Capital budgeting models involve the allocation of limited investment funds among a set of competing investment alternatives. The alternatives available in any given period are each characterised by an investment cost and some benefit associated with it. The determination of investment costs usually is relatively easy. Estimating benefits can be more difficult, especially when projects are characterised by less. tangible returns. The problem is to select the set of alternatives which will maximise overall benefits subject to budgetary constraints and other constraints which may affect the choice of projects.
Example. A central government project of nonconventional sources of energy has 950 crores to give in the form of grants. Six projects submitted by the state governments have been finalised after detailed evaluation by the experts. Estimated benefits from each project in next 10 years are shown in the table below. For example the value of 5· 5 associated with Project A suggests that each rupee invested in this project will return a net (after subtracting the rupee investment) benefit of Rs. 5.50 over the next 10 years. The table also shows the state government request for funding their projects. The government can grant any amount up to the indicated maximum for a given project. The central government wants project E, which is considered very important should get at least 50% of the funds asked for. Also as a policy Projects A and B, being natural priority projects should be given at least 250 crores put together. Formulate the problem as a LP model.
Project 
Net benefit/Rupee invested 
Request for funds (Crores of rupees) 
A 
5.5 
150 
B 
3.5 
250 
C 
2.4 
120 
D 
2.0 
100 
E 
6 
500 
F 
3.1 
160 
of the cost or effort of shipping unit is specified for eachorigin destination combination. This may take the shape of Rupee cost, distance between the two points or time required to move from one point to another. A typical problem is concerned with determining the number of units which should be supplied from each origin to each destination. The objective is to minimise the total transportation or delivery costs while ensuring that
(a) The number of units transported from any origin does not exceed the number of units available at that origin and
(b) The demand at each destination is satisfied.
Example. A town located at high altitude has two locations where kerosene and petrol is stored by Army for use in four different zones during winters when. the highway is closed and no supplied of kerosene and petrol are possible to these locations. The table below provides the cost (Rs) of supplying one kiloliters of kerosene and petrol from each stock location to each zone. In addition, the storing location capacity and normal level of demand for each zone are indicated in kilolitres. Formulate the LP problem

Zone 
Maximum Supply (K.Litre) 

1 
2 
3 
4 

Storage Location 1 
4 
6 
2.50 
3.00 
1000 
Storage Location 2 
5 
2 
3.50 
4.50 
800 
Demand (K litres) 
300 
500 
400 
350 

Sol. In this problem, there are eight decisions to be madehow many K.litres should be transported from each storage location to each zone. In some cases the best decision may be not to transport any units from a particular location to a particular zone.
Let x_{11}, x_{12}, x_{13}, x_{14}…. denote the number ofK. litres supplied by location 1 to zone, 1 to 2, 1 to 3 and 1 to 4 respectively.
Similarly, Let x_{21},x_{22}, x_{23}, x_{24}be the number of K. litres supplied by location 2 to zone 1, 2 to zone 2, 2 to zone 3 and 2 to zone 4.
Total cost = 4 x_{11} + 6x_{12}+ 2· 50 x_{13} + 3· 50x_{14}+ 5 x_{21} + 2 x_{22}+ 300 x_{23}+ 4· 50 x_{24}.
This function has to be minimised.
The constraints are
x_{11}+ x_{12}+ x_{13}+ x_{14} ≤100 (For location 1)
x_{21} + x_{22}+ x_{23}+ x_{24} ≤800 (For location 2)
Also, the constraint of ensuring that each zone receives the quantity demanded.
For zone 1, the sum of the transportation from location I and 2 should be 300 K. litres
orx_{11} + x_{21}= 300.
Example. The problem is to find out the amount of money which can be allotted to each project in order to maximise net benefits, measured in Rs. If x_{1} ,x_{2}, ….x_{6}is the number of crores of rupees allotted to projects A, B, C, D, E &F, then the objective function is
Maximise Z = 5· 5 x_{1}+ 3· 5 x_{2}+ 2· 4 x_{3}+ 2· 0 x_{4}+ 6· 0 x_{5}+ 3.1 x_{6}
The constraints are of the following types
(a) The total budget to be allotted is 950 crores and the total amount allotted cannot exceed this amount or
x_{1}+ x_{2}+ x_{3} + x_{4}+ x_{5}+.x_{6} 950.
(b) There are constraints for different projects. For project A the constraint is
x_{1} ≤150
x_{2≤} 250
x_{3} ≤120
x_{4} ≤100
x_{5} ≤500
x_{6} ≤160
(c) The constraint of the central government allotting atleast 50% to project E
x_{5} ≥50% of 500
≥250.
(d) The constraints of national priority projects A and B.
x_{1} + x_{2} ≥250.
The complete formulation of the problem is
Maximise Z = 5 . 5 x_{1}+ 3.5 x_{2}+ 2.4 x_{3}+ 2.0 x_{4}+ 6.0 x_{5}+ 3.1x_{6}
Subject to x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5}+ x_{6} 950
x_{1} 150
x_{2} 250
x_{3} 120
x_{4} 100
x_{5} 500
x_{6} 160
x_{5} ≥250
x_{1}+x_{2 ≥} 250
x_{1},x_{2}, x_{3}, x_{4},x_{5},x_{6}, ≥ 0.
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