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The current structure of Landmark University (LU) was induced by raising a generation of solution providers through a qualitative and life-applicable training system that focuses on values and creative knowledge by making it more responsive and relevant to the modern-day demands of demonstration, industrialization and development. The challenge facing Landmark University is the question of which of its numerous projects they should invest to give maximum output with minimum input. In this paper, we maximize the Net Present Value (NPV) and maintain the net discount cash overflow of each project per period as contained and extracted as the secondary data of cash inflows of the Landmark University (LU) monthly financial statement and annual reports from 2012 to 2017 of which the documents have been regrouped as small and large scale projects as many enterprises make more use of the trial-and-error method and as such firms have been finding it difficult in allocating scarce resources in a manner that will ensure profit maximization and/or cost minimization with a simple and accurate decision making by the company through an optimization principle in selecting LU project under multi-period capital rationing using linear programming (LP) and integer programming (IP). The annual net cash flow which is the difference between the cash inflows and cash outflows during each period for the project was estimated and recorded. The discount factors were estimated at cost of capital of 10% for each cash flow per period with the corresponding NPV at 10% which revealed that the optimal decision achieves maximum returns of $110 × 102 and this assisted the project manager to select a large number of the variable projects that can maximize the profit which is far better than relying on an ad-hoc judgmental approach to project investment that could have cost 160 × 102 for the same project. Sensitivity analysis on the project parameters are also carried out to test the extent to which project selection is sensitive to changes in the parameters of the system revealed that a little reduction and or addition of reduced cost by certain amount or percentages to its corresponding coefficient in the objective function effect no changes in the shadow prices with solution values for variables (
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_{1}), (
*x*
_{4}), (
*x*
_{5}) and the optimal objective function.

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In this paper, we develop and formulate Linear and Integer Programming models to solve a multi-period capital rationing (MCR) with divisible and indivisible project problems. The model seeks to produce optimum solution quantities (i.e. total NPV) and the shadow cost (i.e. opportunity cost of building constraints).

We consider the following standard form of linear programming:

Maximize F = ∑ j = 1 n C i X j Subjectto ∑ j = 1 n a ( i j ) X j = b i , i = 1 , 2 , ⋯ , n l j ≤ X ≤ u j , j = 1 , 2 , ⋯ , n (1)

where C j is the n objective function coefficient, a ( i j ) and b are parameters in the m linear inequality constraints and l j and u j are lower and upper bounds with l j ≤ u j . Both l j and u j may be positive or negative.

The specified Linear Programming model for the attainment of the objective function is as follows:

Minimize Z = ∑ C j X i (2)

Subject to x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 = b 1 a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 + a 15 x 5 + a 16 x 6 + a 17 x 7 + a 18 x 8 ≤ b 2 a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 + a 2 5 x 5 + a 26 x 6 + a 27 x 7 + a 28 x 8 ≤ b 3 a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 + a 35 x 5 + a 36 x 6 + a 37 x 7 + a 38 x 8 ≤ b 4

a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 + a 45 x 5 + a 46 x 6 + a 47 x 7 + a 48 x 8 b 5 a 51 x 1 + a 52 x 2 + a 53 x 3 + a 54 x 4 + a 55 x 5 + a 56 x 6 + a 57 x 7 + a 58 x 8 b 6 a 61 x 1 + a 62 x 2 + a 63 x 3 + a 64 x 4 + a 65 x 5 + a 66 x 6 + a 67 x 7 + a 68 x 8 b 7 a 71 x 1 + a 72 x 2 + a 73 x 3 + a 74 x 4 + a 75 x 5 + a 76 x 6 + a 77 x 7 + a 78 x 8 b 8

a 81 x 1 + a 82 x 2 + a 83 x 3 + a 84 x 4 + a 85 x 5 + a 86 x 6 + a 87 x 7 + a 88 x 8 b 9 a 91 x 1 + a 92 x 2 + a 93 x 3 + a 94 x 4 + a 95 x 5 + a 96 x 6 + a 97 x 7 + a 98 x 8 b 10 a 101 x 1 + a 102 x 2 + a x 103 3 + a 104 x 4 + a 105 x 5 + a 106 x x + a 107 x 7 + a 108 x 8 b 11 x 1 0 , i = 1 , 2 , 3 , ⋯ , n (3)

