Problem IE426 Set #2
Prof Jeff Linderoth
IE 426 Problem Set #2
Due Date: September 26, 2006. 4:30PM. Note: No late homework will be accepted, as I will be discussing solutions on 9/26
1
LP Knowledge
For each of the problems in this section, you should determine which of the characterizations below best describes the linear program. The linear program may fall into more than one category, in which case write all characterizations that apply. A B C D E 1.1 Problem Unique Optimal Multiple Optima Infeasible Unbounded Degenerate
min
-x1 - x2
s.t. 6x1 + 2x2 8 -x1 + x2 0 x1 0 x2 0 1.2 Problem
max
3x1 + x2
s.t. 6x1 + 2x2 8 -x1 + x2 0 x1 0 x2 0 1.3 Problem
min
Problem 1
x2
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IE426 Problem Set #2
Prof Jeff Linderoth
s.t. x1 1 6x1 + 2x2 8 -x1 + x2 0 x1 0 x2 0
2
Least "Squares."
Warning. It is likely that you will need the XPRESS keyword is free. The set of six equations in four variables (1)--(6) does not have a unique solution.1
8x1 - 2x2 + 4x3 - 9x4 = 17 x1 + 6x2 - x3 - 5x4 = 16 x1 - x2 + x3 = 7 x1 + 2x2 - 7x3 + 4x4 = 15 x3 - x4 = 6 x1 + x3 - x4 = 0
(1) (2) (3) (4) (5) (6)
For each equation i, and values of variables x = (x1 , x2 , x3 , x4 ), let ei be the absolute difference (error) between the left hand side and the right hand side. For example, for i = 2 and x = (-5, 3, 1, 4), the error is e2 = |(1)(-5) + 6(3) - (1)(1) - 5(4) - 16| = | - 24| = 24. 2.1 Problem Write a linear programming instance that will minimize the total absolute error: e1 + e2 + e3 + e4 + e5 + e6 2.2 Problem Create the instance you wrote down for Problem 2.1 in the Mosel modeling language, and solve it. What is the minimum total error that can be achieved? What are the values for x? 2.3 Problem For the same instance, write a linear programming instance that will minimize the maximum error in any one equation. Namely find values of x that will min max{e1 , e2 , e3 , e4 , e5 , e6 } for the definition of error ei I gave above.
1
Most six equations with four variables don't.
Problem 2
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IE426 Problem Set #2
Prof Jeff Linderoth
2.4 Problem Create the instance you wrote down for Problem 2.3 in the Mosel modeling language, and solve it. What is the minimum max-error that can be achieved? What are the values for x? 2.5 Problem What equations have the maximum error value that you found in Problem 2.3? What are the dual variables for these equations?
3
The Name Is Bonds. Barry Bonds.
Broker Barry is currently trying to maximize his future net worth using the bond market. A number of bonds are available for purchase or sale, with the bid and ask price of each bond as shown in Table 2. Barry can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of bonds at the bid price. During each of the next three years, the person who sells a bond will pay the owner of a bond the cash payments shown in Table 3. Note that this means that Barry himself will pay and receive cash payments. Barry's goal is to maximize the amount of money he has at the end of the third period. Barry is not allowed to borrow money to meet his obligations to pay interest on the bonds he sold, so in no period can Barry's cash supply fall below zero. Barry currently has $100,000 to invest in bonds. In order to pay the bills, Barry requires that he generates some income at the end of the first year, and at the end of the second year. He needs $10,000 of income at the end of the first year, and $11,000 of income at the end of the second year. Barry receives an interest rate of 3%/year on all of his cash that he has on hand. For example, Barry could buy or sell nothing. At the end of each year, he would have the following amount of money shown in Table 1. Table 1: Barry's Cash if He Stands Pat Year 0 1 2 3 Barry's Cash On Hand 100000 (1.03)100000 - 10000 = 93000 (1.03)93000 - 11000 = 84790 (1.03)84790 = 87338
3.1 Problem a Formulate linear program to maximize Barry's money at the end of the third year given the requirements and restrictions stated above. 3.2 Problem Write your linear program from Problem 3.1 in the Mosel language, and solve it. What is Barry's optimal course of action, and how much money does he have at the end of the third year?
