ENGM 631 Optimization Ch. 3: Introduction to Linear Programming

Sec. 3.1Alt. Prototype K-Corp manufactures two primary products: a small tractor suitable for an orchard and a tree shaker. K-Corp can make $3,000 per tractor and $5,000 per shaker. Currently, the demand for tractors is 8,000 per year and the demand for shakers is 6,000 per year. Currently, K-Corp is unable to meet total demand for both due to limitations in the assembly area. The assembly area has a capacity of 36,000 hours available. Each tractor takes 3 hours to assemble. Each shaker takes 4 hours.

Sec. 3.1Alt. Prototype Problem Determine the appropriate number of tractors and shakers to make so as to maximize total profit. Constraints Do not exceed demand for either product Do not exceed capacity of assembly area

Sec. 3.1Alt. Model Variables X 1 = number of tractors to manufacture X 2 = number of shakers to manufacture Profit Z = 3X 1 + 5X 2 (in $1,000’s)

Sec. 3.1Alt. Model Constraints Demand for tractors X 1 < 8,000 Demand for shakers X 2 < 6,000 Capacity of Assembly 3X 1 +4X 2 < 36,000 Non-negativity X 1 > 0, X 2 > 0

Sec. 3.1Alt. Model Max Z = 3X 1 + 5X 2 s.t. X 1 < 8,000 X 2 < 6,000 3X 1 + 4X 2 < 36,000 X 1 > 0 X 2 > 0

Sec. 3.1Alt. Rescale Let’s let X 1 = number of tractors in 1,000’s X 2 = number of shakers in 1,000’s

Sec. 3.1Alt. Model Max Z = 3X 1 + 5X 2 s.t. X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 X 1 > 0 X 2 > 0 where Z = profit (in $1,000,000’s)

Sec. 3.1Alt. Graphical Technique Do not exceed demand for tractors X 1 < 8

Sec. 3.1Alt. Graphical Technique Do not exceed demand for tractors, shakers X 2 < 6 X 1 < 8

Sec. 3.1Alt. Graphical Technique Do not exceed demand for tractors, shakers, assembly capacity 3X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8

Sec. 3.1Alt. Graphical Technique Do not exceed demand for tractors, shakers, assembly capacity, can’t make less than zero 3X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8 X 1, X 2 > 0

Sec. 3.1Alt. Feasible Region X 1 + 4X 2 < 36 X 2 < 6 X 1 < 8 X 1, X 2 > 0 Feasible Region

Sec. 3.1Alt. Problem Out of the large (infinite) number of solutions available, which maximizes profit?

Sec. 3.1Alt. Iso-Value Lines Z = 3X 1 + 5X 2 Z = 15

Sec. 3.1Alt. Optimal Solution Z = 3X 1 + 5X 2 = 3(4) + 5(6) = 42 (4,6)

Sec. 3.2 Some Terminology Feasible Solutions

Sec. 3.2 Some Terminology Feasible Region

Sec. 3.2 Some Terminology Boundary

Some Terminology Infeasible Solutions

Sec. 3.2 Aside X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 5X 1 + 3X 2 > 60 No Feasible Solutions

Sec. 3.2 Some Terminology Corner Point Feasible Sol. (CPF) (0,6) (4,6) (8,3) (8,0) (0,0)

Sec. 3.2 Relation: CPF & Optimal (0,6) (4,6) (8,3) (8,0) (0,0) The best CPF solution must be an optimal solution.

Sec. 3.2 Relation: CPF & Optimal (0,6) (4,6) (8,3) (8,0) (0,0) Z = 30Z = 42 Z = 39 Z = 24Z = 0 The best CPF solution must be an optimal solution. If a problem has multiple optimal solutions, at least two must be CPF solutions.

Sec. 3.2 Relation: CPF & Optimal (0,6) (4,6) (8,3) (8,0) (0,0) If a problem has multiple optimal solutions, at least two must be CPF solutions.

Sec. 3.2 Optimal Solution Recall, Z = 3X 1 + 5X 2 (4,6) 3X 1 + 4X 2 < 36

Sec. 3.2 Optimal Solution Suppose, Z = 3X X 2 (4,6) 3X 1 + 4X 2 < 36

Sec. 3.2 Optimal Solution Suppose, Z = 3X X 2 (4,6) 3X 1 + 4X 2 < 36

Sec. 3.2 Optimal Solution Suppose, Z = 3X X 2 (4,6) 3X 1 + 4X 2 < 36 Multiple Optima

Standard Form for LP

Sec. 3.2 Standard Form for LP Decision Variables

Sec. 3.2 Standard Form for LP Objective

Sec. 3.2 Standard Form for LP Functional Constraints

Sec. 3.2 Standard Form for LP Nonnegativity Constraints

Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X Profit = R - C = 8X 1 - 5X 1 = 3X 1

Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X Profit = R - C = 8X 1 - 5X 1 = 3X 1 Valid if costs are variable

Sec. 3.3 Assumptions Proportionality The contribution of each activity to the value of the objective function is proportional to the level of activity. Consider Z = 3X

Sec. 3.3 Assumptions Proportionality Now Suppose fixed costs = 3. Then Profit = R - TC = 8X 1 - (3 + 5X 1 ) = 3X

Sec. 3.3 Assumptions Proportionality Now Suppose fixed costs = 3. Then Profit = R - TC = 8X 1 - (3 + 5X 1 ) = 3X

Sec. 3.3 Assumptions Proportionality Now Suppose we have a decreasing marginal rate of return Profit = R - TC = (8-.025X 1 )X 1 - 5X 1.

