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.

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/Houston

4 Chapter 9 3.Blending problem. You are given a list of possible ingredients to be blended into a product ingredient with a minimum cost for a blend. Let x j be the quantity of ingredient j available in continuous amounts and y k represent ingredients to be used in discrete quantities v k (y k = 1 if used and y k = 0 if not used).

5 Chapter 9 4.Location of oil wells (plant location problem). It is assumed that a specific production-demand versus time relation exists for a reservoir. Several sites for new wells have been designated. The problem is how to select from among the well sites the number of wells to be drilled, their locations, and the production rates from the wells. The integer variables are the drilling decisions (0 = not drilled, 1 = drilled) for a set of n possible drilling locations. The continuous variables are the different well production rates.

6 1.Solve LP without integer constraints (relaxation) 0 < y i < 1 2.If all y i are 0 or 1, you have found the optimum. 3.If not, identify which y j have fractional values. 4.For the first fractional value, set up a branch to two alternative values, where either y j = 0 or y j = 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 Chapter 9

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9 EXAMPLE 13.4 OPTIMAL DESIGN OF A GAS TRANSMISSION NETWORK Suppose that a gas pipeline is to be designed so that it transports a prespecified quantity of gas per time from point A to other points. Both the initial state (pressure, temperature, composition) at A and final states of the gas are known. We need to determine. 1.The number of compressor stations 2.The lengths of pipeline segments between compressor stations 3.The diameters of the pipeline segments 4.The suction and discharge pressures at each station. Chapter 9

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11 The variables. Each pipeline segment has associated with it five variables: (1)The flow rate Q (2)the inlet pressure p d (discharge pressure from the upstream compressor) (3)the outlet pressure p s (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 is assumed to have gas consumed for operation of one-half of one percent of the gas transmitted, only the last four variables need to be determined for each segment. Chapter 9

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13 Chapter 9 The annualized capital costs for each pipeline segment depend on pipe diameter and length, but are assumed to be $870/(in.)(mile)(year). The rate of work of one compressor is

14 The inequality constraints. The operation of each compressor is constrained so that the discharge 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)(c) (e)(e) (h)(h) (d)(d) (f)(f) (g)(g) Chapter 9

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19 Chapter 9 Capital CompressorCompression cost station ratio($/year) Costs reduced from $14 million/yr (initial guess) to $7 million/yr at optimum

20 A microelectronics manufacturing facility is considering six projects to improve operations as well as profitability. Due to expenditure limitations and engineering staffing constraints, however, not all of these projects can be implemented. 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 value subject to the various constraints. Chapter 9

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