1. Session 3a
Decision Models Prof. Juran

2. Decision Models -- Prof. Juran
Overview
Multiperiod Models
Data Processing at the IRS
Arbitrage with Bonds
Transportation Models
Gribbin Brewing
Decision Models Prof. Juran
Decision Models Prof. Juran

3. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

4. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

5. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

6. Managerial Problem Definition
Determine how many workers should be working and how the workers should allocate their hours during the next 5 weeks.
Decision Models Prof. Juran
Decision Models Prof. Juran

7. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

8. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

9. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

10. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

11. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

12. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

13. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

14. Bond Arbitrage Example
Decision Models Prof. Juran
Decision Models Prof. Juran

15. Decision Models -- Prof. Juran
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
Decision Models Prof. Juran
Decision Models Prof. Juran

16. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

17. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

18. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

19. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

20. Decision Models -- Prof. Juran
Formulation
Decision Models Prof. Juran
Decision Models Prof. Juran

21. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

22. Decision Models -- Prof. Juran

23. Decision Models -- Prof. Juran
Solution Methodology
Decision Models Prof. Juran
Decision Models Prof. Juran

24. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

25. Decision Models -- Prof. Juran

26. Decision Models -- Prof. Juran
Optimal Solution
Decision Models Prof. Juran
Decision Models Prof. Juran

27. Decision Models -- Prof. Juran
Conclusions
Buying bonds 1 and 2 today, while selling bond 3, offers an arbitrage opportunity.
Decision Models Prof. Juran
Decision Models Prof. Juran

28. Decision Models -- Prof. Juran
Back to Reality
Decision Models Prof. Juran
Decision Models Prof. Juran

29. Decision Models -- Prof. Juran
Optimal Solution
Decision Models Prof. Juran
Decision Models Prof. Juran

30. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

31. Decision Models -- Prof. Juran
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:
Decision Models Prof. Juran
Decision Models Prof. Juran

32. Decision Models -- Prof. Juran
On a Thursday morning, he has his usual weekly orders from his four loyal customers, as shown :
Decision Models Prof. Juran
Decision Models Prof. Juran

33. Decision Models -- Prof. Juran
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:
Decision Models Prof. Juran
Decision Models Prof. Juran

34. Decision Models -- Prof. Juran
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?
Decision Models Prof. Juran
Decision Models Prof. Juran

35. 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.
Decision Models Prof. Juran
Decision Models Prof. Juran

36. Mathematical Formulation
Decision Variables
Define Xij = Number of kegs shipped from warehouse i to customer j.
Define Cij = Cost per keg to ship from warehouse i to customer j.
i = warehouses 1-3, j = customers 1-4
Decision Models Prof. Juran
Decision Models Prof. Juran

37. Mathematical Formulation
Objective
Minimize Z =
Constraints
Define Si = Number of kegs available at warehouse i.
Define Dj = Number of kegs ordered by customer j.
Do we need a constraint to ensure that all of the Xij are integers?
Decision Models Prof. Juran
Decision Models Prof. Juran

38. Decision Models -- Prof. Juran

39. Decision Models -- Prof. Juran

40. Decision Models -- Prof. Juran

41. Decision Models -- Prof. Juran

42. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

43. Decision Models -- Prof. Juran
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?
Decision Models Prof. Juran
Decision Models Prof. Juran

44. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

45. Decision Models -- Prof. Juran
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?
Decision Models Prof. Juran
Decision Models Prof. Juran

46. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

47. Decision Models -- Prof. Juran
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.
Decision Models Prof. Juran
Decision Models Prof. Juran

48. Decision Models -- Prof. Juran
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?”
Decision Models Prof. Juran
Decision Models Prof. Juran

49. Decision Models -- Prof. Juran
Summary
Multiperiod Models
Data Processing at the IRS
Arbitrage with Bonds
Transportation Models
Gribbin Brewing
Decision Models Prof. Juran
Decision Models Prof. Juran