Chapter 6 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 2 products (lamps and ceiling fans) using 2 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 (per lamp) Fans (per fan) Hours Available Profit Contribution $600$700 Wiring Hours2 hrs3 hrs12 Assembly Hours6 hrs5 hr30

LP Model Summary Max 600 L F ($ of profit) Subject to the constraints: 2L + 3F < 12 (wiring hours) 6L + 5F < 30 (assembly hours) L, F > 0

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 7 stocks to include? Objective: Maximize expected annual return (in $1000’s)

Stock Data

Decision Variables Use the first letter of each stock’s name Example for Trans-Texas Oil: T= 1 if Trans-Texas Oil is included T= 0 if not included

Restrictions Invest up to $3 million Include at least 2 Texas companies Include no more than 1 foreign company Include exactly 1 California company If British Petro is included, then Trans-Texas Oil must also be included

Objective Function(in $1000’s return) Max 50T + 80B + 90D + 120H + 110L + 40S + 75C Subject to the constraints: Invest up to $3 Million 480T + 540B + 680D H + 700L + 510S + 900C < 3000

Include At Least 2 Texas Companies T + H + L > 2 Include No More Than 1 Foreign Company B + D < 1 Include Exactly 1 California Company S + C = 1

If British Petro is included (B=1), then Trans-Texas Oil must also be included (T=1) T=0T=1 B=0ok B=1not okok B < T allows the 3 acceptable combinations and prevents the unacceptable one Go to file 6-3.xls Combinations of B and T

Mixed Integer Models: Fixed Charge Problem Involves both fixed and variable costs Use a binary variable to determine if a fixed cost is incurred or not Either linear or general integer variables deal with variable cost

Fixed Charge Example: Hardgrave Machine Co. Has 3 plants and 4 warehouses and is considering 2 locations for a 4 th plant Decisions: Which location to choose for 4 th plant? How much to ship from each plant to each warehouse? Objective: Minimize total production and shipping cost

Supply and Demand Data Warehouse Monthly DemandPlant Monthly Supply Production Cost (per unit) Detroit10,000Cincinnati15,000$48 Houston12,000 Kansas City 6,000$50 New York15,000Pittsburgh14,000$52 Los Angeles9,000 Total46,00035,000 Note: New plant must supply 11,000 units per month

Production Cost (per unit) Fixed Cost (per month) Seattle$53$400,000 Birmingham$49$325,000 Possible Locations for New Plant

Shipping Cost Data

Decision Variables Binary Variables Y s = 1 if Seattle is chosen = 0 if not Y B = 1 if Birmingham is chosen = 0 if not Regular Variables X ij = number of units shipped from plant i to warehouse j

Objective Function (in $ of cost) Min 73X CD + 103X CH + 88X CN + 108X CL + 85X KD + 80X KH + 100X KN + 90X KL + 88X PD + 97X PH + 78X PN + 118X PL + 113X SD + 91X SH + 118X SN + 80X SL + 84X BD + 79X BH + 90X BN + 99X BL + 400,000Y S + 325,000Y B Subject to the constraints: (see next slide)

Supply Constraints -(X CD + X CH + X CN + X CL ) = -15,000 (Cincinnati) -(X KD + X KH + X KN + X KL ) = - 6,000 (Kansas City) -(X PD + X PH + X PN + X PL ) = -15,000 (Pittsburgh) Possible Locations for New Plant -(X SD + X SH + X SN + X SL ) = -11,000Y S (Seattle) -(X BD + X BH + X BN + X BL ) = -11,000Y B ( B’ham)

Demand Constraints X CD + X KD + X PD +X SD + X BD = 10,000 (Detroit) X CH + X KH + X PH +X SH + X BH = 12,000 (Houston) X CN + X KN + X PN +X SN + X BN = 15,000 (New York) X CL + X KL + X PL +X SL + X BL = 9,000 (L. A.) Choose 1 New Plant Location Y S + Y B =1 Go to File 6-5.xls

Goal Programming Models Permit multiple objectives Try to “satisfy” goals rather than optimize Objective is to minimize underachievement of goals

Goal Programming Example: Wilson Doors Co. Makes 3 types of doors from 3 limited resources Decision: How many of each of 3 types of doors to make? Objective: Minimize total underachievement of goals

Data

Goals 1.Total sales at least $180,000 2.Exterior door sales at least $70,000 3.Interior door sales at lest $60,000 4.Commercial door sales at least $35,000

Regular Decision Variables E = number of exterior doors made I = number of interior doors made C = number of commercial doors made Deviation Variables d i + = amount by which goal i is overachieved d i - = amount by which goal i is underachieved

Goal Constraints Goal 1: Total sales at least $180,000 70E I + 110C + d T - - d T + = 180,000 Goal 2: Exterior door sales at least $70,000 70E + d E - - d E + = 70,000 Note: Each highlighted deviation variable measures goal underachievement

Goal 3: Interior door sales at least $60, I + d I - - d I + = 60,000 Goal 4: Commercial door sales at least $35, C + d C - - d C + = 35,000

Objective Function Minimize total goal underachievement Min d T - + d E - + d I - + d C - Subject to the constraints: The 4 goal constraints The “regular” constraints (3 limited resources) nonnegativity

Weighted Goals When goals have different priorities, weights can be used Suppose that Goal 1 is 5 times more important than each of the others Objective Function Min 5d T - + d E - + d I - + d C -

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 5 th goal Goal 5: Steel usage as close to 9000 lb as possible 4E + 3 I + 7C + d S - = 9000 (lbs steel) (no d S + is needed because we cannot exceed 9000 pounds)

Rank R 1 : Goal 1 Rank R 2 : Goal 5 Rank R 3 : Goals 2, 3, and 4 A series of LP models must be solved 1)Solve for the R 1 goal while ignoring the other goals Objective Function: Min d T - Go to file 6-7.xls

2) If the R 1 goal can be achieved (d T - = 0), then this is added as a constraint and we attempt to satisfy the R 2 goal (Goal 5) Objective Function: Min d S - 3) If the R 2 goal can be achieved (d S - = 0), then this is added as a constraint and we solve for the R 3 goals (Goals 2, 3, and 4) Objective Function: Min d E - + d I - + d C -

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: X 2, 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 3 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 + 2M S + 3S 2 + 2MS + 60P + 3P 2 Subject to the constraints: M + S + P< 200 (patient cap.) M + 3S + P< 560 (x-ray cap.) 3M + 5S + 3.5P< 1000 (marketing $) ( M)x(3M+3S+3P) < 140 (lab hrs) M, S, P > 0

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