MIE376: LP Formulation Problems
A drill and a lathe are used exclusively in the manufacture of two products P1 and P2. The time units necessary for each step
(drilling and turning) in the manufacture of both products are shown in the table below.
MIE376 Network Programming Formulation Problems For each of the following probblems draw the appropriate network model, clearly marking the upper and lower bounds and the cost per unit flow for each directed arc, and whether it is a minimization or maximi
MIE376: DP Problems 1. Consider the network below, where the numbers on each arc represent the distance between the nodes. Find the shortest path in the network using dynamic programming. 5 9 6 7 6 2. The owner of a chain of three grocery stores has purch
A beer company has divided Bloomington into two territories. If x1 dollars
are spent on promotion in territory 1, then 6x11/2 cases of beer can be sold
there; and if x2 dollars are spent on promotion in territory 2, then 4x21/2
cases of beer ca
MIE376: LP Formulation Problems 1. A drill and a lathe are used exclusively in the manufacture of two products P1 and P2. The time units necessary for each step (drilling and turning) in the manufacture of both products are shown in the table below. P1 1
MIE376: Stochastic DP Problems (Winston Ch 19).
For a price of $1/gallon, the Safeco Supermarket chain has purchased 6 gal1ons of milk
from a local dairy. Each gallon of milk is sold in the chain's three stores for $2/gallon. The dairy
must buy back fo
MIE376S Mathematical Programming - Assignment 1
Due: Jan 13, 2011 at the Beginning of Tutorial (2:10pm) For any questions regarding the assignment, email Nick at [email protected] Formulation Only (No need to Solve) Chapter 3 Review Questions:
Generating all possible solutions
Consider only the equations
x1 + 3*x2 + x3
x1 + x2
+x4 = 7
2 equations in 4 unknown
Can only solve 2 x 2 system
How about if we set any 2 variables to zero?
Is every solution that sets 2 variables to zero a bfs?
What if we have thousands of variables?
Problems usually have thousands of variables and constraints.
Need to navigate through a virtual multi-dimensional diagram
Use graphical insights to algebraically find the optimal solution
Equations easier to manipu
Simplex method skeleton
1. Find an initial basic
feasible solution usually
2. Move to the most
promising adjacent basic
3. Repeat step 2 until there is
no better adjacent basic
MIE376 - Mathematical Programmi
Reconsider the Lathe and Drill problem
Let x1 be the number of chairs to be manufactured
Let x2 be the number of tables to be manufactured
Let z be the objective function, i.e. profit
Maximize z = x1+2*x2
subject to the constraints
x1 + 3*x2 16
Which n of the variables should be set to 0?
Set all n original variables to 0
In our example set x1 = x2 = 0
Solve for x3 and x4
Initial Basis: B = cfw_x3, x4
Non-basic variables: N = cfw_x1, x2
MIE376 - Mathematical Programmingt
Is it necessary to go beyond the graph?
Minimal skills are required to convert the
graph into an equivalent LP formulation.
When a problem can be formulated as a
Network Model, the graph says it all!
A Network Modeling problem is completed
once the proper
Simplex Method Variants
Commonly taught in introductory courses
Revised Simplex with product form inverse
Matrix based, closer to commercial implementation
Revised Simplex, with triangular factorization
Equivalence to Linear Programming
Arc from i to j LP variable xij
A constraint for each node i
Cij xij - sum is over all arcs
Node outflows - Node inflows = Node injection
xij - xki = Node i injection
A constraint for each arc
An early LP insight
has at most m basic (non-zero) variables +
at least n non-basic (zero) variables
LP problem with n variables and m constraints
optimal solution has at most m variables > 0
LP model for a factory making 1325 p