Operations Research

… and Mathematical ModelsOriginating during World War II, Operations Research (OR) is the discipline that deals with application of analytical methods to improve decision-making. Although its usage is broad-based, specific uses of this science is in the areas of Supply Chain and Manufacturing planning, Transportation and Logistics, Floor and Network Planning, Allocation, Scheduling and Strategic Planning. Operations Research involves representation of real-world business problems as mathematical formulations that could be solved heuristically or optimally using variety of tools and techniques. This session will touch on the methods and models involved within the science of OR.

https://en.wikipedia.org/wiki/Operations_research

In the World War II era, operational research was defined as "a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control."[7]

[7] "Operational Research in the British Army 1939–1945, October 1947, Report C67/3/4/48, UK National Archives file WO291/1301

Some applications

• Airline, Gate Scheduling• Telecommunications/Road/Rail Network Design• Organization Supply Chain Strategy (DC, Plant,..)• Just-in-Time Manufacturing Planning• Retail Shop floor Layout• Revenue, Pricing and Promotions• Demand Forecasting• Project Planning• Economics – Micro/Macro

COMPLEX WORLD OF SUPPLY CHAINS

Traditional and Modern Supply Chains

Manufacturing Supply Chain

CUSTOMERS

Retail Supply Chain

FAMOUS SUPPLY CHAINS

BOEING

Moving Assembly Line for 7771. http://www.boeing.com/news/releases/2006/q4/061107b_nr.html, 2. http://www.youtube.com/watch?v=AiKIC8ztqhY

http://download.intel.com/newsroom/kits/22nm/pdfs/Global-Intel-Manufacturing_FactSheet.pdf

MATH IN THE MADNESS

Optimization Terminology• Optimal

– finding "best available" values of some objective function given a defined domain

• Heuristics – experience-based techniques for problem solving, learning, and discovery that gives a

solution which is not guaranteed to be optimal. (Ex: Search)

• Decomposition– complex problem or system is broken down into parts that are easier to solve

optimally or otherwise

• Relaxation– approximation of a difficult problem by a nearby problem that is easier to solve.

A solution of the relaxed problem provides information about the original problem.

• Combinatorial– the set of feasible solutions is discrete or can be reduced to discrete, and in

which the goal is to find the best solution

Linear Programming (LP)

• Optimize (Minimize of Maximize) a Linear Objective function (Red Line)

• Subject to Linear equality or inequality constraints (Pink area)

• Optimal solution lies at one of the corners (graphically)

• Simplex method and duality

LP Example: Buying CabinetsYou need to buy some filing cabinets. You know that • Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. • Cabinet Y costs $20 per unit, requires eight square feet of floor space,

and holds twelve cubic feet of files. • You have been given $140 for this purchase, though you don't have to

spend that much. • The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?The question asks for the number of cabinets to buy, so the variables are:x: # of model X cabinets purchased; y: # of model Y cabinets purchased; x > 0 and y > 0. consider costs and floor space (the "footprint" of each unit), while maximizing the storage volume, so costs and floor space will be the constraints, while volume will be the optimization equation.

Buying Cabinets: Solution

MAXIMIZE volume: V = 8x + 12y, Subject to:cost: 10x + 20y < 140, or y < –( 1/2 )x + 7 space: 6x + 8y < 72, or y < –( 3/4 )x + 9

When you test the corner points at (8, 3), (0, 7), and (12, 0), you should obtain a maximal volume of100 cubic feet by buying eight of model X and three of model Y.

LP: Primal and Dual

• Mirror images– Objective Function RHS– ‘<=‘ ‘>’ etc– Every feasible primal cornerpoint/constraint is dual infeasible and vice versa– Optimal is the point where Primal and Dual are feasible– Solving with fewer constraints will be faster. [Large scale problems]

Non-Linear Programming (NLP)• Objective and Constraint

functions are non-linear functions

• Local Maxima and Minima• Branch and Bound Technique

with heuristics• Iterative techniques

Maximize f (x1, x2, . . . , xn),subject to:g1(x1, x2, . . . , xn) b1,......gm(x1, x2, . . . , xn) bm,

http://www.sce.carleton.ca/faculty/chinneck/po/Chapter%2016.pdf

NLP Example: Transportation

• Order – Item(s) that needs to be transported from Origin to Destination, either directly or through other via-points using one or more transportation modes

• Direction - Inbound, Outbound, Return/Reverse• Mode – Truckload, Less-than-Truckload, Express (Air),

Parcel/Package, Shipping/Marine, Rail, Multi-Modal• Points - Origin, Destination, Crossdocks, Distributors (DC),

Zone Skip, Transshipment, Warehouse, Truck stop• Routing – Lanes, Shipping Schedules, Intermodal, In-Transit

Planning, Milk runs, Grocery Store Model, Travelling Salesperson, Minimal Spanning Tree, Shortest Path

Inbound, Outbound, Backhaul• Inbound logistics concentrates on purchasing

and arranging inbound movement of materials, parts and/or finished inventory from suppliers to manufacturing or assembly plants, warehouses or retail stores.

• Multiple Pickups, single delivery (milk runs)• Ex: Automotive, Chip/IC Manufacturing,

Discrete/Process Manufacturing Industries

• Outbound logistics is related to the storage and movement of the final product and the related information flows from the end of the production line to the end user.

