MGTSC 352 Lecture 15: Aggregate Planning Altametal Case Summary of Optimization Modeling

AltaMetal Ltd. (Case 8, pg. 111, and pgs. 87 – 92) Another aggregate planning problem 1,000 products aggregated to 9 groups

AltaMetal Ltd. (Case 8, pg. 111, and pgs. 87 – 92) Is it possible to satisfy demand? If so, how? (production plan by product group) Excel …

Active Learning Pairs, 1 min. Formulate AltaMetal’s problem in English What to optimize, by changing what, subject to what constraints …

To many change-overs … The JIT (“just-in-time”) plan we found may require too many changeovers What if we require a minimum lot size of 30 tons? Daily capacity = 90 tons  At most 3 lots per day Changing cells: Old: # of tons of product X to produce in month Y New: # of __ of product X to produce in month Y Excel …

Tired of Waiting for Solver? Hit Escape key

LINEAR INTEGER NONLINEAR “Programming” MODELS A Summary

CLASSIFICATION Decision Variables Functions Fractional Integer Linear LP ILP Nonlinear NLP INLP

LP SIMPLEX method (linear algebra) Corner point optimality Move from corner-to-corner, improve obj. Very efficient Can solve problems with thousands of variables and constraints

ILP Branch & Bound (divide-and-conquer) Solve the LP, ignoring integer constraints Select a fractional variable, x6 = 15.7 Create two new problems: x6 ≤ 15, x6  16 Solve the new problems Continue until all branches exhausted # of branches is exponential in # of var.

NLP Gradient method (uses derivatives) Repeat until convergence Find an improving direction Move in the improving direction Converges to local optimum Multiple starts recommended

INLP Ignore integer constraints, solve the NLP Use Branch & Bound Solve a series of NLPs Computationally demanding No guarantee of optimality YUCK!

Formulating Optimization Models (pg. 93) Formulate the problem in English Or French, or Chinese, or Icelandic, …  Start with data in spreadsheet Define decision variables – turquoise cells Express performance measure (profit, or cost, or something else) as function of the decision variables Express constraints on decision variables Scarce resources Physical balances Policy constraints

Solving Optimization Problems Try simple values of the decision variables to check for obvious errors Guess at a reasonable solution and see if model is ‘credible’ (sniff test) Look for missing or violated constraints Is profit (cost) in ballpark?

Optimizing with Solver Use Simplex LP method (‘assume linear model’) whenever possible Set Options properly automatic scaling, assume non-negative Watch for diagnostic messages – do not ignore! (infeasible, unbounded) Interpret solution in real-world terms and again check for credibility

Things to Remember The Simplex LP method always correctly solves linear programs Solver is a slightly imperfect implementation of the Simplex method (but you should generally assume that it is correct) The biggest source of errors is in the model building process (i.e., the human)