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 C236 Hw deadline remains Fri 5pm –mailbox or in person in C236 –use Blackboard drop-box for excel files Suggest start reading Ch Will discuss projects on Monday

Absolute Value Trick from Hw 2 error i = predicted by regression - y i penalize overestimates error i |error i | error i penalty i

Sequential Decision Process Discretize Time Variables for each period –for example: #workers W k, inventory level I k period k=12345…

Production Planning (4.12) 1.List time periods –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“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 –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Production Planning (4.12) 1.List time periods –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Variables For each period # hired H k, #fired F k #trained and trainee workers –total #workers W k, #trained workers T k units produced overtime used –R k units produced with regular hours, –O k units produced with overtime inventory I k

Production Planning (4.12) 1.List time periods –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Timeline Period k W k #workers H k #hired F k #fired Production Decision R k #units with regular time O k #units with overtime T k #trained workers I k #units inventory D k #units shipped next periodprev. period

Production Planning (4.12) 1.List time periods –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Constraints Inventory: I 1 =0, I k+1 =I k +R k +O k -D k Meeting Demand: I k+1 ≥ 0 Workforce W 1 =90, W k+1 =W k +H k -F k T k =W k -F k, T 7 =100 Capacity R k ≤18T k +8H k O k ≤(18/4)T k Nonnegativity

Production Planning (4.12) 1.List time periods –maybe add an extra at beginning and end 2.List variables (things to keep track of) –states and actions 3.Make timeline for a single period 4.Add constraints –“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 )