000
Max8X1 5X2
??C1??,?8?10 ??????(320,360)
600
Max 10 X1 5X2
C1??????? 3.75, 10
Max 3.75X1 5X2
(320,360)
??,C2??????? 4, 10.67 (Can you find it ?)
X1
400
600
800
28
???? Reduced cost (p.78)

• ????Xj 0?????RCj???????????????(-DZj)
  ,????????????(Xj gt0)
• ????RCj????Xj??????(DXj1) ,?????????

C12 X(0,600) ? X10 ?C13.75 X(320,360)
?X1320gt0 ? RC1 -?Z1-(3.75-2)-1.75
29
???????????????? (p.79)
X2
1000
?X11 (?X10?X11) ?Z2998.25-3000 -1.75 ?
RC1 -1.75
X1 1
Max 3.75X1 5X2
(1,599.25) Z2998.25
(0,600) Z3000
600
Max 2X1 5X2
X1
500
800
30
(2) ????? ?????? (p.78) Sensitivity Analysis of
Right-Hand Side Values

• ??
• ???????????,??????????,???????????????
• ??????(?????),?????????

31
????? ?????? Sensitivity Analysis of Right-Hand
Side Values

• ??
• ????????(Binding Constraints)????,?????????
• ?????(Non-Binding Constraints)????,?????????(slack
  )???(surplus)??,??????????
• ??????????(Shadow Price)???

32
???? Shadow Prices (p.80)

• ?????????????,???????? ????????????????,??????????
  ??

33
????Shadow Price ???? graphical demonstration
X2
1000
????(320,360)?(320.8,359.4)
2X1 1x2 lt1001
2X1 1x2 lt1000
500
Shadow price 4363.40 4360.00 3.40
??????(???1000?1001)???????
X1
500
34
????? Range of Feasibility (p.81)

• ?????????????
• ??????????????????? ??????????.
• ???????,

????????Change in objective value ??Shadow
price?????Change in the right hand side value
35
???????? Range of Feasibility (p.81)
X2
????????????????????Binding??
1000
2X1 1x2 lt1000
Total Production??? X1 X2 700
500
???????
Production time ???
X1
500
36
???????? Range of Feasibility
X2
???? ??????????????
1000
2X1 1x2 1000
Total Production ??? X1X2 700
???????? ?? 2X1 1X2 2(400)3001100
600
X1 X2 700 3X14X2 2400 ??
X(400,300)????
Production time ??? 3X14X2 2400
X1
500
37
???????? Range of Feasibility
X2
???? ??????????????
1000
Plastic ??? 2X1 1X2 1000
600
???????? ?? 2X1 1X2 2(0)1600600
3X1 4X2 2400 X1 0 ?? X(0,600)????
Production time ??? 3X14X2 2400
X1
500
38
????????? The correct interpretation of shadow
prices (p.83)

• ?????(Sunk costs) ?????????????????????- Shadow
  Price????????????
• ???????????3800?????????

1000?????3 ? Total Cost 3000 Production Time
20/hr ? Total Cost 2040800
?????????????Production Time,30008003800?????
,???????
39
????????? The correct interpretation of shadow
prices (p.83)

• ?????(Included costs)????????????????????-Shadow
  Price???????????????????
• ?p.84??2.5??

????3 ???????3.4 ????????????????6.8(?????)
Production Time 0.33/min (or 20/hr)
, Production Time?????0.4 ?????????Production
Time??? 0.73
40
(3) ???????? (p.84)Other Post - Optimality
Changes

• ????????(Addition of a constraint)
• ???????(Deletion of a constraint)

????????????? Yes, the solution is still
optimal No, re-solve the problem (the new
objective function is worse than the original one)

• ????????????????
• Yes, re-solve the problem (the new objective
  function is better than the original
  one)
• No, the solution is still optimal

41
???????? (p.84)Other Post - Optimality Changes

• ???? (Deletion of a variable)
• ???? (Addition of a variable)-???????(Net
  Marginal Profit)

???????????????0 Yes, the solution is still
optimal No, re-solve the problem (the new
objective function is worse than the original one)
42
???????? (p.85)Other Post - Optimality Changes
???? X3????????
???????3lb???5min ???? ????10
Max 8X1 5X2 10X3 (?????) subject to 2X1
1X2 3X3 1000 (????,Plastic ,Shadow Price
3.4) 3X1 4X2 5X3 2400
(????,Production Time, SP 0.4) X1 X2
X3 700 (????,Total production, SP
0) X1 - X2 350 (??, SP
0) Xjgt 0, j 1,2,3
(???,Nonnegativity)
?????10-(3.4(3)0.4(5)0(1)0(0))
-2.2 lt0 ?????????? ? X(320,360,0) ?????
43
???????? (p.85)Other Post - Optimality Changes

• ???????(Changes in the left - hand side
  coefficients.)

44
??Excel Solver ??????????

• ??Galaxy.xls,???????
• ????\????(Solver),????????.

45
?? Excel Solver

• ??Galaxy.xls,???????
• .

D7D10ltF7F10
46
?? Excel Solver

• ??Galaxy.xls,???????

?Solve?????
D6
Set Target cell
By Changing cells
B4C4
D7D10ltF7F10
47
??Excel Solver ???
48
??Excel Solver ???
Solver ???????????
49
??Excel Solver ???? Answer Report
50
??Excel Solver ???????Sensitivity Report
51
?????????

• ????(Infeasibility) ???????? (p.96)
• ???(Unboundness) ?????????,???????????????????(??
  ????)????(??????) (p.98)
• ???(Alternate solution)?????????????????????(p.98
  )

52
????? Infeasible Model
53
????? Solver ?????
Solver????????????
54
???Unbounded solution
????
55
????? Solver ?????
Solver??Set Cell????????
56
??????? Solver ?????

• Solver ??????????????
• ???????LP??,?????Xj ??????allowable increase or
  allowable decrease?0.
• ?Solver????????????(p.99)
• ???????Xj?

Allowable increase 0, ? Allowable decrease 0.
57
??????? Solver ?????

• ??????? Objective function Current optimal
  value.
• If Allowable increase 0, change the objective
  to Maximize Xj
• If Allowable decrease 0, change the objective
  to Minimize Xj
• Excel???

58
LP???????

• ????????????????
• ??????????????
• ???(Simplex Method) (???????CD3)
• ???(Interior Point Method)
• ?????????????
• ????(Cutting Plane Method)
• ?????(Branch and Bound Point Method)
  (???????CD3)