Business analytics data analysis and decision making 5th by wayne l winston chapter 14

part.

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Business Analytics:

Data Analysis and

Chapter

Decision Making

14
Optimization Models


Introduction
 A wide variety of problems can be formulated as linear programming
models, but there are some that cannot.
 Some models require integer variables, or they are nonlinear in the
decision variables.
 Once the models are formulated, Solver can be used to solve them.
 However, integer and nonlinear models are inherently more difficult to
solve.

 Solver must use more complex algorithms and is not always guaranteed to find
the optimal solution.

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Worker Scheduling Models
 Many organizations use worker scheduling models to schedule
employees to provide adequate service.

 LP can be used to schedule employees on a daily basis, as shown in
the next example.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.1:
Worker Scheduling.xlsx

(slide 1 of 2)

 Objective: To develop an LP model that relates five-day shift schedules
to daily numbers of employees available, and to use Solver on this model
to find a schedule that uses the fewest number of employees and meets
all daily workforce requirements.

 Solution: The number of full-time employees that a post office needs
each day is given in the table on the bottom left.

 Union rules state that each full-time employee must work five
consecutive days and then receive two days off.

 The post office wants to meet its daily requirements using only full-time
employees.

 The variables and constraints for this problem appear in the table on the

bottom right.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.1:
Worker Scheduling.xlsx

(slide 2 of 2)

 Add an integer constraint in Solver to ensure that the number of
employees starting work on some days is not a fraction.

 The spreadsheet model with this integer constraint is shown below.
 When you solve this problem, you might get a different schedule that
is still optimal.

 Such multiple optimal solutions are not at all uncommon and are good
news for a manager, who can then choose among the optimal solutions.

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Modeling Issues
 The postal employee scheduling example is called a static scheduling
model because we assume that the post office faces the same
situation each week.
 In reality, demands change over time. A dynamic scheduling model is
necessary for such problems.

 A scheduling model for a more complex organization has a larger
number of decision variables, and optimization software such as
Solver will have difficulty finding a solution.
 Heuristic methods have been used to find solutions to these problems.
 The scheduling model can be expanded to handle part-time
employees, the use of overtime, and alternative objectives such as
maximizing the number of weekend days off.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Blending Models
 In many situations, various inputs must be blended to produce desired
outputs.
 In many of these situations, blending models can be used to find the
optimal combination of outputs as well as the mix of inputs used to
produce desired outputs.
 Examples of blending problems:

 A company using a blending model would run the model periodically
(each day, for example) and set production on the basis of current
inventory of inputs and the current forecasts of demands and prices.

 Then the model would be run again to determine the next day’s
production.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.2:
Blending Oil.xlsx

(slide 1 of 2)

 Objective: To develop an LP model for finding the revenue-maximizing plan that
meets quality constraints and stays within the limits on crude oil availabilities.

 Solution: Chandler Oil has 5000 barrels of crude oil 1 and 10,000 barrels of
crude oil 2 available.

 Chandler sells gasoline and heating oil, which are produced by blending the two
crude oils together.

 Each barrel of crude oil 1 has a quality level of 10, and each barrel of crude oil 2
has a quality level of 5.

 Gasoline must have an average quality level of at least 8, whereas heating oil
must have an average quality level of at least 6.

 Gasoline sells for $75 per barrel, and heating oil sells for $60 per barrel.
 The variables and constraints required for this model are listed below.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.2:
Blending Oil.xlsx

(slide 2 of 2)

 The spreadsheet model for this problem is shown below.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Logistics Models
 In many situations a company produces products at locations called
origins and ships these products to customer locations called
destinations.

 Each origin has a limited capacity that it can ship, and each
destination must receive a required quantity of the product.

 Logistic models can be used to determine the minimum-cost
shipping method for satisfying customer demands.

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Transportation Models
 In a transportation problem, the only possible shipments are those directly
from an origin to a destination.
 A typical transportation problem requires three sets of numbers:
 Capacities—indicate the most each plant can supply in a given amount of time under
current operating conditions.
 Customer demands—are typically estimated from some type of forecasting model.
 Unit shipping costs—come from a transportation cost analysis.


The unit “shipping” cost can also include the unit production cost at each plant .

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.3:
Transportation 1.xlsx

(slide 1 of 4)

 Objective: To develop an LP model for finding the least-cost way of
shipping the automobiles from plants to regions, staying within plant
capacities and meeting regional demands.

 Solution: Grand Prix Automobile Company manufactures automobiles in
three plants and then ships them to four regions of the country.

 The plants can supply the amounts listed in the right column of the table
below.

 The customer demands by region are listed in the bottom row of the table.
 The unit costs of shipping an automobile from each plant to each region
are listed in the middle of the table.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.3:
Transportation 1.xlsx

(slide 2 of 4)

 The variables and constraints for the problem are listed below.

 A network diagram of the model is
shown to the right.

 A node, indicated by a circle,
generally represents a geographical
location. In this case, the nodes on
the left correspond to plants, and
the nodes on the right to regions.

 An arc, indicated by an arrow,
generally represents a route for
getting a product from one node to
another.

 An arc pointed into a node is called
an inflow, and an arrow pointed out
of node is called an outflow.
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Example 14.3:
Transportation 1.xlsx


(slide 3 of 4)

The spreadsheet model and a graphical representation of the optimal solution are shown below.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.3:
Transportation 1.xlsx


(slide 4 of 4)

It is often useful to model network problems by listing all of the arcs and their corresponding
flows in one long list. Then constraints can be indicated by a separate section of the spreadsheet.



For each node in the network, there is a flow balance constraint.





These flow balance constraints for the basic transportation model are simply the supply and demand
constraints, but they can be more general for other network models .