Linear and Integer Programming Model for the Project Selection ProblemIn this Linear Programming model, we let Q j be the capital available in LU for investment at time period t. Then the problem facing LU is to determine which project or portion of the project it should initiate with Q j . Thus the following algorithms will be strictly follow in determine and solving the challenges.

1) Algorithms for Linear Programming

Step a). Determine the project’s NPV using

β j ( N P V ) = ∑ i = 1 n [ C t ( 1 + r ) t ] (4)

where t = 0 , 1 ; j = 1 , 2 , 3 , 4 , 5 C is the cash flows

We proposed that the NPV of five (5) projects to be initiated as Agriculture (A) = β 1 , Electrification (B) = β 2 , Lecture Hall (C) = β 3 , Lab. Equipment (D) = β 4 , Staff/Student Quarters (E) = β 5 .

Step b). Formulate the Linear Programming problems by defining the objective functions, decision variables and the constraints.

Thus:

Maximize Z = β 1 X A + β 2 X B + β X 3 C + β 4 X D + β 5 X E

While the decision variables ( X j ) are characterized as follows

X A is the proportion of project A to be initiated when j = 1

X B is the proportion of project B to be initiated when j = 2

X C is the proportion of project C to be initiated when j = 3

X D is the proportion of project D to be initiated when j = 4

X E is the proportion of project E to be initiated when j = 5

2) Algorithms for Integer Programming for the project selection problem

For Q j be the capital available in LU for investment at time period t and the problem facing LU is to determine which project or portion of the project it should initiate with Q j . Thus LU must take into consideration that:

a) It cannot invest in all N projects suitable for investment which run for n year.

b) The project characteristics show that ∑ i d ( i , j ) is greater than R j where d ( i , j ) is the least requirement for j projects and R j is the capital for investment.

c) All the projects and the constraints are independent on one another.

d) Equal investment opportunities are assumed for the project for each period.

e) The cash flows, resources and constraints are well known.

Our main decision problem is to determine which project the LU should select in order to maximize the total returns. To formulate this Integer Programming, we follow these algorithms:

Step i). Define the decision variable as follows

Let X j = { 1 , if LU invest in project j 0 , if LU does not invest in project j (5)

j = 1 , 2 , ⋯ , n

where X j are integer variable which takes one of two possible values ( 0 , 1 ) and represents a binary decision.

Step ii). Define the constraints as follows

We let d ( i , j ) be the capital requirement for j project, R j be available capital for j project for each year.

∑ j = 1 N d ( i , j ) X j ≤ R i for j = 1 , 2 , ⋯ , N ; i = 1 , 2 , ⋯ , m (6)

Then the constraints relating to availability of capital funds each year are:

Step. iii) Objective function.

We let the total profit be

∑ j = 1 N P j X j (7)

Maximize

Z = ∑ j = 1 N P j X j (8)

Subject to

∑ j = 1 N d ( i , j ) X j ≤ R j (9)

Since the problem facing LU is to determine which project or portion of the project, it should initiate with Q j and subject to these constraints, they were faced with budgetary limitation. Thus

1) For the capital project at the initial time (t) = 0,

a X ( 1 , 1 ) A + a ( 1 , 2 ) X B + a ( 1 , 3 ) X C + a ( 1 , 4 ) X D + a ( 1 , 5 ) X E ≤ Q 1 (10)

2) For the capital project at the take up time (t) = 1,

a X ( 2 , 1 ) A + a ( 2 , 2 ) X B + a ( 2 , 3 ) X C + a ( 2 , 4 ) X D + a ( 2 , 5 ) X E ≤ Q 2 (11)

3) Then we specified the following proportion constraints to ensure that a project is not accepted more than once or negative projects are not accepted:

X A , X B , X C , X D , X E ≤ 1

X A , X B , X C , X D , X E ≤ 0

where a ( i , j ) are cash flows for each period and for each project.

