Optimization Part I

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Optimization Part I Applications of
Optimization To Operations Management

• For this session, the learning objectives are
• Learn what a Linear Program is.
• Learn how to formulate a Linear Program and solve
  it using Excels Solver.
• Using Solver to solve a Make-or-Buy Problem.
• Using Solver to solve a Transshipment Problem
  (Product Distribution).

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Excels Solver

• Optimization involves the maximization or
  minimization of an objective subject to a set of
  constraints.
• Every copy of Microsoft Excel includes Solver,
  which enables you to solve the following types of
  optimization problems
• a Linear Program,
• an Integer Linear Program,
• a Nonlinear Program.
• The next page summarizes the use of Excels
  Solver.

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A Make-or-Buy Problem
DuPunt, Inc. manufactures three types of
chemicals. For the upcoming month, DuPunt has
contracted to supply its customers with the
following amounts of the three chemicals
DuPunts production is limited by the
availability of processing time in two chemical
reactors. Each chemical must be processed first
in Reactor 1 and then in Reactor 2. The
following table provides the hours of processing
time available next month for each reactor and
the processing time required in each reactor by
each chemical
Because of the limited availability of reactor
processing time, DuPunt has insufficient capacity
to meet its demand with in-house production.
Consequently, DuPunt must purchase some chemicals
from vendors having excess capacity and resell
them to its own customers. The following table
provides each chemicals in-house production cost
and outside purchase cost
DuPunts objective is to fill its customers
orders with the cheapest combination of in-house
production and outside purchases. In short,
DuPunt must decide how much of each chemical to
produce in-house (i.e., make) and how much of
each chemical to purchase outside (i.e.,buy).
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Formulation of the Make-or-Buy Problem as a
Linear Program
Define the following 6 decision variables
Minimize Total Costs
Contracted Sales
Reactor Availabilities
Nonnegativity Constraints
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A Transshipment Problem

• Consider a firm that for simplicity produces a
  single product. The firm has 3 plants (Tokyo,
  Hong Kong, and Bangkok), 2 warehouses (Seattle
  and Los Angeles), and 4 customers (Chicago, New
  York, Atlanta, and Dallas) geographically
  dispersed as diagrammed below. The firm ships
  its product from a plant to a warehouse and then
  on to a customer. In the diagram below
• The number to the left of each plant represents
  the plants supply.
• The number to the right of each customer
  represents the customers demand.
• The number appearing along an arrow from a plant
  to a warehouse or from a warehouse to a customer
  represents the corresponding unit shipping cost.
  For example, the unit shipping cost from Bangkok
  to Seattle is 25 per unit.
• The firm wants to distribute its product at
  minimum cost.

72 Total Supply
Total Demand 70
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Formulation of the Transshipment Problem as an LP
Let AZ denote the amount shipped from location A
to location Z. As examples, TL denotes the
amount shipped from Tokyo to Los Angeles, and SN
denotes the amount shipped from Seattle to New
York.
Min Total Shipping Costs
Supply Constraints
Transshipment Constraints
Demand Constraints
Nonnegativity Constraints
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Solving the Transshipment LP Using Excels Solver
The Spreadsheet Before Optimization
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Solving the Transshipment LP Using Excels Solver
The Spreadsheet After Optimization
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INTRODUCTION TO THE BLENDING PROBLEM
In many businesses and industrial environments,
the goal is to find the optimal recipe for
blending a variety of ingredients to obtain a
product that meets lower and/or upper limits on a
variety of characteristics. The table below
summarizes several applications.
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A Blending Problem
The exercise below is designed to review the
basics of formulating a linear program and
solving it using Solver.

• Harrus Feeding Companys Blending Problem
• The Harrus Feeding Company (HFC) operates a
  feedlot to which cattle are brought for the final
  fattening process. Since HFCs cattle population
  averages about 100,000, it is important for HFC
  to feed the cattle in a way that meets their
  nutritional requirements at minimum cost.
• The mixture HFC feeds the cattle is blend of four
  feedstuffs corn, wheat, barley, and hay. The
  table below provides the relevant dietary and
  cost data per pound of each feedstuff, along with
  a steers daily nutritional requirement. For
  example, for each pound of corn a steer consumes,
  it receives 2 grams of fat, 20 grams of protein,
  4 milligrams of iron, and 200 calories.
• Assuming a steers daily consumption of
  feedstuffs must be exactly 24 pounds, formulate
  and solve a linear program for determining the
  dietary blend that satisfies HFCs daily
  requirements at minimum cost.
• How would you modify your formulation if a
  steers daily consumption of feedstuffs must be
  in the range of 23-25 pounds?
• How would you modify your formulation if there
  were no daily limit on the pounds of feedstuffs
  that a steer must consume?
• Can the formulations in part (a) and part (c)
  result in distinct optimal solutions? Can you
  anticipate a potential problem with the optimal
  solution to the linear program in part (c)?