Problems

I had purchased $20,000 worth of shares at Rs 125 per share in Indian market(1$

=Rs42) on 1st January 200X. I plan to sell sell 75% of such number of shares on 31

st

December 200X at Rs 150 per share. The expected market value 1$ = Rs45. The cost

of capital(Discounting rate) is 10% per annum.

Calculate the following:

a) What is the purchase price in Indian rupee?

b) What is the total sales in dollar?

c) What is the net profit /loss incurred in dollar?

d) If we use cost of capital 10%, what is the present value of selling price in

Indian Rupee?

e) What is the absolute profit/loss in dollars and present value of profit/loss

in dollars?

Production, purchases and sales budgets. 2. The management of AE Manufacturing Co.Ltd. produce a range of components

and products. They are considering next year’s production, purchases and sales

budgets. Shown below are the budgeted total unit costs for two of the components

and two of the products manufactured by the company.

Particulars Component

1

Rs. Per unit

Components

2

Rs. Per unit

Product

P1

Rs.per

unit

Product

P2

Rs.per unit

Direct Material

Direct Labour

Variable overhead

Fixed overhead

18

16

8

20

62

26

4

2

5

37

12

12

6

15

45

28

24

12

30

94

Components 1 and 2 are incorporated into other products manufactured and sold by

the company, but they are not incorporated into the two products shown above.

It is possible to purchase components 1 and 2 from other companies for Rs.60nper

unit and Rs.30 per unit respectively.

The current selling prices of products P1 and P2 respectively.

Required:

a) Evaluate, clearly indicating all the assumptions you make as to whether it would

be profitable in the year ahead for the company to

i) Purchase either of the components

ii) Sell either of the above products.

b) Prepare statements for management with supporting explanations as to how the

following additional information would affect your evaluation in a) above if next

year’s production requirements for the components are 7,000 units of component

1 and 6,000 units of component 14 and the budgeted sales for the two products

P1 and P2 are 5000 units and 4,000 units respectively when a special machine,

AMC is required.

The AMC machine is needed exclusively for these two components and two

products because of specific customer requirements but for technical reasons the

machine can only be used for a maximum of 80,000 hours in the year.

The budgeted AMC machine usage for any one year is 80,000 hours and

requirements per unit for the various items are as follows:

Component 1 8 Machine hours

Component 2 2 machine hours

Product P1 6 machine hours

Product P2 12 machine hours

The operating costs of the AMC machine have been included in the unit costs shown

in (a) above.

Answer:a)

Particulars Component

1

Rs. Per unit

Components

2

Rs. Per unit

Product

P1

Rs.per

unit

Product

P2

Rs.per unit

Purchase /Selling price

Variable cost

Saving/contribution

60

42

(18)

30

32

2

33

30

3

85

64

21

(i) The company should continue to manufacture component 1

but should purchase component 2 from other companies, at a

saving of Rs.2 per unit.

Assumptions:-

1. No fixed over head would be saved if the production of

any component was ceased.

2. Variable costs vary in direct proportion of output.

3. The quality and reliabllity of external supplies is

acceptable

4. The moral of staff would would be adversely affected by

purchasing externally.

5. Suppliers prices and variable use for the capacity which

will be stable for the year ahead.

6. There is no more profitable use for the capacity which

will be used to manufacture component 1.

ii)The company should continue to produce and sell both products since

each of them makes a contribution towards fixed over head and profit.

• 1. After spending Rs. 20,00,000 for market study a BPO Company has

identified two places to set up its business operations. They request you to choose

any one of the places based on the following information.

• Place I:- Koramangala:

• 80% of the employees are staying in 8 kilometres radious and 20% of the

employees are staying in 12 kilometres radius. Cabs are arranged for the

employees both the ways.10% of the total number of employees do not take the

transport facilities and are paid by Rs.3 per Kilometre for 10 kilometres basis per

day trip irrespective of the number of kilometres traveled by them. A cab can

bring 4 employees at a time. The cabs while leaving the first shift employees in

their houses pick up the second shift employees to the company.

