Session 3a - PowerPoint Presentation

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Session 3aSlide2

Decision Models -- Prof. Juran2

Overview

Multiperiod Models

Data Processing at the IRS

Arbitrage with Bonds

Transportation Models

Gribbin Brewing Slide3

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IRS Example

To process income tax forms, the Internal Revenue Service (IRS) first sends each form through the data preparation (DP) department, where information is coded for computer entry.

Then the form is sent to data entry (DE), where it is entered into the computer. Slide4

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IRS Example

During the next 3 weeks, the following numbers of forms will arrive: week 1, 40,000; week 2, 30,000; week 3, 60,000.

All employees work 40 hours per week and are paid $500 per week.

Data preparation of a form requires 15 minutes, and data entry of a form requires 10 minutes. Slide5

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IRS Example

Each week an employee is assigned to either data entry or data preparation.

The IRS must complete processing all forms by the end of week 5 and wants to minimize the cost of accomplishing this goal.

Assume all employees are capable of performing data preparation or data entry, but must be assigned to one task for an entire week at a time. Slide6

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Managerial Problem Definition

Determine how many workers should be working and how the workers should allocate their hours during the next 5 weeks.Slide7

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Formulation

Decision Variables

Numbers of workers for two tasks over five weeks (10 decisions) and numbers of forms completed on each task in each week (10 decisions).

Objective

Minimize total cost.

Constraints

Forms arrive at a fixed schedule.

All work must be completed in five weeks.

Data prep task cannot begin until the forms arrive.

Data entry task cannot begin until data prep task is finished.

The plan can’t call for more labor than is available for either task in any week.Slide8

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FormulationSlide9

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FormulationSlide10

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Solution MethodologySlide11

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Solution MethodologySlide12

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Solution MethodologySlide13

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Optimal Solution

Minimum total cost is $677,073.

All work could be done in four weeks.

Note that the balance equations are not constraints in the usual sense (i.e. specified in Solver). We build them into the model, linking the tasks and weeks together.

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Bond Arbitrage ExampleSlide15

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Formulation

Given the current price structure, the question is whether there is a way to make an infinite amount of money. To answer this, we look for an arbitrage.

An arbitrage exists if there is a combination of bond sales and purchases today that yields

A positive cash flow today

Non-negative cash flows at all future dates

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Formulation

If such a strategy exists, then it is possible to make an infinite amount of money.

For example, if buying 10 units of bond 1 today and selling 5 units of bond 2 today yielded, say, $1 today and nothing at all future dates, then we could make $k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today. Slide17

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Formulation

Decision Variables

How much to buy or sell of each bond. (Selling a bond is conceptually the same as buying a negative amount.)

Objective

Maximize cash flow at the end of the first period (today).

Constraints

Non-negative cash flow at the end of all future periods.Slide18

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FormulationSlide19

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FormulationSlide20

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FormulationSlide21

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Solution MethodologySlide22

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Solution MethodologySlide24

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Solution Methodology

This is actually good news! It indicates an “unbounded” problem; one in which there are no constraints that limit the value of the objective function. In the context of this problem, it means that there is no limit on the amount of cash flow in the first period. In other words, there is an arbitrage opportunity.

Unfortunately, because Solver couldn’t solve the problem, we don’t know which bonds to buy and sell. We can get around this by playing a little trick; we introduce a new constraint limiting the objective function artificially.Slide25

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Optimal SolutionSlide27

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Conclusions

Buying bonds 1 and 2 today, while selling bond 3, offers an arbitrage opportunity.

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Back to RealitySlide29

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Optimal SolutionSlide30

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Conclusions

This result indicates that no arbitrage opportunity exists.