We then transform the formula into the compact form as:

Maximize Z = β 1 X 1 + β 2 X 2 + β X 3 3 + β 4 X 4 + β 5 X 5 Subject to:

a ( 1 , 1 ) X 1 + a ( 1 , 2 ) X 2 + a ( 1 , 3 ) X 3 + a ( 1 , 4 ) X 4 + a ( 1 , 5 ) X 5 ≤ Q 1 a ( 2 , 1 ) X 1 + a ( 2 , 2 ) X 2 + a ( 2 , 3 ) X 3 + a ( 2 , 4 ) X 4 + a ( 1 , 5 ) X 5 ≤ Q 2 X 1 ≤ 1 X 2 ≤ 1 X 3 ≤ 1 X 4 ≤ 1 X 5 ≤ 1 a ( 1 , 1 ) X 1 + a ( 1 , 2 ) X 2 + a ( 1 , 3 ) X 3 + a ( 1 , 4 ) X 4 + a ( 1 , 5 ) X 5 ≤ Q 1 a ( 2 , 1 ) X 1 + a ( 2 , 2 ) X 2 + a ( 2 , 3 ) X 3 + a ( 2 , 4 ) X 4 + a ( 1 , 5 ) X 5 ≤ Q 2 X 1 ≤ 0 X 2 ≤ 0 X 3 ≤ 0 X 4 ≤ 0 X 5 ≤ 0 } (12)

Generally, we then have the following form of equation:

Maximize Z = ∑ j = 1 N B j X j (13)

Subject to ∑ j = 1 N a ( i , j ) X j = Q i 0 ≤ X j ≤ 1 , i = 1 , 2 , ⋯ , m ; j = 1 , 2 , ⋯ , n } (14)

where ( a ( i , j ) , Q j , β j ) are given and n is the number of the projects to be invested and Z constitutes the objective function.

Information about cash inflows of the LU small and capital projects with distribution of capital requirements for the Small Project from 2012-2017 as shown in

From Equations (7) and (8) we applied LP model to LU capital rationing data in

Maximize Z = ∑ j = 1 N B j X j

Subject to ∑ j = 1 N a ( i , j ) X j = Q i 0 ≤ X j ≤ 1 , i = 1 , 2 , ⋯ , M ; j = 1 , 2 , ⋯ , N }

Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | NPV@10% | P.1 |
---|---|---|---|---|---|---|---|---|

Project period | 0 | 1 | 2 | 3 | 4 | 5 | ||

Agriculture (x_{1}) | 100 | 100 | 200 | 400 | 600 | 500 | 264 | 0.83 |

Electrification (x_{2}) | 400 | 500 | 1000 | 1200 | 1400 | 1200 | 719 | 0.80 |

Lecture hall (x_{3}) | 250 | 200 | 360 | 500 | 400 | 0 | 237 | 0.92 |

Lab. equipment (x_{4}) | 30 | 50 | 60 | 60 | 150 | 90 | 217 | 0.33 |

Staff/std quarters (x_{5}) | 10 | 20 | 10 | 0 | 30 | 50 | 72 | 0.72 |

Discount factors | 1.00 | 0.909 | 0.826 | 0.751 | 0.683 | 0.621 | ||

Capital Limitation Q_{1} | 550 | 500 | 450 | 400 | 650 | 700 |

Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | Capital returns |
---|---|---|---|---|---|---|---|

