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Mixed-Integer Programming for Optimization Problems

This chapter discusses the use of mixed-integer programming to solve optimization problems with binary and integer variables. It covers different examples such as the traveling salesman problem, blending problem, oil well location, and the branch and bound procedure.

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Mixed-Integer Programming for Optimization Problems

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  1. Chapter 9 Mixed-Integer Programming Chapter 9

  2. Chapter 9 Enumeration approach for 20 objects (0,1): 2 20 possibilities, evaluate each case for satisfying constraint.

  3. The traveling salesman problem. The problem is to assign values • of 0 or 1 to variables yij, where yij 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, Chapter 9 subject to the 2n constraints Example: Austin/San Antonio/El Paso/Dallas/Houston

  4. 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 xj be the quantity of ingredient j available in • continuous amounts and yk represent ingredients to be used • in discrete quantities vk (yk = 1 if used and yk = 0 if not used). Chapter 9

  5. 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. Chapter 9

  6. Branch and Bound Procedure (MILP) for Binary Variables • 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 yjhave fractional values. • 4. For the first fractional value, set up a branch to two • alternative values, where either yj= 0 or yj= 1. • Solve each case using LP to find the minimum for the • two cases. This terminates that node. • 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. Chapter 9

  7. Chapter 9

  8. Chapter 9

  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. • The number of compressor stations • The lengths of pipeline segments between compressor stations • The diameters of the pipeline segments • The suction and discharge pressures at each station. Chapter 9

  10. Chapter 9

  11. The variables. Each pipeline segment has associated with it five variables: • The flow rate Q • 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 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

  12. Chapter 9

  13. 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 Chapter 9

  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 (c) and the compression ratio does not exceed some prespecified maximum limit K (d) In addition, upper and lower bounds are placed on each of the four variables Chapter 9 (e) (f) (g) (h)

  15. Chapter 9

  16. Chapter 9

  17. Chapter 9

  18. Chapter 9

  19. Capital Compressor Compression cost station ratio ($/year) 1 1.44 70.00 2 1.40 70.00 3 1.12 70.00 4 1.00 70.00 5 1.00 70.00 6 1.00 70.00 7 1.00 70.00 8 1.26 70.00 9 1.00 70.00 10 1.00 70.00 Chapter 9 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

  21. Chapter 9

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