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

utdallas.edu /~ metin Page 2 Whitt Window Company - Setting

utdallas.edu /~ metin Page 3 Whitt Window Company - Formulation

utdallas.edu /~ metin Page 4 Hotdogs and Buns - Setting

utdallas.edu /~ metin Page 5 Hotdogs and Buns - Formulation

utdallas.edu /~ metin Page 6 Linear Programming formulations can be used to select a desirable bond portfolio. u ProMax has raised $8,000,000 to invest into four bonds. u The average annual return, the worst-case annual return and the duration of each bond is below: Average ReturnWorst-case ReturnDuration Bond 114%5%2 Bond 29%8%1 Bond 312%10%3 Bond 415%12%3 Portfolio Optimization - Setting

utdallas.edu /~ metin Page 7 Portfolio Optimization - Formulation

utdallas.edu /~ metin Page 8 Financial institutions use debt to increase their investment capital. u ProMax has raised $8,000,000 from a private equity fund to invest into four bonds. u The average annual return, the worst-case annual return and the duration of each bond is below: u 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 u Formulate a linear program to maximize the average annual return of a portfolio over a year. Average ReturnWorst-case ReturnDuration Bond 17%3%1 Bond 25%4%1 Bond 310%1%1 Bond 412%-2%1 Portfolio Optimization with Debt - Setting

utdallas.edu /~ metin Page 9 Portfolio Optimization with Debt - Formulation

utdallas.edu /~ metin Page 10 Production Planning 1-period - Setting Input and 1 hr: A 2As and 2 hrs: B 1B and 3 hrs: CSell C at $100 Sell B at $60 Sell A at $15 u A company produces products A, B, C u Products A, B, C are sold at prices $15, $60 and $100 in unlimited numbers u 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 u A total of 60 labor hours are available u Formulate a linear program to maximize profit

utdallas.edu /~ metin Page 11 Production Planning 1-period - Formulation

utdallas.edu /~ metin Page 12 Production Planning 1-period – Formulation  Setting Input and ? hr: Product 1 ? Product 1 and ? hrs: Product 2 ? Product 2 and ? hrs: Product 3 Sell Product 3 at $? Sell Product 2 at $? Sell Product 1 at $? u A total of ? labor hours are available

utdallas.edu /~ metin Page 13 Production Planning - Setting

utdallas.edu /~ metin Page 14 Production Planning - Formulation

utdallas.edu /~ metin Page 15 Metalco Blends Alloys - Setting

utdallas.edu /~ metin Page 16 Metalco Blends Alloys - Formulation

utdallas.edu /~ metin Page 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.

utdallas.edu /~ metin Page 18 Fagersta Steelworks Formulation

utdallas.edu /~ metin Page 19 Fagersta Steelworks Complete Setting: Diagram M1 M2 S1 S2 P Capacity Demand

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