Project period | 0 | 1 | 2 | 3 | 4 | 5 | |

Machineries (x_{1}) | 50 | 30 | 20 | 60 | 40 | 20 | 50 |

Refuse facility (x_{2}) | 10 | 80 | 20 | 20 | 30 | 60 | 30 |

Borehole water (x_{3}) | 15 | 15 | 30 | 40 | 60 | 0 | 50 |

Shield (x_{4}) | 10 | 40 | 10 | 10 | 0 | 10 | 10 |

Bus stop (x_{5}) | 10 | 0 | 10 | 20 | 50 | 10 | 20 |

Available capital | 80 | 145 | 90 | 100 | 165 | 80 |

Decision variables | Solution variables | Unit cost or profit | Total contribution | Shadow price | Reduction cost |
---|---|---|---|---|---|

Agriculture (x_{1}) | 1.00 | 264 | 264 | 0 | 0 |

Electrification (x_{2}) | 0.662 | 719 | 474.56 | 1.438 | 0 |

Lecture hall (x_{3}) | 0 | 237 | 0 | 0 | 50.6 |

Lab. equipment (x_{4}) | 1.00 | 217 | 217 | 0 | 0 |

Staff/Std. quarters (x_{5}) | 1.00 | 72 | 72 | 0 | 0 |

Pmax. | 1,509 | 1,027.56 |

Thus we have:

Maximize Z = 264 X 1 + 719 X 2 + 237 X 3 + 217 X 4 + 72 X 5

Subject to 100 X 1 + 400 X 2 + 250 X 3 + 30 X 4 + 10 X 5 ≤ 550

100 X 1 + 500 X 2 + 200 X 3 + 50 X 4 + 20 X 5 ≤ 500

200 X 1 + 1000 X 2 + 360 X 3 + 60 X 4 + 10 X 5 ≤ 450

400 X 1 + 1200 X 2 + 500 X 3 + 60 X 4 ≤ 400

600 X 1 + 1400 X 2 + 400 X 3 + 150 X 4 + 30 X 5 ≤ 650

500 X 1 + 1200 X 2 + 90 X 4 + 50 X 5 ≤ 700

0 ≤ X j ≤ 1 , j = 1 , 2 , 3 , 4 , 5

Here the stability or robustness of the model is tested by a slight change in the technological coefficients in order to determine the redundancy or otherwise of one of the constraints, this helps make better recommendations and reduce errors in making decisions. The redundancy of a constraint is also put into test and the solution compared to the original LP problem as shown in _{3}) shows the amount by which the objective function coefficient for the variable (x_{3}) should be change to make it a non-zero. Hence the coefficient of (x_{1}) in the objective function is altered by −50.6 and the LP problem will be resolved to yield.

Addition of the reduced cost of 50.6 on the row of variable (x_{2}) to its corresponding coefficient in the objective function effect no changes in the shadow prices with solution values for variables (x_{1}), (x_{4}), (x_{5}) and the optimal objective function. However, there were sharp variations in some optimal solution values. The coefficient of variables (x_{2}) decreased from 0.662 to 0.38042 while (x_{3}) increases from 0 to 0.69895, increasing the NPV per unit on variable (x_{3}), impact a sharp change on the optimal solution. Given the sensitivity analysis of one or more of the key factors of project like this, the LU management’s task is to decide whether the project is commendable and worthwhile.

From

Maximize Z = ∑ j = 1 N P j X j

Subject to ∑ j = 1 N d ( i , j ) X j ≤ R j , X j = 0 ; j = 1 , 2 , ⋯ , N

0 ≤ X j ≤ 1 , i = 1 , 2 , ⋯ , m ; j = 1 , 2 , ⋯ , n

Thus:

