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3_Integer_Programming_(Student)

# 3_Integer_Programming_(Student) - Integer Programming(IP...

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1 Chapter 5 Integer Programming (IP) 2 Chapter Outline ± To introduce different types of Integer Programming (IP) Models ± To learn the model formulation of IP ± To use computer software to solve IP problems 3 Total Integer Model: All decision variables required to have integer solution values. e.g. product mix 0-1 Integer Model: All decision variables required to have integer values of zero or one. e.g. assignment problem Mixed Integer Model: Some of the decision variables (but not all) required to have integer values. Integer Programming Models Types of Models 4 Machine Required Floor Space (ft. 2 ) Purchase Price Press Lathe 15 30 \$8,000 4,000 The owner of a machine shop plans to expand by purchasing new machines: presses and lathes. Marginal profitability: each press \$100/day; each lathe \$150/day. Resource constraints: \$40,000 budget, 200 sq. ft. floor space. Machine purchase prices and space requirements: Total Integer Model Example 1 Problem Statement (1 of 2)

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5 Total Integer Model Example 1 IP Model Formulation (2 of 2) Let x 1 and x 2 be the number of presses and lathes purchased Maximize Z = \$100x 1 + \$150x 2 subject to: 8,000x 1 + 4,000x 2 40,000 15x 1 + 30x 2 200 x 1 , x 2 0 and integer 6 Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution . A feasible solution is ensured by rounding down non- integer solution values but may result in a less than optimal (sub-optimal) solution . Integer Programming Models Solution 7 Maximize Z = \$100x 1 + \$150x 2 subject to: 8,000x 1 + 4,000x 2 \$40,000 15x 1 + 30x 2 200 ft 2 x 1 , x 2 0 and integer Optimal Solution: Z = \$1,055.56 x 1 = 2.22 presses x 2 = 5.55 lathes Figure 5.1 Feasible Solution Space with Integer Solution Points Integer Programming Models Graphical Solution of Total Integer Model Example 1 8 Å Z-value without Integer constraints Optimal solution with Integer constraints “GIN” statement is used to specify general integer variables. A general integer variable can take on any non-negative integer value (0,1,2,…). The ”GIN” statements should appear after the “END” statement in the model. Integer Programming Models Computer Solution of Total Integer Model Example 1
9 Total Integer Model Example 2 Problem Statement (1 of 2) Textbook publishing company is developing two new regions. Planning to transfer some of its 10 salespeople into new regions.

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3_Integer_Programming_(Student) - Integer Programming(IP...

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