Transcript of "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.
2.
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
12.
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
13.
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
14.
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.
15.
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.
16.
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]
17.
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
18.
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
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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
20.
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
21.
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
22.
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,
23.
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
24.
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
25.
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
26.
DEEP DIVE –
AN EXAMPLE
Optimize Gates, Trucks/Trips,
Machines/Forklift,
People/Shifts, minimize cost
27.
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
28.
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
29.
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
…
…
30.
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
31.
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
32.
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)
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