• The cab is always full. The cabs are taken on a daily rental of Rs.800 irrespective

of the kilometres for 12 hours duration. For 18 hours duration Rs.1200 is charged

per cab(both shifts and transport time)ie.Rs. 400 charged the company for second

shift employees to be dropped after second shift.

• The rent of the building is Rs. 25per month per square feet for the first year, Rs.28

Per month for the second year and Rs.30 per month for the third year.

• Place II:- Rajaji Nagar:-

• 40% of the employees are staying 10 kilometres radius and 60% of the

employees are staying 14 Kilometres radius. All employees prefer to take the

transport facilities. A cab is taken on a daily rental of Rs.900 per day irrespective

of the kilometres and Rs. 400 is charged per cab to drop the second shift

employees in their houses.

• The rent of the building is Rs. 15 per month per square feet for the first year,

Rs.20 Per month for the second year and Rs.25 per month for the third year.

• Other informations:

• The company has a fixed order of 5,54,400 accounts per month to be processed.

There are two shifts in 24 hours a day. The working hours are 9 hours duration

but effective working hours are 8 hours per day. Each account is estimated to be

processed in a 5 minutes duration.

• On an average there are 22 working days in a month.

• Irrespective of the place selected, the company makes a contract for three years

only. Each employee occupies on an average of 40 square feet including all other

facilities such as canteen, toilet etc.

. wage per hour is Rs.60. The company charges $4,$4.5 and $5 per account for the fist

year, second year and third year respectively. The expected currency value for three

years respectively $1=Rs39 in the first year $1=Rs.38 in the second year; and

$1=Rs35.0.

• Discounted rate is 12%.

• Required:-

• A) Choose the correct place to set up their business in Bangalore.

• B) Having chosen the place If the company expects Rs.20,00,000 per month as

profit. Calculate Break even employees per year and Margin of safety employees

to get a profit of Rs.20,00,000 and also calculate break even accounts per year and

margin of safety accounts to be processed.

• C) Should the company run in a single shift or double shift? Advice.

Rules of Merger

A LTD AMALGAMATES WITH B LTD

AS ON 2007

NO CAPITAL

GAIN TAX &

ACCUMULATED

LOSSES &

UNABSORBED

DEPERICIATION

CAN BE

CARRIED

FORWARD

DOES NOT

ATTRACT

CAPITAL GAIN

FOR A BUT NO

GAIN FOR B

NO BENEFIT

TO A & B

A MERGES WITH

B (A GOES OUT)

SATISFIES

BOTH 2(1B) & 72

A

SATISFIES 2(1B)

BUT DOES NOT

SATISFY 72 A

DOES NOT

SATISFY SEC

2(1B) & 72 A

PARTICULARS

CONDITIONS OF AMALGAMATION UNDER INCOME TAX ACT SEC 2 (1B)

1.ALL ASSETS AND LIABILITIES OF TRANSFEROR CO. TO BE THE

ASSETS OF THE TRANSFREE CO.

2.SHARE HOLDERS HOLDING NOT LESS THAN 3/4TH IN VALUE OF

SHARES OTHER THAN SHARES ALREADY HELD SHOULD BECOME

SHARE HOLDERS OF AMALGAMATED COMPANY

EX. NO. OF SHARES OF Altd CO. 1,00,000

NO. OF SHARES HELD BY Bltd IN Altd IS 20,000

NOMINAL VALUE OF SHARE IS RS.10

ASSUME Altd MERGE WITH Bltd THEN 75% OF 1,00,000- 20,000 = 60,000

TO BE THE SHARE HOLDES OF B CO.

NOTE:SHARE HOLDERS MAY BE EQUITY OR PREFERNCE SHARE

HOLDERS

Other conditions

•THE AMALGAMATED CO. IS AN INDIAN CO.

EXCEPTION

1.IF SHARES OF INDIAN CO.HELD BY FOREIGN BEFORE MERGER AND

SUCH FOREIGN CO. TAKEN OVER BY ANOTHER FOREIGN CO.

