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Chapter 9

Mixed-Integer Programming

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Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint.

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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

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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

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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.

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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

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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

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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.

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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

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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

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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

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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.

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