Operations Control

1. Operations Control Mathematical Optimization Models 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, …. )

2. Outline • Basics • Example • Mathematical optimization

3. 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.

4. 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.

5. 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?

6. Example • Can they make 9 of each? • Oven used time = 20 x 9 + 32 x 9 = 468 • Prep time = 30 x 9 + 20 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.

7. Mathematical optimization Mathematical Programming • 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.

8. Mathematical optimization Mathematical Programming • 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.

9. Mathematical optimization Mathematical Programming • 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.

10. 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

11. Ex. 25 20 15 c = cakes 10 5 5 10 15 20 25 p= pastries

12. Ex. 25 20 15 c = cakes 10 5 5 10 15 20 25 p= pastries

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

14. 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.

15. 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

16. 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.

17. 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).

18. 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.

19. Types of MO problems • Production • Marketing • Blending • Financial • Capital Budgeting/project selection • Assignment • Trans-shipment • Location Web resource http://www.swlearning.com/economics/mcguigan/mcguigan9e/web_chapter_b.pdf