2.ATLEAST 25% OF THE FOREIGN CO. (BEFORE MERGER) TO BE SHARE

HOLDERS OF THE NEW FOREIGN CO.

? WHAT IS THE BENEFIT TO THE AMALGAMATED CO. AMALGAMATING

CO.(OLD CO.)

•

•NO CAPITAL GAIN ON TRANSFER ON CAPITAL ASSETS BY THE

TRANSFEROR CO. UNDER SEC 47(VI) OF I.T ACT

? CAN NEW CO. CARRY FORWAD AND SET OF LOSS AND

DEPRECIATION

SEC 72 A of Income tax Act

1.ACCUMULATED LOSSES REMAIN UNABSORBED FOR 3 OR MORE

YEARS

2.75% OF BOOK VALUE TO BE HELD ATLEAST FOR 2 YEARS BEFORE

AMALGAMATION

3.THE AMALGAMATED CO. CONTINUES TO HOLD 3/4TH OF BOOK

VALUE ATLEAST FOR 5 YEARS

4.NEW CO. SHOULD CONTINUE FOR ANOTHER 5 YEARS

5.NEW CO. SHOULD ACHIEVE ATLEAST 50%OF INSTALLED CAPACITY

BEFORE END OF 5 YEARS AND SHOULD CONTINUE FOR 5 YEARS

THE NEW AMALGAMATED CO. SHOULD FURNISH TO ASSESSING

OFFICER ABOUT PARTICULARS OF PRODUCTION

BENEFIT

•THIS SCHEME IS ALSO APPLICABLE TO BANKING INSTITUTIONS

•?TATA VOLTAS & KELVINATOR HYDERABAD DIVISION vs. CBDT•

Rules of Merger

A LTD AMALGAMATES WITH B LTD

AS ON 2007

NO CAPITAL

GAIN TAX &

ACCUMULATED

LOSSES &

UNABSORBED

DEPERICIATION

CAN BE

CARRIED

FORWARD

DOES NOT

ATTRACT

CAPITAL GAIN

FOR A BUT NO

GAIN FOR B

NO BENEFIT

TO A & B

A MERGES WITH

B (A GOES OUT)

SATISFIES

BOTH 2(1B) & 72

A

SATISFIES 2(1B)

BUT DOES NOT

SATISFY 72 A

DOES NOT

SATISFY SEC

2(1B) & 72 A

PARTICULARS

EXERCISE

4070MARKET PRICE

810P/E RATIO

57EPS

7,50020,000NO. OF SHARES

37,5001,40,000EAT

CO. BCO. APARTICULARS

•

Co. A is acquiring co. B Exchanging one share for every 1.5 shares of B ltd & p/e ratio

will continue even after merger

? Are they better or worse of than they were before in merger

? Determine the range of minimum & maximum ratio between the two firms

? A is an Indian co.

? A is a foreign co.

? A merges with T & formed a new co. AT ltd

? What are the tax planning required before & after merger

•

EXAMPLE

7.58P/E RATIO(TIMES)

18,75,00050,00,000TOTAL MARKET

VALUE (N*MPS) OR

(EAT*P/E RATIO)

18.7525MARKET PRICE PER

SHARE(MPS)

2.53.125EPS

1,00,0002,00,000NO. OF SHARES

2,50,0006,25,000EAT

FIRM BFIRM APRE MERGER

SITUATION

•

? IF EXCHANGE RATIO IS 2.5:1 WHO GAINS WHO LOSES

? IF EXCHANGE RATIO IS 1:1 WHO GAINS WHO LOSES

? HOW TO CALCULATE TOLERABLE SHARE EXCHANGE RATIO

•

Answer:

7.58P/E RATIO

(ASSUMED TO BE THE

SAME)

21.8253.125*8=25MPS

65,47,50070,00,000TOTAL MARKET VALUE

8,75,000/3,00,000=2.91/8.75/2.8=3.125EPS

2,00,000+1,00,000=3,00,0

00

2.8 lakhsNO. OF SHARES

8,75,0006.25+2.5=8.75EAT(COMBINED FIRM)

