A mathematical model is a description of a system using mathematical concepts. Business mathematical model discusses about the decisions of making choices between leasing and gaining of capital assets in a manufacturing company or industry. Mathematical techniques are adopted in much simpler way to solve problems regarding choices of leasing and gaining of capital assets in a company. It uses mathematics as a tool in financial investments.

A business model is a sustainable way of doing business and sustainability stresses the ambition to survive over time and creates a successful, perhaps even profitable, entity in the long run. The problems related to financial investments can be made easier using programming language. The codes were developed in java programming language and are executed on computer. More accurate solutions can be obtained with the aid of computers.

Salvage value is the estimated resale value of an asset at the end of its useful life. Salvage value is subtracted from cost of a fixed asset to determine the amount of the asset cost that will be depreciated .Thus; salvage value is used as a component of the depreciation calculation. When the cost of an asset less its accumulated depreciation equals its salvage value, no more depreciation may be taken .Whenever purchasing of a commodity is not quite possible then the next possibility for an individual and commodities comes down to leasing and renting .

Both of renting and leasing are similar, getting access to an asset for a limited period, there are significant differences.
A lease is a contract to rent an asset, be it land, building, or machinery, for a particular period of time or a calendar year for set payment terms. Leases often come with many conditions in terms of the allowed use of the asset and even required maintenance terms. A typical lease is often long term, ranging from 1 year to as many as 10 or 20 years.

Renting typically involves a shorter time period, often one year maximum with the option to extend after the term at description of both parties. If the cost o renting or leasing the asset is high, it can be rented for a short period only when absolutely needed and then returned.

## USE OF MATHEMATICAL MODELS IN BUSSINESS

There are several ways in which mathematical models are used in the field of business. Some of them include;

1. DECISION MAKING

In business the most crucial activity is making decisions, which involves multiple participants with different views .These types of models are using input variables and some conditions to help the company arrive at a decision.

2. MAKING PREDICTIONS

Businesses includes predicting of some factors such as growth rate, leasing or renting, cost and revenue,e.t.c.Predictions are made using historical data and use probability distribution as input for predicting the future values. Regression analysis is one of the most commonly used techniques for predicting models.

3. OPTIMIZING

Businesses optimize certain variables to control costs and ensure maximum efficiency. Such variable include some planning like human resource planning, route planning, capacity planning e.t.c.Optimization of mathematical models are typically used for such problems.

## PROBLEM STATEMENT

A manufacturing company is considering either buying or leasing machinery. Le it buy the machinery for # x.00 or leasing it for #y.00 per month. The life of the machinery is M years, after which the salvage value become #z.00. Suppose that money is worth r% for jth period of time. It may be compounded annually,semiannually,quarterly or monthly. Assume that the company purchases a maintenance contract for #k.00 per month. Our aim is to advice the company either they should buy or lease the machinery.

1. Let us take the interest conversion period be j.
2. The total number of the interest periods is n=mj , where m is the asset’s life span.
3. If i% is the interest rate in decimal form. Then it can be represented as i=r/100.

We have y &lt; z &lt; x and m, n &gt; 0.

RESULT:

(1)If y ≥ x, then the company will just buy the machinery for #x.00 instead of leasing it because leasing will be costlier.
(2)If x ≥ y, then the company will lease the machinery for #y.00 instead of buying it because buying will be costlier.

DERIVATION OF THE MODEL

In-order to get a solution for the problem as described in the problem statement , the present or salvage value #z.00 which is to be received at the end of M years and to find the result from x.If y ≥ x then the company will buy the machinery for #x.00 instead of leasing it. Hence it is dented by PC .

P = A (1+i)-n (1)

In-order to get the original present value, subtract the result from x. This represents the present value of the cost of owning the machinery.
This implies that,

PC = x – {z (1+r/100)-n} (2)
PC = x –z (1+i)-n
PC = x – z/ (1+i)n (3)

RP represents the present value rent for M years. The formula for the annuity doesn’t include a payment at the beginning of the term and RP would be,

RP = y + y {(1+i)n-1-1/ (1+i)n-1} (4)
RP = y {1+ ((1+i)n-1)-1/i (1-i)n-1} (5)

MC represents the maintenance cost of the machinery which may also be included in the rental price for the same period of time that is M years.Suppose the company could purchase a maintenance contract or other miscellaneous expenses for servicing the machinery then the present value for MC would be;

MC = K+K{((1+i)n-1)-1/i(1-i)n-1} (6)
MC = K {1 + (( 1 + i )n-1) – 1 / I (1- i)n-1 } (7)

K can be found from the formula for depreciation.We have , PC + MC = TC . This represents the total cost of buying and maintaining the machinery .

TC = { x – { z/(1+i)n } } + { K { 1+((1+i)n-1)-1 / I(1-i)n-1} } (8)

### CONCLUSIONS:

If TC &lt; RP then the company would be advised to buy the machinery .
If TC &gt; RP then the company would be advised to lease the machinery .
If TC = RP then the company can take any of the choices as they are same.

Consider that a company wishes to buy a machinery for Rs.10,000 or lease it for Rs.1,000 per month . Let us assume that the money is worth 12% compounded monthly and the life of the machinery is 6 years , after which time the salvage value will be Rs.20,000 . Suppose the company would purchase a maintenance contract for Rs.500 . Our aim is to advice the company on whether they should buy or lease the machinery .

SOLUTION TO THE REAL LIFE CASE

In-order to find the present value of Rs.10,000 which is to be received in 6 years , n=6*12=72 ,
Since the money is worth 12% compounded monthly . Here r=12/12%=1% per month.

Now, i=1/100 = 0.01
Using equation(1) gives;

PC = Rs. 20,000 (1+.01)-72 = Rs.9769.922

Meanwhile , the difference of the cost and leasing of the machinery which is obtained thus ,
Rs.10,000 – Rs.9769.922 = Rs.230.078

This represents the present value of the cost of owning the machinery . Using equation (4) or (5)to find the present value of the rent for 6 years , the RP would be ;

RP = Rs.1000 + Rs.1000 {(1+.01)72-1-1/(.01)(1+.01)72-1)}
= Rs.1000 + Rs.1000 [1.02683/0.02027]
= Rs.51661.846

This is in line since the formula for annuity doesn’t include a payment at the beginning of the term . This certainly seems to indicate that the company should buy the machinery . This doesn’t however consider other factors such as maintenance, which would be included in the rental price RP. Suppose the company could purchase a maintenance contract for Rs.500 per month.Then the present value of the maintenance contract would be ,

MC = Rs.500 + Rs.500{(1.01)71-1/(.01)(1.01)71} = Rs.500 + Rs.500[1.026831/0.0202681] = Rs.25830.9477

Therefore , TC = MC + PC
Which is the total cost of buying and maintaining the machinery,gives;

TC = MC + PC = Rs.25830.9477 + Rs.230.0783 = Rs.26061.026

Hence, we get TC &lt; RP that is Rs.26061.026 &lt; Rs.51661.846. So we conclude by advising the company to buy the machinery instead of leasing it .