• Single pickup, multiple delivery (grocery store)• Ex: Retail, FMCG sectors

• Backhaul includes hauling some cargo back instead of driving empty

Integer Programming

• Discrete Solution space• NP – Non-Deterministic

Polynomial time• Solution approaches

– Linear Relaxation– Branch and Bound– Heuristics – Tabu search, Simulated

Annealing, Hill Climbing, Ant Colony• Mixed Integer Problem

IP Example: Services Business

You have to decide how many resources to put on projects A and B (ProjA, ProjB):• Revenue per project (RevA, RevB)• Each project requires resources (ResA, ResB)• You have constrained resources (ResTotal)• Cost of each resource costs (CostA, CostB)

ProjA, ProjB are 0,1 Integer VariablesResA, ResB are Integer Variables

IP Formulation

Objective:Maximize Revenue: ProjA*RevA + ProjB*RevB

Constraints: Resources: ProjA*ResA + ProjB*ResB <= ResTotal

Variables that does not affect the above Objective:

Cost: ProjA*CostA + ProjB*CostBSlack Variables: Cost of not using a Resource,

Network Flow• Given a “Source” (A) and a

“Sink” (G), determine the maximum quantity that can flow, given edge capacities

• Maximum telephone calls on a network, maximum vehicles on a road

• Project Planning and Scheduling

• Two-directional flow capacity

Capacity in A-> D direction

Capacity in D->A direction

Practical Application of Maximum Flow• Tyson Foods, IBP Merger in 2001

– Combine Transportation Networks– Optimize Fleet Carriers (Strategic), Residual Carriers (Contract and

Spot Carriers)• Approach

– Mine trips data from previous 3 years– Generate Aggregates (Min, Max, Avg) for each Lane/Start

DOW/End DOW– Run Flow problem iteratively for each start location and start day of

week [Modeled as a single node]• Result (Cost optimization)

– Propose Routing Loops for Fleet Carriers– Propose Residual Flow and suggested rates for Contract

Negotiations

Stochastic Programming

• Uncertain outcomes, Probabilistic models• Time Series – Change in value over time• Two-stage – Optimal Certain stage 1, followed

by recourse for random event• Ex: Stock, Exchange Rate, Heart Rate• Pricing and Promotions

– Target Pricing, Decoy Pricing, Freemium, Psychological Pricing, Pay-as-you-want, value pricing

DEEP DIVE – AN EXAMPLE

Optimize Gates, Trucks/Trips, Machines/Forklift, People/Shifts, minimize cost

Math Modelling – Location Constraints• Problem Statement – A small Warehouse location

has the following constraints– The location is open 10:00 AM - 5:00 PM Mon– It can accommodate only 2 Trucks every hour– It can only serve 2000 KG of material each hour

• Given: – 10 Trips picking up goods from the warehouse, each

with different potential start times and corresponding costs. Each trip carriers 500 KG

• Objective:– Date/Time Schedule the 10 trips with the lowest overall

cost such that all location constraints are honored

Objective Function• Minimize total cost of solution Minimize obj: 1000 x1 + 1234 x2 + 1343 x3 ….+ 2123 x10 + ….[Cost of unique Trip and Start Time combination][All variables are made linear: 0 ≤ xi ≤ 1. Fractional results are converted to its closest

integer{0,1}. This makes the problem easier to solve. This is called Linear Programming (LP) Relaxation.]

• Add Above (50,000 - Soft) and Below (3,000 - Soft) Target Penalty variables

3000 x221 + 50000 x222 + 3000 x223 + 50000 x224 +……* Integer programs are complicated to solve. Linear programs can be solved in polynomial time

Trip Constraints

• A trip can only start at one of its possible start times

Subject ToC1: x1 + x21 + x33 + x112 = 1 [x1, x21, x33 and x112 represent

4 different times trip can pickup at the location]

C2: x2 + x22 + x34 + x 113 = 1……

Capacity Constraints• Location can serve two trips in each hour bucket• Location can serve trips totaling 2000 KG in each hour bucket

C101: x2 + x26 + x32 + x64 + x76 =2[Each constraint corresponds to a specific 1-hour bucket. Each of the variables correspond to a trip at a particular

pickup time that falls in that one hour bucket. If that trip is selected, it contributes a trip count of 1 to the Bucket Capacity of 2 trips]

C102: 500x1 + 500x13 + 500x55 + 500x84 + 500x96 =2000[Each constraint corresponds to a specific 1-hour bucket. Each of the variables correspond to a trip at a particular

pickup time that falls in that one hour bucket. If that trip is selected, it contributes 500 KG to the Bucket Capacity of 2000 KGS]

……

* Concept of Slack variables to accommodate lesser quantity than capacity

Balancing Constraints

• To create a balanced workload and prevent peaks. Helps ramp-up and ramp-down labor resources

C201: x187 – x188 – x189 + x190 >=0[The difference between adjacent time bucket assignments should be kept low].

Creates a pattern such as this ….

9 10 11 12 1 2 3 4 50

0.5

1

1.5

2

2.5

Series1

Some key aspects in use at iLabs

• Linear Regression Models• Naïve Bayesian• Attribute Selection• MetaHeuristics• Resource Scheduling• Project Scheduling with Profit

Optimization – Set Covering Problem• Microsoft Excel Solver Add-in – Can

solve Linear (Simplex), Non-Linear and Evolutionary algorithms (http://www.wikihow.com/Use-Solver-in-Microsoft-Excel)