Problem 3 Page 3
IE426 Problem Set #2
Prof Jeff Linderoth
Table 2: Bid and Ask Prices for Bonds (in $/bond) Bond 1 2 3 4 Bid Price 980 970 960 940 Ask Price 990 985 972 954
Table 3: Payouts for Bonds (in $) Year 1 2 3 Bond 1 100 110 1100 Bond 2 80 90 1120 Bond 3 70 80 1090 Bond 4 60 50 1110
3.3 Problem What does the LP suggest if Barry has no limits on the number of bonds he can buy or sell? 3.4 Problem What are the dual variables values on the cash flow constraints in each year? Can you give me an economic interpretation of these dual variables?
4
Sometimes You Feel Like a Nut
Ned's Nuts is a small, Springfield-based, company in the business of making and distribution a variety of different mixed-nut products. We're going to help Ned's Nuts. Nut Peanuts Walnuts Almonds Price ($/lb.) 0.2 0.35 0.5 Availability (lbs.) 300 Unlimited 200
Table 4: Price and Availability of Nuts
4.1 Problem Ned's Noble Nut Blend (NNNB) is required to contain no more than 25% peanuts, and must contain at least 40% almonds. The demand for NNNB this month is forecast at 600 pounds, and NNNB sells for $0.80/lb. The prices and availability of the individual nuts this month is shown in Table 4. Formulate a linear program that will tell Ned how to maximize his profit in producing NNNB.
Problem 4 Page 4
IE426 Problem Set #2
Prof Jeff Linderoth
4.2 Problem Build your instance from Problem 4.1 in the Mosel programming language and solve it. What is the solution and maximum profit that Ned can make on NNNB this month? 4.3 Problem Ned would like to incorporate into your analysis from Problem 4.1 a second mixed nut line: Ned's Imperial Nut Blend (NINB). NINB can contain no more than 60% peanuts and must contain at least 20% almonds. NINB sells for $0.40/lb. Ned forecasts that the total demand for nuts (NINB + NNNB) will be 700 pounds this month. Update your linear program from Problem 4.1 so that you can tell Ned how to maximize his profit on the two nut lines: NINB and NNNB. 4.4 Problem Build your instance from Problem 4.3 in the Mosel programming language and solve it. What is the solution and maximum profit that Ned can make on NNNB and NINB this month? 4.5 Problem
T HIS ONE IS H ARD
Ned is high-diddily impressed with your vast knowledge of Nut Mixing, and he would like you to help him out on his Nut Mixing and Scheduling over the next three months. Being a highly volatile market, the price of nuts varies month-to-month, and the prices of the nuts (in $/lb.) over the next three months is shown in Table 5. The total demand (in lb.) that Ned expects for his nut blends in each month is shown in Table 5 as well. As before, there is still (the same) limits on availability of individual nuts (seen in Table 4) each month. Month 2 3 0.15 0.40 0.50 0.30 0.45 0.55 500 1000
Nut Peanuts (Price) Walnuts (Price) Almonds (Price) Nut Mix Demand
1 0.2 0.35 0.5 700
Table 5: Price of Nuts per Month
Ned can choose to store plain nuts (not mixed) in inventory for a cost of $0.02/lb. Ned currently has only 20 pounds of walnuts on hand, and nothing else in inventory. Because of a contractual obligation, Ned must produce at least 200 pounds of NNNB and 100 pounds of NINB each month. Formulate a linear program that will help Ned meet his contractual obligations, and maximize his profit in mixing NINB and NNNB. 4.6 Problem Build your instance from Problem 4.5 in the Mosel programming language and solve it. What is the solution and maximum profit that Ned can make on NNNB and NINB over the next three months.
Problem 4 Page 5