The treasurer has forecast the cash needs for the next 5 years. He has cash requirements of 10000 at the beginning of each of the first 2 years and 20000 at the beginning of the remaining 3 years.

The treasurer is considering the following investments: - intermediate bonds have a return, after 2 years, of 1.47 for each 1.00 invested. - long-term bonds have a return, after 3 years, of 1.78 for each 1.00 invested. - short-term bonds have a return, after 1 year, of 1.20 for each 1.00 invested.

1. Supposing the corporation currently has 100000 available cash, the treasurer wishes to maximize the amount of cash at the end of the planning period, at which time no investment will be outstanding.

   ?- cash(100000, Out), linear_maximize(Out, Max).

   Max = 0r18297283/147 = 124471

2. Using the same model, determine the minimal start capital necessary to meet the cash requirements.

   ?- cash(In, _), linear_minimize(In, Min).

   Min = 0r100833685000/1923201 = 52430

3. Using the same model, determine the minimal start capital the company must have in order to have at the end of the planning period the same amount of cash (while meeting the cash requirements during the 5 years).

   ?- cash(In, In), linear_minimize(In, Min).

   Min = 0r100833685000/1188201 = 84862

:- module(cash).
:- export([cash/3]).
:- begin_module(cash).
:- import(const_linear).

cash(In, Out1) :-
	V = [	In, C1, C2, C3, C4, Out,
		Profit1, Profit2, Profit3, Profit4, Profit5,
		I1, I2, I3, I4, I5,
		I11, I12, I13,
		I21, I22, I23,
		I31, I32, I33,
		I41, I42,
		I51
	    ],
	all_positive(V),
	C1 + 10000 + I1 $= In + Profit1,
	C2 + 10000 + I2 $= C1 + Profit2,
	C3 + 20000 + I3 $= C2 + Profit3,
	C4 + 20000 + I4 $= C3 + Profit4,
	Out + 20000 + I5 $= C4 + Profit5,

	I1 $= I11 + I12 + I13,
	I1 + 10000 $=< In,
	Profit1 $= I11 * 0r120/100,

	I2 $= I21 + I22 + I23,
	I2 + 10000 $=< C1,
	Profit2 $= I21 * 0r120/100 + I12 * 0r147/100,

	I3 $= I31 + I32 + I33,
	I3 + 20000 $=< C2,
	Profit3 $= I31 * 0r120/100 + I22 * 0r147/100 + I13 * 0r178/100,

	I4 $= I41 + I42,
	I4 + 20000 $=< C3,
	Profit4 $= I41 * 0r120/100 + I32 * 0r147/100 + I23 * 0r178/100,

	I5 $= I51,
	I5 + 20000 $=< C4,
	Profit5 $= I51 * 0r120/100 + I42 * 0r147/100 + I33 * 0r178/100,
	linear_maximize(Out,Out1),
   	write(V).