Formulation Problems

1. Formulation Problems Outline • Whitt Window Company • Hotdogs and Buns • Portfolio Optimization • No debt • With debt • Production Planning 1-period • Forward: Setting  Formulation • Backward: Formulation  Setting • Production Planning • Metalco Blends Alloys • Fagersta Steelworks • Backward: Incomplete Setting, Formulation  Complete Setting

2. Whitt Window Company - Setting

3. Whitt Window Company - Formulation

4. Hotdogs and Buns - Setting

5. Hotdogs and Buns - Formulation

6. Portfolio Optimization - Setting Linear Programming formulations can be used to select a desirable bond portfolio. • ProMax has raised $8,000,000 to invest into four bonds. • The average annual return, the worst-case annual return and the duration of each bond is below: • ProMax wants to maximize the expected return from its bond investments • The worst-case return of bond portfolio must be at least 9%. The average duration of a portfolio can be computed as a weighted average, e.g., 3 million, 2.5 million and 2.5 million investments to bonds 1,2,3, yield average duration of =2 years • To achieve diversification, at most 40% can be invested in a single bond. • The average duration of bond portfolio must be at least 2 years. The average duration of a portfolio can be computed as a weighted average, e.g., 3 million, 2.5 million and 2.5 million investments to bonds 1,2,3, yield average duration of =2 years • Formulate a linear program to maximize the average annual return of a portfolio.

7. Portfolio Optimization - Formulation 1. Decision Variables: is the money invested in bond in million dollars 2. Objective function: 3. Constraints • The worst-case return of bond portfolio must be at least 9%. • To achieve diversification, at most 40% can be invested in a single bond. • The average duration of bond portfolio must be at least 2 years. The average duration of a portfolio can be computed as a weighted average, e.g., 3 million, 2.5 million and 2.5 million investments to bonds 1,2,3, yield average duration of • Is there a missing constraint? • How to make sure that the total money available for investing is 8,000,000?

8. Portfolio Optimization with Debt - Setting Financial institutions use debt to increase their investment capital. • ProMax has raised $8,000,000 from a private equity fund to invest into four bonds. • The average annual return, the worst-case annual return and the duration of each bond is below: • ProMax wants to maximize the expected return from its bond investments • The worst-case return of bond portfolio must be at least 2%. • To achieve diversification, at most 35% can be invested in a single bond. • ProMax can borrow extra money to invest into bonds 1 & 2 and bonds 3 & 4 • High leverage: Bonds 1 & 2 have relatively high worst-case return. ProMax leverages its high worst-case return investments by a factor of 4. At least 20% (=1/(1+4)) of investments in high return investments must come from ProMax raised funds. The rest, at most 80%, can come from debt. • Low leverage: Bonds 3 & 4 have relatively low worst-case return. ProMax leverages its low worst-case return investments by a factor of 1. At least 50% (=1/(1+1)) of investments in high return investments must come from ProMax raised funds. The rest, at most 50%, can come from debt. • Borrowed money must be returned in a year with 1% interest • Formulate a linear program to maximize the average annual return of a portfolio over a year.

9. Portfolio Optimization with Debt - Formulation 1. Decision Variables: is the money invested in bond in million dollars is the money borrowed to invest in high worst-case return bonds is the money borrowed to invest in high worst-case return bonds 2. Objective function: 3. Constraints • The worst-case return of bond portfolio must be at least 2%. • To achieve diversification, at most 35% can be invested in a single bond. • Investment in bonds must be less than or equal to money available to invest • Leverage constraints and

10. Production Planning 1-period - Setting • A company produces products A, B, C • Products A, B, C are sold at prices $15, $60 and $100 in unlimited numbers • Products require the following • A: Input material and 1 labor hour • B: 2 product As and 2 labor hours • C: 1 product B and 3 labor hours Input and 1 hr: A Sell A at $15 Sell B at $60 2As and 2 hrs: B 1B and 3 hrs: C Sell C at $100 • A total of 60 labor hours are available • Formulate a linear program to maximize profit

11. Production Planning 1-period - Formulation

12. Production Planning 1-period – Formulation  Setting Input and ?hr: Product 1 Sell Product 1 at $? ? Product 1 and ?hrs: Product 2 Sell Product 2 at $? ? Product 2 and ?hrs: Product 3 Sell Product 3 at $? • A total of ? labor hours are available

13. Production Planning - Setting

14. Production Planning - Formulation

15. Metalco Blends Alloys - Setting

16. Metalco Blends Alloys - Formulation

17. Fagersta Steelworks Incomplete Setting, Formulation  Complete Setting A friend of yours is given a formulation problem and a diagram. The problem is as follows: Your friend provides the formulation on the next page but loses the diagram. Using the information on the next page re-generate the diagram.

18. Fagersta Steelworks Formulation

19. Fagersta Steelworks Complete Setting: Diagram Capacity S1 M1 Demand P Capacity S2 M2

20. Summary • Whitt Window Company • Hotdogs and Buns • Portfolio Optimization • No debt • With debt • Production Planning 1-period • Forward: Setting  Formulation • Backward: Formulation  Setting • Production Planning • Metalco Blends Alloys • Fagersta Steelworks • Backward: Incomplete Setting, Formulation  Complete Setting