Operations Control Key Sources: Data Analysis and Decision Making (Albrigth, Winston and Zappe) An Introduction to Management Science: Quantitative Approaches to Decision Making (Anderson, Sweeny, Williams, and Martin), Essentials of MIS (Laudon and Laudon), Slides from N. Yildrim at ITU, Slides from Jean Lacoste, Virginia Tech, …. ) Mathematical Optimization Models 1

Outline Basics Example Mathematical optimization 2

Basics Goal is to maximize (or minimize) a real function by systematically choosing input values from within an allowed set and computing the value of the function. We will focus on mathematical programming (which is not related at all with computer programming). – Linear and integer functions. 3

Example Just started a home business baking/ “decorating” all natural/low calorie cakes. – Each cake takes 30 minutes to prepare/setup/finish and 20 minutes in the oven. One item in the oven at a time. While baking work on the prep/…. – A cake generates a profit contribution of $14 ( post materials and other production costs ). – Available work time is 8 hours per day (480 mins). – What is the profit per week? Based on the prep/setup/finish time limit, it can output 16 cakes per day. Week profits = 16c/d x 5d x $14/c = $1,120. 4

Example Considering switching into all natural/low calorie pastries. – Profit per pastry = $15 – Each pastry takes 20 minutes to prepare/…/ and 32 in the oven. – Should they? Now the oven is the constraint. A maximum of 15 pastries per day. Week profits = 15p/d x 5d x $15/p = $1,125. So, not much of an improvement. Is there a better option? a combination? 5

Example Can they make 9 of each? – Oven used time = 20 x x 9 = 468 – Prep time = 30 x x 9 = 450 – Yes. Can they make 10 of each? – Not without breaking the oven limit. – There is a mathematical method to find the optimal solution. 6

Mathematical optimization All MP problems have constraints that limit the degree to which the objective can be pursued. – Budgets, inventories, materials. – Resources (people, equipment, knowledge). – Customers and demand. – Time. A feasible solution satisfies all the problem's constraints. – A problem could have many feasible solutions. – Some feasible solutions could be very poor. 7 Mathematical Programming

Mathematical optimization An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing). – Typically only one, but could be a few. – However, as we will discuss later, there are multiple criteria in most business problems. – No optimal decisions but tradeoffs. 8 Mathematical Programming

Mathematical optimization A problem can have no feasible solutions. – Constraints are too many / too tight. – One or more constraints must be relaxed/changed. We want to invite 100 people to the wedding. Each seat costs $100. The budget is $8,0000. A problem could be unbounded. Typically there is something wrong in the definition of the problem. – Always a limit of space, money, time. 9 Mathematical Programming

Example Decision variables – c = number of cakes – p = number of pastries Objective function (to be maximized) – Profits = 14c + 15p Constraints – Oven time : 20c + 32p ≤ 480 – Prep time: 30c + 20p ≤ 480 – c ≥ 0 and p ≥ 0 = we cannot make negative amounts 10

Ex. p= pastries c = cakes

Ex. p= pastries c = cakes

Example Optimal solutions in the vertices. Here, given an integer number of cakes and pastries, “close” to them. 13 cakespastriesweekly profit , , , , , ,295

Mathematical optimization Linear Programming Both the objective function and the constraints are linear functions. – Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0). – Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant. Integer Programming One or more variables can only take integer values. 14

Model formulation The process of transforming a business problem (its description) into a mathematical model. Key steps – Identify what can be controlled: the decision variables (DV). – Define the objective function. Maximize or minimize? How it connects to the DV ? Write in terms of the DV. – Define the constraints. What is the bound? How each C connects to the DV ? Write in terms of the DV 15

Slack /Surplus variables Helps understand the level of “unused” resources or the level generated above a minimum. – Slack : the amount of an available resource that is not used, for example budget not used. – Surplus : the amount of “something” above a minimum requirement, for example units made of type above the demand. – For the first example, slacks are? Binding constraints = those with no slack/surplus. 16

Solving with Excel’s Solver We will use Excel to setup and solve demo problems. Add-in called Solver. – A low level solution engine. – Optimality is not guaranteed. – Small problems (few variables and constraints). 17

Sensitivity analysis In MO optimization problems we perform what if analysis to determine effect on the values of the decision variables. – The effect of the RHS constraints. – The effect of the objective function coefficients. – The effect of constraint coefficients. 18

Types of MO problems Production Marketing Blending Financial Capital Budgeting/project selection Assignment Trans-shipment Location 19 Web resource