1 : 12.5:3.125=.8EXCHANE RATIO/ SWAP

RATIO (ASSUMING)

SITUATION 2SITUATION 1

(BASED ON CURRENT

MARKET PRICE

POST MERGER

TOTAL MV

LESS: MINIMUM TO BE GIVEN TO B

75,00,000

10,00,000

NET BENEFIT TO A

65,00,000

NO. OF SHARES OF A TO A CO. SHARE HOLDERS

1,00,000

DESIRED POST MERGER MPS

65 PER SHARE

NO. OF EQUTY SHARES TO BE ISSUED BASED ON DESIRED MARKET PRICE

10,00,000/65 = 15,385 SHARES

TOLERANCE SHARE EXCHANGE RATIO

50,000/15385 = 3.25 SHARES OF FIRM B, 1 SHARE IN FIRM A

1:3.25

CONCLUSIONS-1

•EXCHANGE AT EPS – NO EFFECT ON EPS AFTER MERGER

•EXCHANGE MORE THAN EPS RATIO – COMPANY WITH LOWER EPS GAINS

•IF LESS THAN EPS RATIO – COMPANY WITH HIGHER EPS BEFORE MERGER

GAINS

Conclusion-2

•IF SHARES ARE EXCHANGED BASED ON CURRENT MARKET PRICE PER

SHARE , POST MARKET PRICE SHARE INCREASED AT HIGHER RATE THAN

EXCHANGED BELOW THIS RATIO

•Boot strap effect

Conclusion-3

•FIRM WITH HIGHER P/E RATIO CAN ACQUIRE FIRM WITH LOWER P/E

RATIO WHICH WILL INVARIABLY INCREASES MARKET VALUE AFTER

MERGER

Conversion of sole proprietorship into a company

conditions

•All assets and Liabilities of the sole proprietarily concern leading to the business

immediately before the succession shall become the A/L of the company

•Sole proprietor should hold not less than 50% of the total voting power in the company

•The sole proprietor should continue for a minimum period of 5 years.

•The sole proprietor should receive the consideration only in the form of shares in the

company

Consequences if not fulfilled

•Withdrawal of exemption U/S 47A(3)

•The capital gain which wad not taxed earlier will become taxable in the hands of the

company

Conversion of Firm into a company

•Conditions:

•1. All assets and Liabilities of the firm leading to the business immediately before the

succession shall become the A/L of the company.

•2. All the partners of the firm become the shareholders of the company in the same

proportion of their capital account stood before the succession.

•3.Every thing should be received in Shares of the company by the partners.

•4.Not less than 50% of voting power in the company by all the partners and hold such

shareholdings for a period of 5 years from the date of succession.

Failed to fulfill the conditions

•Withdrawal of exemption U/S 47A(3)

•The capital gain which wad not taxed earlier will become taxable in the hands of the

company

•

•If Shareholder Y transfers his shares to Z on 5th

March 2011 what is the consequencesto

the firm and Company and partners(shareholders)?

•How do you compute capital gain tax?

•Suppose the firm has a land worth Rs. 50 crores and sells which attracts 8 crores

Income tax. Is there tax planning to avoid tax liability?

Exercise-1-Case study

Plant 80,000

House property 2,00,000

(acquired in 1982-83)

Stock in trade 40,000

Debtors 70,000

Bank 10,000

Capital

X 1,00,000

Y 2,00,000

Sundry credi 1,00,000

AssetsLiabilities

Balance Sheet as on 1st April 2008

•A Ltd. is incorporated on April 1st 2008 which takes over the assets and liabilities at the

agreed valuation of X Co. as follows:

•Plant-2,80,000, House property-10,00,000, stock-60,000, debtors-70,000,Bank –10,000.

???

•How is it treated as per IT?

•If firm sells the whole business at the agreed value what is the tax implication?

•Calculate the total consideration the company is willing to give the partners and also

find out the number of shares alloted to each partner?

