SCM 460 Midterm

Network Model - Capacitated Transshipment Model

- Flow on arcs decision variables - Objective Function is minimized - Balance Constraints * Supply Node - Net outflow does not exceed supply * Intermediate Node - Flow into the node is equal to the flow out * Demand Node - Net inflow is equal to the demand - Bound constraints on each are. Flow can't exceed capacity on arc.

Summary of Graphical Solutions to LP problems

1.Plot the boundary line of each constraints 2.Identify the feasible region 3.Locate the optimal solution by either: a. plotting level curves b. Enumerating the extreme points

Linear Programming

5 steps to solve LP models: 1. Understand the problem 2.Identify the decision variables 3.State the objective as a linear function 4.State the constraints as a linear function 5.Identify upper and lower bounds of variables (non-negativity constraints)

Types of Solutions

Alternate Optimal Solutions - Some LP models can have more than one optimal solution Infeasible - the model cannot be solved with a value inside of the feasible region (constraints never intersect, no feasible regions) Unbounded - the feasible region has no barrier, can still find an optimal solution depending on the direction of the unbounded region and your objective function, the directions must be opposite. Multiple-Optimal Solutions?

For Minimum Cost Network Flow Problems Where: * Total Supply > Total Demand * Total Supply = Total Demand * Total Supply < Total Demand

Apply This Balance-of-Flow Rule at Each Node: * Inflow-Outflow ≥ Supply or Demand * Inflow-Outflow = Supply or Demand * Inflow-Outflow≤ Supply or Demand

When to optimize

Determine Product Mix Manufacturing Routing and Logistics Financial Planning

Classical Transportation Problem

Binary Decisions - do we us the link or not?

Balance of Flow Rules:

Card 24

Mathematical Model Categories:

Descriptive: Relation ship between x and y is well defined. Independent variables are unknown/unclear

Branch and Bound Method

Don't add integrality constraints. We don't add them because it makes the problem harder to solve a. Can be used to solve ILP problems b.Requires the solution of a series of LP problems termed "candidate problems" c. Maximization problem=upper bound d. Minimization=lower

Value of Decision Making

Good Decision Good Outcome = Deserved Success Good Decision Bad Outcome = Bad Luck Bad Decision Good Outcome = Dumb Luck Bad Decision Bad Outcome = Poetic Justice Good decisions don't always lead to good outcomes. A structured approach can help make good decisions but guarantee good outcomes.

Demand and Supply effect on Constraints

In generalized network flow problems, gains and/or losses associated with flows across each arc effectively increase and/or decrease the available supply. Ensures no excess volume in the model Only move what is necessary

Unbounded Constraint

Increases the value of the objective function in a max problem without leaving the feasible region limits (positive infinity)

Flow constraints

Inflow - outflow = 0

Binary Variables

Integer variables are integer variables that can assume only two values: 0 or 1 These variables can be useful in a number of practical modeling situations Uses: Indicate whether you are using a resource (warehouse) or not. Linking constraints - ? cant have constraints with variables on RHS - move to left

Types of Math Programming Models

LP - Linear Programming NLP- Nonlinear Programming ILP - Integer Linear Programming MIP - Mixed Integer Programming

Arcs

Lines connecting nodes in figure 5.1. Indicates valid routes between nodes in a network flow problem. Called directed arcs when arrows indicate a direction.

Basic Models

Mental (arranging furniture) Visual (blueprints, maps) Physical/scale (buildings) Mathematical (studying in this class)

Objective Functions

Min or Max?

Minimal Spanning Tree Problem

Minimal spanning tree problem involves determining the set of arcs that connects all the nodes at minimum cost.

Linear Functions

No exponents, objective has a constant slope

Unfeasibility

Not possible, will give you red bar in solver

Descriptive Model

Objective is to describe the outcome or behavior of a given operation or system. Relationship between x and y is well defined. Independent variables are unknown/unclear

Maximum Flow Problem

Objective is to determine the maximum amount of flow that can occur through a network. Arcs in these problems have upper and lower flow limits. Ex. How much water can flow through a network of pipes?

Redundant Constraints

Removing this constraint wont change the optimization problem, doesn't touch feasible region or affect optimal solution

Heuristic Models

Rule of thumb for making decisions that might work well in some instances but is not guaranteed to produce optimal solutions or decision.

Net Supply Concept

Supply must equal demand so flows balance out to 0

Prescriptive model

Tells decision maker what steps to take Relationship between x and y is well defined. Independent variables are known or under decision makers control

Production Models

The hot tub example we did, To max profit how many of each tub type do we make?

Predictive Model

The objective is to predict or estimate what value the dependent variable Y will take on when the independent variables x1 x2... take on specific values. Relationship between x and y is unknown/unclear. Independent variables are known or under decision makers control

Nodes

Use them to build balance flow constraints (circles in book figure 5.1) Supply nodes - sending nodes (negative number represents available outflow) Demand nodes - receiving nodes (Positive number represents demand) Transshipment nodes - can both send and receive

Decision Variables

What we are looking to optimize. Ex. Number of Hyrdolux spas and number of Aqualux spas

Integer Programming

When some or all of the decision variables in an LP problem are restricted to assuming only integer values.

Binding Constraints

You are using all resources available. If you use one more item, it will change the objective function. Will use all resources. LHS=RHS

Mathematical Models

field of management science that fins the optimal way of using limited resources to achieve the objectives of an individual of a business

Concept of Optimization

finds the optimal/most efficient way of using limited resources to achieve objectives of an individual in a business (mathematical programming

Feasible Region

within constraint area. If in there