utdallas.edu /~ metin page 1 formulation problems outline u whitt window company u hotdogs and buns...
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utdallas.edu /~
metin
Page 1Formulation 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
utdallas.edu /~
metin
Page 6
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.
Average Return Worst-case Return Duration
Bond 1 14% 5% 2
Bond 2 9% 8% 1
Bond 3 12% 10% 3
Bond 4 15% 12% 3
Portfolio Optimization - Setting
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metin
Page 7
Is there a missing constraint? How to make sure that the total money available for investing is 8,000,000?
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
Portfolio Optimization - Formulation
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metin
Page 8
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.
Average Return Worst-case Return Duration
Bond 1 7% 3% 1
Bond 2 5% 4% 1
Bond 3 10% 1% 1
Bond 4 12% -2% 1
Portfolio Optimization with Debt - Setting
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metin
Page 9
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
Portfolio Optimization with Debt - Formulation
utdallas.edu /~
metin
Page 10Production Planning 1-period - Setting
Input and 1 hr: A
2As and 2 hrs: B
1B and 3 hrs: C Sell C at $100
Sell B at $60
Sell A at $15
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
A total of 60 labor hours are available Formulate a linear program to maximize profit
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metin
Page 11
Let ix be the number of units produced corresponding to A, B, C;
Objective function: Max 15( BA xx 2 )+60( CB xx )+100 Cx
Constraints: CBA xxx 32 60
BA xx 2 0
CB xx 0
0ix , CBAi ,,
Production Planning 1-period - Formulation
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metin
Page 12
Let ix be the number of units produced, 3,2,1i corresponding to products;
Objective function: Max 8( 21 3xx )+56( 32 2xx )+128 3x
Constraints: 321 42 xxx 40
21 3xx 0
32 2xx 0
0ix , 3,2,1i
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 $?
A total of ? labor hours are available
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metin
Page 14
Let ix be the number of products produced in Month i , 4,3,2,1i ;
Let iy be the number of ending inventory produced in Month i , 4,3,2,1i ;
Objective function: Min 4321 7485 xxxx +2( 321 yyy )-6 4y
Constraints: 00 y
iiii dxyy 1 , 4,3,2,1i
0ix , 0iy , 4,3,2,1i
501 d , 652 d , 1003 d , 704 d
Production Planning - Formulation
utdallas.edu /~
metin
Page 17Fagersta SteelworksIncomplete 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 19Fagersta SteelworksComplete Setting: Diagram
M1
M2
S1
S2
P
Capacity
Capacity
Demand
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metin
Page 20Summary
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
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