A more general network model of the Grand Prix problem is shown below.

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Modeling Issues


The customer demands in typical transportation problems can be handled in one of two
ways:
 Think of forecasted demands as minimal requirements that must be sent to the customers.
 Consider demands as maximal sales quantities, the most each region can sell.



If all the supplies and demands for a transportation problem are integers, the optimal Solver
solution automatically has integer-valued shipments.
 Explicit integer constraints are not required, so the “fast” simplex method can be used .



Shipping costs are often nonlinear due to quantity discounts.



There is a streamlined version of the simplex method, called the transportation simplex
method, that is much more efficient than the ordinary simplex method for transportation
problems.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Other Logistics Models
(slide 1 of 2)

 The general logistics model is like the transportation model, except for
two possible differences:

 Arc capacities are often imposed on some or all of the arcs.
 They become simple upper-bound constraints in the model.
 There can be inflows and outflows associated with any node.
 An origin node is a location that starts with a certain supply.
 A destination is the opposite; it requires a certain amount to end up there.
 A transshipment point is a location where goods simply pass through.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Other Logistics Models
(slide 2 of 2)

 The best way to think of the node categories is in terms of net inflow
and net outflow.
 Net inflow for any node is defined as total inflow minus total outflow for that
node.

 Net outflow is the negative of this, total outflow minus total inflow.
 An origin is a node with positive net outflow, a destination is a node with
positive net inflow, and a transshipment point is a node with net outflow
(and net inflow) equal to 0.

 There are two types of constraints in logistics models:
 Arc capacity constraints, which are simple upper bounds on the arc flows
 Flow balance constraints, one for each node:
 For an origin: Net Outflow = Capacity (or possibly Net Outflow ≤ Capacity)
 For a destination: Net Inflow ≥ Demand (or possibly Net Inflow = Demand)
 For a transshipment point: Net Inflow = 0 (equivalent to Net Outflow = 0)

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.4:
RedBrand Logistics1.xlsx

(slide 1 of 3)

 Objective: To develop an LP model for finding the minimum-cost way
to ship the tomato product from suppliers to customers, possibly
through warehouses, so that customer demands are met and supplier
capacities are not exceeded.

 Solution: RedBrand Company produces a tomato product at three
plants. The cost of producing the product is the same at each plant.

 The product can be shipped directly to the company’s two customers,
or it can first be shipped to the company’s two warehouses and then
to the customers.

 The production capacity of each plant (in tons per year) and the
demand of each customer are shown in the graphical representation
 Nodes 1, 2, and 3 represent the plants
below.
(denoted by S for supplier).



Nodes 4 and 5 represent the warehouses
(denoted by T for transshipment).



Nodes 6 and 7 represent the customers
(denoted by D for destination).

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.4:
RedBrand Logistics1.xlsx

(slide 2 of 3)

 The cost (in thousands of dollars) of shipping a ton of the product between each pair of locations is
listed in the table below, where a blank indicates that RedBrand cannot ship along that arc.

 The most that can be shipped between any two nodes is 200 tons. (This is the common arc capacity.)
 The variables and constraints for RedBrand’s model are listed below.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.4:
RedBrand Logistics1.xlsx

(slide 3 of 3)

 The spreadsheet model and a graphical representation of the optimal solution are shown below.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Modeling Issues
 Solver uses the simplex method to solve logistics models, but the
network simplex method is much more efficient and can solve large
logistics problems.
 If the given supplies and demands for the nodes are integers and all
arc capacities are integers, the logistics model always has an optimal
solution with all integer flows.
 This integer benefit is guaranteed only for the basic logistics model.
 When the model is modified in certain ways, such as by adding a shrinkage
factor, the optimal solution is no longer guaranteed to be integer-valued.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Aggregate Planning Models
 Models where we determine workforce levels and production
schedules for a multiperiod time horizon are called aggregate
planning models.
 There are many variations, depending on the detailed assumptions made.
 A number of inputs are required for this type of problem:
 Initial inventory, holding costs, and demands
 Data on the current number of workers, regular hours per worker per
month, regular hourly wage rates, overtime hourly rate, and maximum
number of overtime hours per worker per month

 Costs for hiring and firing a worker
 Unit production cost, which is a combination of two inputs:

 Raw material cost per unit
 Labor hours per unit

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Example 14.5:
Aggregate Planning 1.xlsx

(slide 1 of 3)

 Objective: To develop an LP spreadsheet model that relates workforce and
production decisions to monthly costs, and to find the minimum-cost solution that
meets forecasted demands on time and stays within limits on overtime hours and
production capacity.

 Solution: SureStep Company must meet (on time) the following demands for pairs
of shoes: 3000 in month 1; 5000 in month 2; 2000 in month 3; and 1000 in month
4.

 At the beginning of month 1, 500 pairs are on hand, and Sure Step has 100
workers.

 A worker is paid $1500 per month
 Each worker can work up to 160 hours per month before he or she receives
overtime. A worker can work up to 20 hours of overtime per month and is paid $13
per hour for overtime labor.

 It takes four hours of labor and $15 of raw material to produce a pair of shoes.
 At the beginning of each month, workers can be hired or fired. Each hired worker

costs $1600, and each fired worker costs $2000.

 At the end of the month, a holding cost of $3 per pair of shoes left in inventory is
incurred.

 Production in a given month can be used to meet that month’s demand.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.


Example 14.5:
Aggregate Planning 1.xlsx

(slide 2 of 3)

 The variables and constraints for this aggregate planning model are
listed below.

 The most difficult aspect of modeling this problem is knowing which
variables the company gets to choose—the decision variables—and which
variables are determined by these decisions.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.