QM B Linear Programming

Overview What is linear programming (LP)? Formulating LPs The Stratton company Graphical insight Using Excel Solver to solve LPs Mile-High Microbrewery

What is linear programming? It is NOT computer programming. Programming here means planning. Mathematical technique Optimization technique A decision needs to made Our goal is to determine the ‘best’ or ‘optimum’ decision There are scarce resources available and/or specified requirements for achieving our goal.

Proctor and Gamble North American Product Sourcing 60 plants 10 distribution centers 1000 customer zones Save $200 million dollars annually Which products should be produced in which plants? Which plants should supply which distribution centers? Which distribution centers should supply which customer zones.

What does an LP look like? There is a goal: objective function Maximized or minimized Written as a linear equation There are scarce resources, restrictions and/or requirements: constraints Limits your ability to achieve the goal

Formulating an LP: Stratton Co. Produces two basic types of plastic pipes Three resources have been identified as critical to pipe output Pipe extrusion hours Packaging hours Special additive mix

Stratton Company Data Product Resource Availability Resource Type 1 Extrusion 4 hrs. 6 hrs. 48 hrs. Packaging 2 hrs. 18 hrs. Additive Mix 2 lbs. 1 lbs. 16 lbs. Profit $34 $40 All data given is for a package of pipe – 100 feet

Stratton Company (cont) Formulate an LP model to determine how much of each type of pipe should be produced to maximize profit.

Three questions to formulate an LP: What is the decision to be made? Stratton Company How much of pipe 1 to produce How much of pipe 2 to produce Defines the variables (if you are specific enough). P1 – number of packages of Pipe 1 to produce P2 – number of packages of Pipe 2 to produce

Question 2 for Formulating an LP: What is the goal? Stratton Company Maximize profit Defines the objective function MAX 34 P1 + 40 P2

Question 3 for Formulating an LP: What are the limited resources or requirements? Extrusion hours Packaging hours Additive mix 4P1 + 6P2  48 2P1 + 2P2  18 2P1 + 1P2  16

LP for Stratton Company Objective Function LP for Stratton Company MAX 34 P1 + 40 P2 Subject to: 4 P1 + 6 P2  48 Extrusion hours 2 P1 + 2 P2  18 Packaging hours 2 P1 + 1 P2  16 Additive supply P1  0 and P2  0 Non-negativity Constraints

Solving LPs ‘What if’ analysis (go to Excel) Graphical analysis For insight Simplex method Solver – an Excel add-in Computer packages designed for linear optimization

Graphical analysis – non-negativity

Graphical analysis – Extrusion constraint 4 P1 + 6 P2  48

Graphical analysis – Packaging Constraint 2 P1 + 2 P2  18

Graphical analysis – Additive supply constraint Feasible Region 2 P1 + 1 P2  16

Which is the optimal solution? Feasible Region Set of all solutions that satisfy all of the constraints Infinite number of solutions Which is the optimal solution? One of the solutions at the corner points

Corner point solutions Feasible Region MAX 34 P1 + 40 P2 (go to Excel)

Stratton Company – Summary Optimal solution P1 = 3 P2 = 6 Max = $342 The optimal product mix is 3 packages of Pipe 1 and 6 packages of Pipe 2. This provides a maximum profit of $342.

Setting up Excel Solver to solve LPs Solver is an add-in to Excel Not automatically ready To get solver ready: In Excel Tools -> Add ins Scroll down to Solver Add in Check the box Click on OK Only need to do this one time

Mile-High Microbrewery Mile-High Microbrewery makes a light beer and a dark beer. Mile-High has a limited supply of barley, limited bottling capacity, and a limited market for light beer. Profits are $0.20 per bottle of light beer and $0.50 per bottle of dark beer. Formulate an LP to maximize profits and determine how many bottles of each product should be produced per month.

Mile-High Microbrewery Data

Think-pair-share: Three questions: What are the decisions to be made? What is the goal? What are the limited resources or requirements?

What are the decisions to be made? L – Number of bottles of light beer to produce D – Number of bottles of dark beer to produce

What is the goal? Maximize profit MAX 0.20 L + 0.50 D

What are the limited resources or requirements? Barley supply 0.10 L + 0.60 D  2000 Bottling capacity 1 L + 1 D  6000 Market capacity 1 L  4000

An aside for SUMPRODUCT function 2 groups of cells Both in a row or both in a columns Wish to multiply the corresponding entries then sum the products =sumproduct(a2:c2, a3:c3) = 2*5 + 3*6 + 4*7

Think-pair-share: SUMPRODUCT function =SUMPRODUCT(B6:C6,B10:C10) = 2*2 + 1*4 = 8

To solve an LP using Excel Solver Setup the spreadsheet TYPE data in one place (go to Excel) CREATE Cells for decisions variables ENTER formulas to calculate LHS of constraints ENTER formulas to calculate Objective Function Open solver box Tools -> Solver

Excel Solver Dialog Box Click on cell that calculates objective function Select Max or Min Click & drag to select decision variables Click add to add the constraints

Excel solver – constraints dialog box Select cell(s) with LHS Select cell(s) with RHS Select symbol (, , =) Remember – Non-negativity constraints

Go to the Options Dialog box Click options to assume linear model

Last dialog box - options Click OK Check the Assume linear models box Check the Assume Non-negative box

Now SOLVE Click solve to find optimal solution

Solver found a solution Click on Answer and Sensitivity Click OK

Answer and sensitivity reports