Maximize Z = 50 X 1 + 30 X 2 + 50 X 3 + 10 X 4 + 20 X 5

Subject to 50 X 1 + 10 X 2 + 15 X 3 + 10 X 4 + 10 X 5 ≤ 80

X 1 + 80 X 2 + 15 X 3 + 40 X 4 + 0 X 5 ≤ 145

20 X 1 + 20 X 2 + 30 X 3 + 10 X 4 + 10 X 5 ≤ 90

20 X 1 + 20 X 2 + 40 X 3 + 10 X 4 + 20 X 5 ≤ 100

60 X 1 + 30 X 2 + 60 X 3 + 0 X 4 + 50 X 5 ≤ 145

40 X 1 + 60 X 2 + 0 X 3 + 0 X 4 + 10 X 5 ≤ 80

X j = 0 or 1 , j = 1 , 2 , 3 , 4 , 5

The above LP was solved using MATLAB and the results of the binary decision are shown in

Decision variables | Solution variables | Unit cost or profit | Total contribution | Shadow price | Reduction cost |
---|---|---|---|---|---|

Agriculture (x_{1}) | 1.00000 | 264 | 264 | 0 | 0 |

Electrification (x_{2}) | 0.38042 | 719 | 273.52 | 1.438 | 0 |

Lecture hall (x_{3}) | 0.69895 | 287.6 | 201.20 | 0 | 0 |

Lab. equipment (x_{4}) | 1.00000 | 217 | 217 | 0 | 0 |

Staff/Std. quarters (x_{5}) | 1.00000 | 72 | 72 | 0 | 0 |

Pmax. | 1,027.72 |

Decision variables | Solution variables | Unit cost or profit | Total contribution | Reduction cost |
---|---|---|---|---|

Machineries (x_{1}) | 1 | 50 | 50 | 0 |

Refuse facility (x_{2}) | 0 | 30 | 0 | 30 |

Borehole water (x_{3}) | 1 | 50 | 50 | 0 |

Shield (x_{4}) | 1 | 10 | 10 | 0 |

Bus stop (x_{5}) | 0 | 20 | 0 | 20 |

Pmax. | 160 | 110 |

The optimal decision is to choose (x_{1}), (x_{3}), (x_{4}), while LU can provide (x_{2}), (x_{5}) with n capital for the next five years unless the LU investment is reviewed. The optimal decision achieves maximum returns of 110 × 10^{2}. It is evident that the model has assisted the project manager to select a large number of the variable projects that can maximize profit. This is larger than relying on an ad-hoc judgmental approach to project investment that could have cost 160 × 10^{2} for the same project.

In this paper we have successfully examined optimization principles and its applications in selecting potential projects in LU in order to maximize the returns and the profits from the batch of projects by maximizing the Net present Value (NPV) and maintain the net discount cash overflow for each project per period as contained in data collected from LU monthly financial statement and annual report from 2011 to 2016 revealed that LU will incur 1509 × 10^{2} as unit cost or profit for a total contribution of 1027.56 × 10^{2}.

The discount factors were estimated at cost of capital of 10% for each cash flow per period with the corresponding NPV at 10% which revealed that the optimal decision achieves maximum returns of $110 × 10^{2} and this will help the project manager to select a large number of the variable projects that can maximize the profits which is far better than relying on an ad-hoc judgmental approach to project investment that could have cost 160 × 10^{2} for the same project.

Sensitivity analysis on the project parameters revealed that a little reduction and/or addition of reduced cost by certain amount or percentages to its corresponding coefficient in the objective function effect changes in the shadow prices with solution values for variables (x_{1}), (x_{4}), (x_{5}) and the optimal objective function. However, there were sharp variations in some optimal solution values where the coefficient of variables (x_{2}) decreased while (x_{3}) increased and an increase in NPV per unit on variable (x_{3}), has a sharp change on the optimal solution.

This will give some guidance to the firm management in their consideration of many options with regards to the limited resources and for the decision-making process.

The author declares no conflicts of interest regarding the publication of this paper.

Oladejo, N.K. (2019) Application of Optimization Principle in Landmark University Project Selection under Multi-Period Capital Rationing Using Linear and Integer Programming. Open Journal of Optimization, 8, 73-82. https://doi.org/10.4236/ojop.2019.83007^{ }