Previously in IEMS 310 Notation of optimization problems

Previously in IEMS 310… • Notation of optimization problems • Linear Programs – Sensitivity Analysis / Duality – Assignment and Network Flow Problems • Tricks: – Piecewise linear functions

Agenda • Another trick: absolute value • Another LP… – Sequential Decision Processes – … shortest path (Ch 5) and other dynamic programs

Logistics • TA OH now Thurs 1: 30 -3: 30 C 236 • Hw deadline remains Fri 5 pm – mailbox or in person in C 236 – use Blackboard drop-box for excel files • Suggest start reading Ch 5. 1 -5. 5 • Will discuss projects on Monday

Absolute Value Trick • from Hw 2 • errori = predicted by regression - yi |errori| • penalize overestimates errori penaltyi errori

Sequential Decision Process • Discretize Time period k=1 2 3 4 5 … • Variables for each period – for example: #workers Wk, inventory level Ik

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Problem Summary • • Producing snow tires Monthly demand: Oct-March Goal: cheaply meet demand Decisions: – hire or fire, overtime, production quantity • Inventory cost, trainees are less productive

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Variables For each period • # hired Hk, #fired Fk • #trained and trainee workers – total #workers Wk, #trained workers Tk • units produced • overtime used – Rk units produced with regular hours, – Ok units produced with overtime • inventory Ik

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Timeline Period k Ik #units inventory Production Decision Rk #units with regular time Ok #units with overtime prev. period next period Dk #units shipped Wk #workers Hk #hired Fk #fired Tk #trained workers

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Constraints • Inventory: I 1=0, Ik+1=Ik+Rk+Ok-Dk • Meeting Demand: Ik+1 ≥ 0 • Workforce W 1=90, Wk+1=Wk+Hk-Fk Tk=Wk-Fk, T 7=100 • Capacity Rk≤ 18 Tk+8 Hk Ok ≤(18/4)Tk • Nonnegativity

Production Planning (4. 12) 1. List time periods 1. maybe add an extra at beginning and end 2. List variables (things to keep track of) 1. states and actions 3. Make timeline for a single period 4. Add constraints 1. “laws of motion”: constraints connecting a period to the next 5. Add objective 6. Solve

Objective • Hiring / Firing costs $3000*(H 1+…+H 7) $7000*(F 1+…+F 7) • Compensation $2600*(W 2+…+W 7) $2600*1. 5*(O 1+…+O 7)/18 • Inventory $40*(I 1+…+I 7)