Operations Research and Mathematical Modeling
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Operations research (OR) deals with applying analytical methods to improve decision-making and originated during World War II. OR involves representing real-world business problems as mathematical models that can be solved optimally or heuristically using various tools and techniques. Some common applications of OR include supply chain planning, transportation and logistics, manufacturing scheduling, and strategic planning. The goal of this session is to discuss the methods and models used in the science of operations research.
![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](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-3-320.jpg)
![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]](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-18-320.jpg)
![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](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-26-320.jpg)
![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](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-30-320.jpg)
![Trip Constraints
• A trip can only start at one of its possible start
times
Subject To
C1: 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
…
…](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-31-320.jpg)
![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](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-32-320.jpg)
![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 ….
0
0.5
1
1.5
2
2.5
9 10 11 12 1 2 3 4 5
Series1](https://image.slidesharecdn.com/247operationsresearchandmathematicalmodelling-140412113013-phpapp02/85/Operations-Research-and-Mathematical-Modeling-33-320.jpg)
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Operations Research and Mathematical Modeling
- 1. Operations Research … and Mathematical Models Originating 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.
- 3. 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
- 4. 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
- 5. COMPLEX WORLD OF SUPPLY CHAINS
- 6. Traditional and Modern Supply Chains
- 10. BOEING Moving Assembly Line for 777 1. http://www.boeing.com/news/releases/2006/q4/061107b_nr.html, 2. http://www.youtube.com/watch?v=AiKIC8ztqhY
- 13. MATH IN THE MADNESS
- 14. 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
- 15. 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
- 16. LP Example: Buying Cabinets You 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.
- 17. 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.
- 18. 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]
- 19. 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
- 20. 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
- 21. 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
- 22. 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
- 23. 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 Variables ResA, ResB are Integer Variables
- 24. 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*CostB Slack Variables: Cost of not using a Resource,
- 25. 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
- 26. 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
- 27. 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
- 28. DEEP DIVE – AN EXAMPLE Optimize Gates, Trucks/Trips, Machines/Forklift, People/Shifts, minimize cost
- 29. 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
- 30. 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
- 31. Trip Constraints • A trip can only start at one of its possible start times Subject To C1: 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 … …
- 32. 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
- 33. 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 …. 0 0.5 1 1.5 2 2.5 9 10 11 12 1 2 3 4 5 Series1
- 34. 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)