Sec. 3.3 Assumptions Proportionality Now Suppose we have a decreasing marginal rate of return Profit = R - TC = (8-.025X 1 )X 1 - 5X 1.

Sec. 3.3 Assumptions Additivity Every function in the LP model is the sum of the individual contributions of the respective activities Z = 3X 1 + 5X 2 + 2X 1 X 2 Interaction terms not allowed

Sec. 3.3 Assumptions Divisibility Decision variables, X i i=1,2,... n, are allowed to have any values, including non- integer values, that satisfy the functional and nonnegativity constraints.

Sec. 3.3 Divisibility For K-Corp, is it This or this ?

Sec. 3.3 Assumptions Certainty The value assigned to each parameter of a linear programming model is assumed to be a known constant. Max Z = 3X 1 + 5X 2 s.t. X 1 < 8 X 2 < 6 3X 1 + 4X 2 < 36 X 1 > 0, X 2 > 0 How certain are we of this demand?

Sec. 3.4Alt Power Generation A regional power system has three generating stations: A, B, and C. Each station serves a local area. Three outlying areas are also served by the system: X, Y, and X. The power demand at areas X, Y, and Z is 25, 50, and 30 MW respectively. The maximum generating capacity beyond local requirements and the cost of generation at the three stations is shown below.

Sec. 3.4Alt Power Generation StationExcess Cap.Cost of Gen. A B C Power can be transmitted between any pair of generating stations, but 5% of the amount shipped will be lost. Power can be transmitted from some of the generating stations to outlying areas, but 10% of the amount shipped is lost.

Sec. 3.4Alt Power Generation Model Lines exist from stations A & C to X, from B & C to Y, and from A & B to Z. A C B X Z Y Formulate as a linear program!

Sec. 3.4Alt Power Generation Model Problem: Min Cost of satisfying outlying areas s.t. Do not exceed capacity at any plant Exactly serve each outlying area

Sec. 3.4Alt Power Generation Model Decision Variables X ij = power shipped from plant i to outlying area j Objective Function Min Cost Min Z = 500X AX + 500X AZ + 700X BY + 700X BZ + 400X CX + 400X CY

Sec. 3.4Alt Power Generation Model Do not Exceed Capacity X AX + X AZ < 100 X BY + X BZ < 75 X CX + X CY < 200

Sec. 3.4Alt Power Generation Model Shipped - Losses satisfy outlying areas X AX -.1X AX + X CX -.1X CX = 25 X BY -.1X BY + X CY -.1X CY = 50 X AZ -.1X AZ + X BZ -.1X BZ = 30 or.9X AX +.9X CX = 25.9X BY +.9X CY = 50.9X AZ +.9X BZ = 30

Sec. 3.4Alt Power Generation Model Min Z = 500X AX + 500X AZ + 700X BY + 700X BZ + 400X CX + 400X CY s.t. X AX + X AZ < 100 X BY + X BZ < 75 X CX + X CY < 200.9X AX +.9X CX = 25.9X BY +.9X CY = 50.9X AZ +.9X BZ = 30 X i,j > 0, i = A, B, C j = X, Y, Z

3.4 Additional Examples Design of Radiation therapy Regional Planning of Kibbutzim Controlling Air Pollution Reclaiming Solid waste Personnel Scheduling at Union Airways (United Airlines 24/7 40? $6m/yr saved) Distributing Goods in a Network (Like our Power Example)

3.5 Spreadsheet Solution Range name Data cells Target cell Output cells Changing cells Solver

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig. 3.15

3.5 Spreadsheet Fig. 3.16

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig Old

3.5 Spreadsheet Fig Old

3.6 Large Models Modeling Languages Older edition Citgo (petroleum products distributor) had 3000 functional constraints and decision variables Need for a modeling language You can learn MPL and Lindo with links from the book’s website (not part of our course this year)

3.7 Conclusions LP is powerful LP is “standard” For allocating resources in any social organization Sometimes LP is not applicable; instead do other things

Cases 3.1 Auto assembly (Crusier &Thrillseeker) 3.2 (Online) Cutting Cafeteria costs 3.3 (Online) Staffing a Call Center at California Children’s Hospital 3.4 (Online) Promoting a Breakfast Cereal