Question

09) I now have $100 and three choices of investment during the next four years. For every dollar invested: Investment A yields $0.10 a year after the investment and yie years after the investment, Investment B yields $0.20 a year S1.10 two years after the investment, Investment C yields $1.50 th Investments A and C are available in the beginning of year I and 2 while B is available in the lds $1.30 three after the investment and yields ree years after the investment. During each year, uninvested casl funds, which yield 6% interest per year. At most $50 may be placed in each of investments A, B, and C in each year. Formulate an LP to maximize my cash on hand four years from now.

Answer 1

Let,

Bi = beginning cash at year-i beginning
Ei = ending cash at year-i end
Ai = investment in A in year-i beginning
Bi = investment in B in year-i beginning
Ci = investment in C in year-i beginning
Mi = investment in money market in year-i beginning
Si = total investment at the year-i beginning
Ri = total income at the year-i end

The situation is best depicted in the following tabular form.

	Year
	1	2	3	4
Beginning cash (Bi)	B1 = 100	B2 = E1	B3 = E2	B4 = E3
Invest in,
A	A1	A2	A3	A4
B	B1	B2	B3	B4
C	C1	C2	C3	C4
Money market	M1	M2	M3	M4
Total invested (Si)	S1	S2	S3	S4
Income from,
A	0.1A1	0.1A2	0.1A3 + 1.3A1	0.1A4 + 1.3A2
B	0.2B1	0.2B1 + 1.1B1	0.2B3 + 1.1B2	0.2B4 + 1.1B3
C	0	0	1.5C1	1.5C2
Money market	0.06M1	0.06M2	0.06M3	0.06M4
Total income (Ri)	R1	R2	R3	R4
Ending cash (Ei)	E1=B1+R1-S1	E2=B2+R2-S2	E3=B3+R3-S3	E4=B3+R4-S4

LP formulation

Maximize. Z = E4

Subject to,

B1 = 100
B_i = E_i-1 for i=2,3,4
Si = Ai + Bi + Ci + Mi for i=1,2,3,4
Ei = Bi + Ri - Si for i=1,2,3,4

R1 = 0.1A1 + 0.2B1 + 0.06M1
R2 = 0.1A2 + 0.2B1 + 1.1B1 + 0.06M2
R3 = 0.1A3 + 1.3A1 + 0.2B3 + 1.1B2 + 1.5C1 + 0.06M3
R4 = 0.1A4 + 1.3A2 + 0.2B4 + 1.1B3 + 1.5C2 + 0.06M4

Bi ≥ Si

Ai ≤ 50; Bi ≤ 50; Ci ≤ 50

Bi, Ei, Si, Ai, Bi, Ci, Mi, Ri ≥ 0