1. Weingartner’s Horizon Model

2. Primal Model Description

3. Dual Model Description

4. Numerical Example  Budget Limits: $3,000, $5,000, and $4,800 at time = 0,1,and 2, respectively.  Lending and Borrowing Rate: 20% without limit on both.  Planning Horizon: 2 years. Year123456 0($1,200) ($2,000)($2,500)($3,000)$1,000 1-2,000-2,400-2,100-1,300900-700 22,0002,5003,0002,0001,400-700 32,9003,5672,6212,0001,6000 402,0002110 5 02,29600 $2,4172,9722,1844,3841,4800

5. Primal Model Formulation

6. Solution to the Example Variable Type Objective Function $17,786 Project Selection x 1 1.0 x 2 1.0 x 3 0.0 x 4 1.0 x 5 0.0 x 6 0.0 Lending v 0 0.0 v 1 0.0 v 2 $8,012 Borrowing W 0 $1,700 W 1 2,740 W 3 0.0 Budget Constraint Dual Variable ρ 0 1.44 ρ 1 1.20 ρ 2 1.00 Project Upper Bound Dual Variable μ 1 577 μ 2 865 μ 3 0 μ 4 1,224 μ 5 0 μ 6 0

7. Interpretation ● ρ n *: a compound factor reflecting the value, at t=N, of an additional dollar at t=n.