The only way to have non-negative cash flows in the first period and zero cash flows in all future periods is not to invest at all.Slide31

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Gribbin Brewing

Regional brewer Andrew Gribbin distributes kegs of his famous beer through three warehouses in the greater News York City area, with current supplies as shown:Slide32

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On a Thursday morning, he has his usual weekly orders from his four loyal customers, as shown : Slide33

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Tracy Chapman, Gribbin’s shipping manager, needs to determine the most cost-efficient plan to deliver beer to these four customers, knowing that the costs per keg are different for each possible combination of warehouse and customer:Slide34

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What is the optimal shipping plan?

How much will it cost to fill these four orders?

Where does Gribbin have surplus inventory?

If Gribbin could have one additional keg at one of the three warehouses, what would be the most beneficial location, in terms of reduced shipping costs?

Gribbin has an offer from Lu Leng Felicia, who would like to sublet some of Gribbin’s Brooklyn warehouse space for her tattoo parlor. She only needs 240 square feet, which is equivalent to the area required to store 40 kegs of beer, and has offered Gribbin $0.25 per week per square foot. Is this a good deal for Gribbin? What should Gribbin’s response be to Lu Leng?Slide35

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Managerial Problem Formulation

Decision Variables

Numbers of kegs shipped from each of three warehouses to each of four customers (12 decisions).

Objective

Minimize total cost.

Constraints

Each warehouse has limited supply.

Each customer has a minimum demand.

Kegs can’t be divided; numbers shipped must be integers.Slide36

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Mathematical Formulation

Decision Variables

Define

X

ij

= Number of kegs shipped from warehouse

i

to customer

j

.

Define

C

ij

= Cost per keg to ship from

warehouse

i

to customer

j

.

i

= warehouses 1-3,

j

= customers 1-4Slide37

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Mathematical Formulation

Objective

Minimize

Z

=

Constraints

Define

S

i

= Number of kegs available at warehouse

i

.

Define

D

j

= Number of kegs ordered by

customer

j

.

Do we need a constraint to ensure that all of the

X

ij

are integers?

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Where does Gribbin have surplus inventory?

The only supply constraint that is not binding is the Hoboken constraint. It would appear that Gribbin has 45 extra kegs in Hoboken.

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If Gribbin could have one additional keg at one of the three warehouses, what would be the most beneficial location, in terms of reduced shipping costs? Slide44

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According to the sensitivity report,

One more keg in Hoboken is worthless.

One more keg in the Bronx would have reduced overall costs by $0.76.

One more keg in Brooklyn would have reduced overall costs by $1.82.Slide45

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Gribbin has an offer from Lu Leng Felicia, who would like to sublet some of Gribbin’s Brooklyn warehouse space for her tattoo parlor. She only needs 240 square feet, which is equivalent to the area required to store 40 kegs of beer, and has offered Gribbin $0.25 per week per square foot.

Is this a good deal for Gribbin?

What should Gribbin’s response be to Lu Leng?Slide46

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Assuming that the current situation will continue into the foreseeable future, it would appear that Gribbin could reduce his inventory in Hoboken without losing any money (i.e. the shadow price is zero).

However, we need to check the sensitivity report to make sure that the proposed decrease of 40 kegs is within the allowable decrease.

This means that he could make a profit by renting space in the Hoboken warehouse to Lu Leng for $0.01 per square foot.Slide47

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Lu Leng wants space in Brooklyn, but Gribbin would need to charge her more than $1.82 for every six square feet (about $0.303 per square foot), or else he will lose money on the deal.

Note that the sensitivity report indicates an allowable decrease in Brooklyn that is enough to accommodate Lu Leng.Slide48

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As for the Bronx warehouse, note that the allowable decrease is zero. This means that we would need to re-run the model to find out the total cost of renting Bronx space to Lu Leng.

A possible response from Gribbin to Lu Leng:

“I can rent you space in Brooklyn, but it will cost you $0.35 per square foot. How do you feel about Hoboken?”

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Summary

Multiperiod Models

Data Processing at the IRS

Arbitrage with Bonds

Transportation Models

Gribbin Brewing