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

Example 14.1: 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

 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.

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.

Example 14.1:

 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.

Example 14.1:

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.

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.

 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.

Example 14.2:

 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.

Example 14.2:

 The spreadsheet model for this problem is shown below.

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.

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

Example 14.3:

 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.

Example 14.3:

 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

 The variables and constraints for the problem are listed below

Example 14.3:

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

Example 14.3:

 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.

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.

Other Logistics Models

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 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.

Other Logistics Models

 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)

Example 14.4:

 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 below  Nodes 1, 2, and 3 represent the plants

(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).

Example 14.4:

 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.

Example 14.4:

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

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.

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

Example 14.5:

 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.

Example 14.5:

 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.

Example 14.5:

 The spreadsheet model is shown below.

The Rolling Planning Horizon Approach

 In reality, an aggregate planning model is usually implemented via a rolling planning horizon.

 To implement the SureStep model in the rolling planning horizon

 Next, observe month 1’s actual demand and replace the demands in the

Demand range with updated forecasts for the next four months.

 Rerun Solver and use the production levels and hiring and firing

recommendations in column B as the production level and workforce policy for month 2.

Model with Backlogging Allowed

 In many situations, backlogging of demand is allowed—that is, customer demand can be met at a later date.

 There are several modeling approaches to the backlogging problem; the most natural approach for the SureStep model is shown below.

 This is a nonlinear model using IF functions.

 However, you should avoid these nonsmooth functions in optimization models if at all possible, because Solver often has trouble handling them.

Example 14.6:

 Objective: To develop an LP model that relates investment decisions to total ending cash, and to use Solver to

find the strategy that maximizes ending cash and invests no more than a given amount in any one investment.

 Solution: At the beginning of year 1, Barney-Jones Investment Corporation has $100,000 to invest for the next

four years It wants to limit the amount put into any investment to $75,000

 There are five possible investments, labeled A through E.

 Investment A: Invest at the beginning of year 1, and for every dollar invested, receive returns of $0.50 and

$1.00 at the beginnings of years 2 and 3.

 Investment B: Invest at the beginning of year 2, receive returns of $0.50 and $1.00 at the beginnings of years 3 and 4.

 Investment C: Invest at the beginning of year 1, receive return of $1.20 at the beginning of year 2.

 Investment D: Invest at the beginning of year 4, receive return of $1.90 at the beginning of year 5.

 Investment E: Invest at the beginning of year 3, receive return of $1.50 at the beginning of year 4.

Example 14.6:

 At the beginning of any year, Barney-Jones can invest only cash on hand, which includes returns from previous investments.

 Any cash not invested in any year can be put in a short-term money market account that earns 3% annually.

 The variables and constraints for this investment model are listed in the table below.

Example 14.6:

 The spreadsheet model for this investment problem is shown below.

Example 14.7:

 Objective: To develop an LP model that relates initial allocation of money and bond purchases to future cash

availabilities, and to minimize the initial allocation of money required to meet all future pension fund payments.

 Solution: Armco Incorporated’s pension fund must be sufficient to make the payments listed in the table below.

 Each payment must be made on the first day of the year, and the payments will be financed by purchasing bonds.