Managerial decision modeling with spreadsheets by stair render chapter 06

Chapter Integer, Goal, and Nonlinear Programming Models © 2007 Pearson Education Variations of Basic Linear Programming • Integer Programming • Goal Programming • Nonlinear Programming Integer Programming (IP) Where some or all decision variables are required to be whole numbers • General Integer Variables (0,1,2,3,etc.) Values that count how many • Binary Integer Variables (0 or 1) Usually represent a Yes/No decision General Integer Example: Harrison Electric Co Produce products (lamps and ceiling fans) using limited resources Decision: How many of each product to make? (must be integers) Objective: Maximize profit Decision Variables L = number of lamps to make F = number of ceiling fans to make Lamps Fans (per lamp) (per fan) Profit Contribution $600 $700 Hours Available Wiring Hours hrs hrs 12 Assembly Hours hrs hr 30 LP Model Summary Max 600 L + 700 F ($ of profit) Subject to the constraints: 2L + 3F < 12 (wiring hours) 6L + 5F < 30 (assembly hours) L, F > Graphical Solution Properties of Integer Solutions • 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 Using Solver for IP • IP models are formulated in Excel in the same way as LP models • The additional integer restriction is entered like an additional constraint int - Means general integer variables bin - Means binary variables Go to file 6-1.xls Binary Integer Example: Portfolio Selection Choosing stocks to include in portfolio Decision: Which of stocks to include? Objective: Maximize expected annual return (in $1000’s) Goal Constraints Goal 1: Total sales at least $180,000 70E + 110I + 110C + dT- - dT+ = 180,000 Goal 2: Exterior door sales at least $70,000 70E + dE- - dE+ = 70,000 Note: Each highlighted deviation variable measures goal underachievement Goal 3: Interior door sales at least $60,000 110 I + dI- - dI+ = 60,000 Goal 4: Commercial door sales at least $35,000 110C + dC- - dC+ = 35,000 Objective Function Minimize total goal underachievement Min dT- + dE- + dI- + dCSubject to the constraints: • The goal constraints • The “regular” constraints (3 limited resources) • nonnegativity Weighted Goals • When goals have different priorities, weights can be used • Suppose that Goal is times more important than each of the others Objective Function Min 5dT- + dE- + dI- + dC- Properties of Weighted Goals • Solution may differ depending on the weights used • Appropriate only if goals are measured in the same units Ranked Goals • Lower ranked goals are considered only if all higher ranked goals are achieved • Suppose they added a 5th goal Goal 5: Steel usage as close to 9000 lb as possible 4E + 3I + 7C + dS= 9000 (lbs steel) (no dS+ is needed because we cannot exceed 9000 pounds) • • • Rank R1: Goal Rank R2: Goal Rank R3: Goals 2, 3, and A series of LP models must be solved 1) Solve for the R1 goal while ignoring the other goals Objective Function: Min dTGo to file 6-7.xls 2) If the R1 goal can be achieved (dT- = 0), then this is added as a constraint and we attempt to satisfy the R2 goal (Goal 5) Objective Function: Min dS3) If the R2 goal can be achieved (dS- = 0), then this is added as a constraint and we solve for the R3 goals (Goals 2, 3, and 4) Objective Function: Min dE- + dI- + dC- Nonlinear Programming Models • Linear models (LP, IP, and GP) have linear objective function and constraints • If a model has one or more nonlinear equations (objective or constraint) then the model is nonlinear • Example nonlinear terms: X2, 1/X, XY Characteristics of Nonlinear Programming (NLP) Models • Difficult to solve • Optimal solutions are not necessarily at corner points • There are both local and global optimal solutions • Solution may depend on starting point • Starting point is usually arbitrary Nonlinear Programming Example: Pickens Memorial Hospital Patient demand exceeds hospital’s capacity Decision: How many of each of types of patients to admit per week? Objective: Maximize profit Decision Variables M = number of Medical patients to admit S = number of Surgical patients to admit P = number of Pediatric patients to admit Profit Function Profit per patient increases as the number of patients increases (i.e nonlinear profit function) Constraints • Hospital capacity: 200 total patients • X-ray capacity: 560 x-rays per week • Marketing budget: $1000 per week • Lab capacity: 140 hours per week Objective Function (in $ of profit) Max 45M + 2M2 + 70S + 3S2 + 2MS + 60P + 3P2 Subject to the constraints: M+S+P < 200 (patient cap.) M + 3S + P < 560 (x-ray cap.) 3M + 5S + 3.5P < 1000 (marketing $) (0.2+0.001M)x(3M+3S+3P) < 140 (lab hrs) M, S, P > Using Solver for NLP Models • Solver uses the Generalized Reduced Gradient (GRG) method • GRG uses the path of steepest ascent (or descent) • Moves from one feasible solution to another until the objective function value stops improving (converges) Go to file 6-8.xls ... all decision variables are required to be whole numbers • General Integer Variables (0,1,2,3,etc.) Values that count how many • Binary Integer Variables (0 or 1) Usually represent a Yes/No decision. .. products (lamps and ceiling fans) using limited resources Decision: How many of each product to make? (must be integers) Objective: Maximize profit Decision Variables L = number of lamps to make F =... Selection Choosing stocks to include in portfolio Decision: Which of stocks to include? Objective: Maximize expected annual return (in $1000’s) Stock Data Decision Variables Use the first letter of