MATHEMATICAL PROGRAMMING
Models for
Integrated Supply Chain
Management
• Descriptive modeling - forecasting, data mining,
activity-based costing, performance metrics,
simulation, systems dynamics
• Prescriptive modeling - optimization models
(mathematical programming combined with
heuristic methods)
Scope
Supply Chain
Modeling System
Hierarchy
Top-Down View
Strategic
Analysis
Tactical Optimization
Modeling System
Long-Term
Tactical
Analysis
Production Planning Logistics Optimization
Optimization Modeling
Modeling System
System
Production Scheduling
Optimization
Modeling Systems
Short-term
Tactical
Analysis
{
{
Demand
Forecasting
and Order
Management
System
Strategic Optimization
Modeling System
Distribution Scheduling
Optimization
Modeling Systems
Transactional IT
Operational
Analysis
Formulate LP Model
• Identify the parameters
(activities/values you cannot control)
• Identify the decision variables
(activities/values that you can control and need to make
a decision on)
• Identify the objective function
(function of the decision variables for
minimization/maximization)
• Identify the constraints
(limitations you cannot control)
LP Terminology
•
•
•
•
•
The allocation of limited resources to competing
activities
for maximizing the value of these activities
Activities, n
Resources, m
Decision variables – or level of activities, x
Objective function – or value of activities, Z
Constraints
– functional
– non-negativity
LP Terminology
•
•
•
•
Feasible region, feasible solution
Infeasible solution
Optimal solution
Extreme-point – or corner-point feasible
solutions
• Parameters
Standard (Canonical) Form of an
LP Model
Maximize Z = c1x1 + c2x2
+ … + cnxn
subject to
a11x1
+ a12x2 + … +
a1nxn ≤ b1
a21x1
+ a22x2 + … + a2nxn
≤ b2
…
am1x1
+ am2x2 + … + amnxn
≤ bm
Other Forms that can be Converted
into Standard Form
• Objective function:
Minimize Z=cx
(instead of Maximize Z=cx)
• Functional constraints:
Ax ≥ b or Ax = b
(instead of Ax ≤ b)
• Non-negativity constraints:
x unrestricted in sign
(instead of x ≥ 0)
Assumptions of Linear
Programming
• Proportionality
• Additivity
• Divisibility
• Certainty
A Transportation Example
• A company has 2 plants and 3 warehouses
• Supply at plants
100 units in Plant 1, 200 units in Plant 2
• Sales potential at warehouses
150 units, 200 units, and 350 units at Warehouses 1, 2
and 3, respectively
• Revenue
12 $/unit, 14 $/unit and 15 $/unit at Warehouses 1, 2
and 3, respectively
Warehouse
• Cost of manufacturing
one unit at plant i
and shipping to w/h j:
Plant
1
2
3
1
8
10
12
2
7
9
11
Mixed 0-1 Integer Programming
max cx  dy
Ax  Gy  b
x j  {0,1}, l j  y j  u j
Why Use MIP ?
•
•
•
•
•
•
Indivisible commodities
Binary choices
Logical relations
Start up or fixed costs
Economies of scale
Combinatorial modeling
Applicability
• Tremendous increase of the use of MIPs
in the last decade
• Large-scale MIPs are solvable (provably
good solutions) on PCs with commercially
available software
Recent Applications
•
•
•
•
•
Transportation planning and operations
Facility location
Production scheduling
Supply-chain management
Many others
Transportation planning and
operations
• Aircraft arrival slot allocation at
American. Network optimization used
to allocate arrival slots of canceled
flights, leading to reduced delays,
translates into direct annual operating
cost savings of $5.2M (1991)
• Optimizing airline scheduling of planes
and crews. Delta's Cold Start solves its
fleet assignment with expected savings
of $300M over three years (1994)
Facility location
• Base closing in Germany (US Department
of Defense). 0-1 IP used to determine
base closures and optimal stationing plan
for US troops in Europe after force
reductions, annual savings of up to $58M
(1996)
Supply-chain management
• Global supply chain management (Digital).
MIP used to design production, distribution
and vendor network so as to minimize cost
of production and distribution times, saved
$100M (1995)
• Restructuring the supply chain (Proctor
and Gamble). MIP is used to help
restructure the supply chain (1997)
LP Based Branch and Bound
• Implicit enumeration
tree
• At every node, an LP
relaxation is solved
• Upper bounds come
from LP solutions;
lower bounds come
from MIP feasible
solutions
LP Based Branch and Bound
Initialization
zip=-, xip=
List empty?
Choose problem
Solve LPzlp,xlp
If infeasible,
then fathom
If zlp zip,
then fathom
If xlp integral,
then zip zlp,xip xlp
and fathom
Branch
xip optimal
How to improve performance?
1. Faster computer
2. Faster linear programming optimizer
3. Smaller zlp
4. Larger zip
5. Improved branching
6. Smaller linear programs
Formulations
• Most MIPs have many correct
formulations, but while one formulation
may be easy to solve, another may be
very hard to solve
• Failure to solve a MIP may not be the fault
of the algorithm, but the result of a “bad”
formulation
• Smaller size formulations (number of
variables and constraints) may not be
better and can be much worse