•If Shareholder Y transfers his shares to Z on 5th

March 2011 what is the consequencesto

the firm and Company and partners(shareholders)?

•How do you compute capital gain tax?

•Suppose the firm has a land worth Rs. 50 crores and sells which attracts 8 crores

Income tax. Is there tax planning to avoid tax liability?

Answer

1.Short term capital gain on plant and Machinery as a depreciated asset=2,00,000

House property – Long term as no depreciation provided use index costof acquision

Stock in trade 60,000-40,000=20,000 is business income.

•If all the conditions fulfilled as per 47(xiii)

•No capital gain tax.However business income arises on stock which attracts tax as

business income and to be paid by the firm.

•Stock is a current asset which is used as a stock in trade does not amount to capital asset.

•If Y transfers his shares to Z the capital gain earlier exempted will be taxed as it was

originally calculated.Long term capital gain on house property and short term capital gain

on plant and machinery.

Bond valuation

C M

Bond valuation ———— + ——————

t =1 (1+i)t (1+i) n

Where:

C1.. Cn = period coupon payment from year 1 to n

i = market interest rates, prevailing

n = period to maturity

M = Principal with / without redemption premium

Yield to maturity

This term popularly known as YTM connotes redemption yield and is very useful for

Treasury Managers whose investment horizon is long term. YTM can be interpreted as

the bond’s average compounded rate of return if the bond is bought at the current asked

price and held until it matures and the face value is repaid. That is, YTM can be defined

as the discount rate that equates present value of all cash flows to the present market price

of the Bond. Future cash flows includes interest and capital gain/loss. This can be

algebraically expressed as follows: Let the Bond with a face value of ‘A’ of coupon ‘C’

with a term to maturity of ‘n’ years is quoted/traded at a market price of P, then

C C C (C + A)

P = ———— + ———— + ————+ ———— + ————

(1+y)1 (1+y)2 (1+y)3 (1+y)n

Where ‘y’ is the discount rate (to be found by trial & error method ) at which the cash

flows are discounted so that the right hand side of the above equation tallies/equates with

the Price P (left hand side) of the Bond.

The 'y' so derived would be the Yield to maturity (YTM) of the bond. It implies that, if

the Bond is held till maturity and the Coupons/Cash flows received are reinvested at the

'y' rate itself, the overall yield on the Bond will be 'y', which is its YTM.

An example would further help to understand the mechanics of the YTM. Suppose the

market value of Rs 100 (face value) bond carrying coupon of 13 per cent p.a. maturing

after 7 years is quoted Rs 109.45 in the market. The YTM of the bond is found by

discounting the yearly coupon flows of Rs 13 in the next 6 years and Rs 113 (Principal of

Rs 100 + coupon of Rs 13) at the end of 7 year at a rate (to be found by trial & error

method), say ‘r’ so that the Present value of such cash flows sums to Rs 109.45 Rs 13

(PVIFA) + Rs 100 (PVIF) = Rs 109.45 PVIFA being the Price Value Interest Factor for

the 7 year Annuity and PVIF the Price Value Interest Factor for 7 years to be taken from

the PVIFA table and PVIF table (available in all standard Finance Text Books) for a 7

year term, by trial and error method.

Accordingly for 7 years (PVIFA) at 11% = 4.712

and for 7 years (PVIF) at 11% = 0.482

Then LHS of the equation becomes 13 x (4.712) + 100 x (0.482) = Rs 109.45

Then 11 per cent is said to be the YTM of the bond, also described as the Internal Rate of

Return, (IRR). In other words, in the above example, if the above bond is held by the

buyer till maturity the overall return from the Bond will be 11 per cent. However as the

above process will be time consuming, YTM can be found by approximation as follows.

C + (A – P)/n

YTM = ————————— X 100

(A + P) /2

Where C = coupon

A = Face Value/maturity Value

P = Price paid for the Bond

n = term to maturity

Applying this in the above example,

13 + (100 –109.45)/ 7 13 + (– 9.45/7)

YTM = ————————————— = —————————

(100 + 109.45)/ 2 104.725

13 – 1.35

= —————— X 100 = 11.12%

104.725

However underlying assumption in the YTM concept is that the coupons/cash flows

received during the tenure of the bond is reinvested at YTM rate, which may not be true

since the market interest rates will always be changing from time to time.

