Question

As part of the settlement for a class action lawsuit, Hoxworth Corporation must provide sufficient cash to make the following annual payments (in thousands of dollars):

Year	1	2	3	4	5	6
Payment	195	220	270	320	350	470

The annual payments must be made at the beginning of each year. The judge will approve an amount that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to savings (at 3.5% annually) and government securities, at prices and rates currently quoted in The Wall Street Journal.

Hoxworth wants to develop a plan for making the annual payments by investing in the following securities (par value = $1000). Funds not invested in these securities will be placed in savings.

Security	Current Price	Rate (%)	Years to Maturity
1	$1045	6.85	3
2	$1000	5.525	4

Assume that interest is paid annually. The plan will be submitted to the judge and, if approved, Hoxworth will be required to pay a trustee the amount that will be required to fund the plan.

1. Use linear programming to find the minimum cash settlement necessary to fund the annual payments.
   

   Let
   	F = total funds required to meet the six years of payments
   	G1 = units of government security 1
   	G2 = units of government security 2
   	Si = investment in savings at the beginning of year i

   
   Note: All decision variables are expressed in thousands of dollars.
   
   If required, round your answers to five decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
   

   Min	F
   s.t.
   	______F	+	_____G1	+	_____G2	+	_____S1	=	______
   	_____G1	+	_____G2	+	_____S1	+	_____S2	=	______
   	_____G1	+	_____G2	+	_____S2	+	_____S3	=	______
   	_____G1	+	_____G2	+	_____S3	+	_____S4	=	______
   			_____G2	+	_____S4	+	_____S5	=	______
   					_____S5	+	_____S6	=	______

   
   Round your answer to the nearest dollar. If an amount is zero, enter "0".
   

   Current investment required	$_____
   Investment in government security 1	$ _____
   Investment in government security 2	$ ____
   Investment in savings for year 1	$_____
   Investment in savings for year 2	$ _____
   Investment in savings for year 3	$ _____
   Investment in savings for year 4	$ _____
   Investment in savings for year 5	$ _____
   Investment in savings for year 6	$ ____

2. Use the dual value to determine how much more Hoxworth should be willing to pay now to reduce the payment at the beginning of year 6 to $400,000. Round your answer to the nearest dollar.
   
   $ _____
   
3. Use the dual value to determine how much more Hoxworth should be willing to pay to reduce the year 1 payment to $150,000. Round your answer to the nearest dollar.
   
   Hoxworth should be willing to pay anything less than $ _____ .
   
4. Suppose that the annual payments are to be made at the end of each year. Reformulate the model to accommodate this change.
   
   Note: All decision variables are expressed in thousands of dollars.
   
   If required, round your answers to five decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
   

   Min	F
   s.t.
   1)	_____F	+	_____G1	+	_____G2	+	_____S1	=	________
   2)	____G1	+	_____G2	+	_____S1	+	_____S2	=	________
   3)	____G1	+	_____G2	+	_____S2	+	_____S3	=	_______
   4)	____G1	+	_____G2	+	_____S3	+	_____S4	=	_______
   5)			_____G2	+	_____S4	+	_____S5	=	_______
   6)					_____S5	+	_____S6	=	_______
   7)					_____S6	+	_____S7	=	_______

   
   How much would Hoxworth save if this change could be negotiated? Round your answer to the nearest dollar.
   
   $ _____

Answer 1

a) LP model and solution is as follows

b) Sensitivity report is as follows

Shadow price of year 6 payment is 0.77618. Therefore to reduce the year 6 payment from 470 to 400 (1000s), Hoxworth should be willing to pay additional = (470000-400000)*0.77618 = $ 54333

c) Shadow price of year 1 payment is 1. Therefore to reduce year 1 payment from 195000 to 150000, Hoxworth should be willing to pay now = (195000 - 150000)*1 = 45000

d) revised LP model is following

Saving = 1579578 - 1506164 = $ 73414