1 chapter 9 mixed-integer programming. 2 chapter 9 enumeration approach for 20 objects (0,1): 2 20...
DESCRIPTION
3 Chapter 9 2.The traveling salesman problem. The problem is to assign values of 0 or 1 to variables y ij, where y ij is 1 if the salesman travels from city i to city j and 0 otherwise. The salesman must start at a particular city, visit each of the other cities only once, and return to the original city i to city j, subject to the 2n constraints Example: Austin/San Antonio/El Paso/Dallas/HoustonTRANSCRIPT
![Page 1: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/1.jpg)
1
Cha
pter
9
Chapter 9
Mixed-Integer Programming
![Page 2: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/2.jpg)
2
Cha
pter
9
Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint.
![Page 3: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/3.jpg)
3
Cha
pter
9
2. The traveling salesman problem. The problem is to assign valuesof 0 or 1 to variables yij, where yij is 1 if the salesman travels fromcity i to city j and 0 otherwise. The salesman must start at aparticular city, visit each of the other cities only once, and return tothe original city i to city j,
subject to the 2n constraints
1 1
min ( ) ,n n
ij i ji j
f y c y
1 1
1, 1n n
i j i ji j
y y
0,1 , 1,...,
0 i j
i j
y i j n
y i k
Example: Austin/San Antonio/El Paso/Dallas/Houston
![Page 4: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/4.jpg)
4
Cha
pter
93. Blending problem. You are given a list of possible ingredients
to be blended into a product ingredient with a minimum cost fora blend. Let xj be the quantity of ingredient j available incontinuous amounts and yk represent ingredients to be usedin discrete quantities vk (yk = 1 if used and yk = 0 if not used).
Minimize : j j k k kj k
c x d v y
Subject to : (weight)
(composition)
I uj k k
j k
l ui ij j ik k k i
j k
W x v y W
A a x a v y A
![Page 5: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/5.jpg)
5
Cha
pter
9
4. Location of oil wells (plant location problem). It is assumed thata specific production-demand versus time relation exists for areservoir. Several sites for new wells have been designated. Theproblem is how to select from among the well sites the numberof wells to be drilled, their locations, and the production ratesfrom the wells.
The integer variables are the drilling decisions (0 = not drilled,1 = drilled) for a set of n possible drilling locations. The continuousvariables are the different well production rates.
![Page 6: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/6.jpg)
6
1. Solve LP without integer constraints (relaxation)0 < yi < 1
2. If all yi are 0 or 1, you have found the optimum.3. If not, identify which yj have fractional values.4. For the first fractional value, set up a branch to two
alternative values, where either yj = 0 or yj = 1.5. Solve each case using LP to find the minimum for the
two cases. This terminates that node.6. Repeat step 4 for the next fractional value and check
its optimum. Continue.7. Find the best termination point out of all the nodes
examined, which is the MILP optimum.
Branch and Bound Procedure (MILP)for Binary Variables
Cha
pter
9
![Page 7: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/7.jpg)
7
Cha
pter
9
![Page 8: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/8.jpg)
8
Cha
pter
9
![Page 9: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/9.jpg)
9
EXAMPLE 13.4 OPTIMAL DESIGN OF A GAS TRANSMISSION NETWORK
Suppose that a gas pipeline is to be designed so that it transports aprespecified quantity of gas per time from point A to other points.Both the initial state (pressure, temperature, composition) at A andfinal states of the gas are known. We need to determine.
1. The number of compressor stations2. The lengths of pipeline segments between compressor stations3. The diameters of the pipeline segments4. The suction and discharge pressures at each station.C
hapt
er 9
![Page 10: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/10.jpg)
10
Cha
pter
9
![Page 11: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/11.jpg)
11
The variables. Each pipeline segment has associated with it five variables:
(1) The flow rate Q(2) the inlet pressure pd (discharge pressure from the upstream
compressor)(3) the outlet pressure ps (suction pressure of the downstream compressor)(4) the pipe diameter D, and(5) the pipeline segment length L.
Because the mass flow rate is fixed, and each compressor isassumed to have gas consumed for operation of one-half of onepercent of the gas transmitted, only the last four variables need to bedetermined for each segment.
Cha
pter
9
![Page 12: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/12.jpg)
12
Cha
pter
9
![Page 13: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/13.jpg)
13
Cha
pter
9
The annualized capital costs for each pipeline segment depend onpipe diameter and length, but are assumed to be $870/(in.)(mile)(year).The rate of work of one compressor is
![Page 14: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/14.jpg)
14
The inequality constraints. The operation of each compressor is constrained so that thedischarge pressure is greater than or equal to the suction pressure
and the compression ratio does not exceed some prespecified maximum limit K
In addition, upper and lower bounds are placed on each of the four variables
(c)
(e)
(h)
(d)
(f)
(g)
1
1, 1, 2,...,d
s
p i np
1
, 1, 2,...,di
s
pK i n
p
min maxi i id d dp p p
min maxi i is s sp p p
min maxi i iL L L
min maxi i iD D D
Cha
pter
9
![Page 15: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/15.jpg)
15
Cha
pter
9
![Page 16: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/16.jpg)
16
Cha
pter
9
![Page 17: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/17.jpg)
17
Cha
pter
9
![Page 18: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/18.jpg)
18
Cha
pter
9
![Page 19: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/19.jpg)
19
Cha
pter
9Capital
Compressor Compression cost station ratio ($/year)
1 1.44 70.002 1.40 70.003 1.12 70.004 1.00 70.005 1.00 70.006 1.00 70.007 1.00 70.008 1.26 70.009 1.00 70.00
10 1.00 70.00
Costs reduced from $14 million/yr (initial guess) to $7 million/yr at optimum
![Page 20: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/20.jpg)
20
A microelectronics manufacturing facility is consideringsix projects to improve operations as well as profitability.Due to expenditure limitations and engineering staffingconstraints, however, not all of these projects can beimplemented. The following table gives projected cost,staffing, and profitability data for each project
A new or modernized production line must be provided(project 1 or 2). Automation is feasible only for the new line.Either project 5 or project 6 can be selected, but not both. Determine which projects maximize the net present valuesubject to the various constraints.
Cha
pter
9
![Page 21: 1 Chapter 9 Mixed-Integer Programming. 2 Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint](https://reader036.vdocuments.us/reader036/viewer/2022062905/5a4d1b177f8b9ab059991f84/html5/thumbnails/21.jpg)
21
Cha
pter
9