Integer, Goal, and Nonlinear Programming Models 6 Integer, Goal, and Nonlinear Programming Models

LEARNING OBJECTIVES Formulate integer programming (IP) models. Set up and solve IP models using Excel’s Solver. Understand the difference between general integer and binary integer variables.

LEARNING OBJECTIVES Understand the use of binary integer variables in formulating problems involving fixed costs. Formulate goal programming (GP) problems and solve them using Excel’s Solver. Formulate nonlinear programming (NLP) problems and solve them using Excel’s Solver.

Introduction Relax the basic assumptions Integer Programming Fractional value One objective Linear equations Integer Programming Goal Programming Nonlinear Programming

Models Integer Models Goal Models General integer variables Binary variables Pure IP problems Mixed IP problems Goal Models More than one objective

Models Nonlinear Models Objective function Constraints Maximize profit = 25X – 0.4X 2 + 30Y – 0.5Y 2 Constraints

Integer Models Rounding off the LP solution might not yield the optimal IP solution The IP objective function value is usually worse than the LP value IP solutions are usually not at corner points

General Integer Variables Harrison Electric Company Ornate lamps Old-fashioned ceiling fans Lamps Fans Hours Wiring 2 3 12 Assembly 6 5 30 Profit $600 $700

Harrison Electric Decision Variables L = number of lamps F = number of ceiling Integer values

Harrison Electric Objective function Maximize profit = $600L + $700F subject to 2L + 3F ≤ 12 (wiring hours) 6L + 5F ≤ 30 (assembly hours) L, F ≥ 0

Graphical Solution + + L F 6 – 5 – 4 – 3 – 2 – 1 – – | | | | | | | | | | | | 0 1 2 3 4 5 6 Figure 6.1 + = Integer Valued Point 6L + 5F ≤ 30 + Rounded-off IP Solution (L = 4, F = 2, Infeasible) + Optimal IP Solution (L = 3.75, F = 1.50, Profit = $3,300) 2L + 3F ≤ 12 Nearest Feasible Rounded-off IP Solution (L = 4, F = 1, Profit = $3,100)

Integer Solutions LAMPS (L) CEILING FANS (F) PROFIT ($600L + $700F) 0 0 $ 0 1 0 $ 600 2 0 $1,200 3 0 $1,800 4 0 $2,400 5 0 $3,000 0 1 $ 700 1 1 $1,300 2 1 $1,900 3 1 $2,500 4 1 $3,100  Nearest feasible rounded-off solution 0 2 $1,400 1 2 $2,000 2 2 $2,600 3 2 $3,200  Optimal IP solution 0 3 $2,100 1 3 $2,700 0 4 $2,800 Table 6.1

Solving the Problem Screenshot 6-1

Solving the Problem Screenshot 6-1

Solver Options Screenshot 6-2A

Solver Options Screenshot 6-2B

Binary Variables Only two possible values (0, 1) Selection problems Set covering problems

Simkin and Steinberg Oil stock portfolios EXPECTED COST FOR COMPANY NAME ANNUAL RETURN BLOCK OF SHARES (LOCATION) (IN THOUSANDS) (IN THOUSANDS) Trans-Texas Oil (Texas) $ 50 $ 480 British Petro (Foreign) $ 80 $ 540 Dutch Shell (Foreign) $ 90 $ 680 Houston Drilling (Texas) $120 $1,000 Lone Star Petro (Texas) $110 $ 700 San Dieago Oil (California) $ 40 $ 510 California Petro (California) $ 75 $ 900 Table 6.2

Simkin and Steinberg Decision Variables T = 1 if Trans-Texas Oil is included in the portfolio = 0 if Trans-Texas Oil is not included in the portfolio Similarly B (British Petro), D (Dutch Shell), H (Houston Oil), L (Lone Star Petro), S (San Diego Oil), and C (California Petro)

Simkin and Steinberg Objective function Maximize ROI = $50T + $80B + $90D + $120H + $110L + $40S + $75C subject to $480T + $540B + $680D + $1,000H +700L + $510S + $900C ≤ $3,000 (investment limit) T + H + L ≥ 2 (Texas co‘s) B + D ≤ 1 (foreign co‘s) S + C = 1 (California co‘s) B ≤ T (Trans-Texas and British Petro) All variables = 0 or 1

Binary Requirements Screenshot 6-3

Binary Requirements Screenshot 6-3

Sussex County Build health care clinics Table 6.3 TO FROM A B C D E F G 15 20 35 45 40 3 50 30 Table 6.3

Sussex County Build health care clinics COMMUNITY COMMUNITIES WITHIN 30 MINUTES A A, B, C B A, B, D C A, C, D, G D B, C, D, F, G E E, F F D, E, F G C, D, G Table 6.4

Sussex County Decision Variables A = 1 if a clinic is located in community A = 0 if a clinic is not located in community A Similarly B (community B), C (community C), D (community D), E (community E), F (community F), and G (community G)

Sussex County Objective function Minimize total number of clinics = A + B + C + D + E + F + G subject to A + B + C ≥ 1 (community A is covered) A + B + D ≥ 1 (community B is covered) A + C + D + G ≥ 1 (community C is covered) B + C + D + F + G ≥ 1 (community D is covered) E + F ≥ 1 (community E is covered) D + E + F ≥ 1 (community F is covered) C + D + G ≥ 1 (community G is covered) All variables = 0 or 1

Solving the Problem Screenshot 6-4