Yield on Discounted instruments:

The issue price of a discounted instrument is calculated as follows:

F

D = ———————————————

1 + {( r x n)/36500}

where,

D = Discounted value of the instrument

F = Maturity Value

r = Effective rate of interest per annum

n = Tenure of the instrument ( in days)

Conversely to find out the yield from a discounted instrument, the following formula can

be derived from the above one,

(F –D) 365

r = —————— X ——————— X 100

D n

where,

D = Discounted value of the instrument

F = Maturity Value

r = Effective rate of interest per annum

n = Tenure of the instrument ( in days)

REPO Transactions—calculations:

Assume Bank ‘A’ borrows from Bank ‘B’ an amount of Rs 10 crores for a period of 14

days from 10.10.2005 to 24.10.2005, at an interest rate of 8 per cent against its holding of

11.50 per cent GOI 2007 (Interest Payment dates of this stock are 5th April and 5th

October of the year). As already stated earlier, the transaction involves 2 legs—First

leg/Ready leg and Second leg/Forward leg. The calculation for both legs are explained

below:

Working

(Note: While calculating interest accrued on Government securities, 360 days are

considered for an year.)

FIRST LEG/READY LEG on 10.10.2005: (Bank A sold 11.50 per cent GOI 2007 to

Bank B)

Calculation for first leg is as if Bank A is selling the security (11.5 per cent GOI 2007)

outright to Bank B at the market price of Rs 100. This is as follows:

Principal (Rs 10 crs. @ 100.00) = Rs 10,00,00,000.00

Accrued int.on the stock

= (10 crs x 11.5% x 5/360) = Rs 1,59,722.22

First/Ready leg settlement amount...(1) = Rs 10,01,59,722.22

(It may be understood from the above transaction, that Bank A borrowed Rs

10,01,59,722.22 from Bank B)

FORWARD/SECOND LEG on 24.10.2005: (Bank A bought back the stock from Bank

B)

Though the second leg transaction is to be calculated as if Bank A is buying outright the

security from Bank B, to arrive at the buying rate/price, the calculation has to be done on

the reverse way, as follows:

1. Calculate the settlement amount Bank A has to pay Bank B which is = Amount

borrowed + interest @ 8% for 14 days (Repo rate)

= Rs 10,01,59,722.22 + Rs 3,07,339.42

Settlement amount = Rs 10,04,67,061.64

2. From this subtract accrued interest on the stock till date.

Accrued interest on the stock

= 10,00,00,000 x 11.5% x 19

---------

360

= Rs 6,06,944.44

Settlement amt. – Accrued interest = 10,04,67,061.64

6,06,944.44

Rs 9,98,60,117.20

3. Resulting amount of Rs 9,98,60,117.20 is the principal amount for the Rs 10 crore

value stock. Hence to get rate of repurchase, divide this value by nominal value

i.e. 9,98,60,117.20

------------------ = 99.860117

10,00,00,000

Now based on this rate, the accounting is done as follows:

Principal (Rs 10 crs. @99.860117) = Rs 9,98,60,117.20

Accrued int. on the stock

= (10 crs x 11.5% x19/360) = Rs 6,06,944.44

Forward/second leg settlement amt

= (1) + int @ 8% for 14 days = (2) = Rs 10,04,67,061.64

3.Dilip Company currently produces two products. The cost per unit of Product are as

follows:

Product Y Product Z

Selling price 110

Less : Variable Cost

Material ( 8 units at Rs.4) 32

Labour ( 6 hrs at Rs.10) 60

Variable Overheads ( 4 Machine hrs at Rs.1) 4

96

Contribution 14

Selling price 118

Less : Variable Cost

Material ( 4 units at Rs.4) 16

Labour ( 8 hrs at Rs.10) 80

Variable Overheads ( 6 Machine hrs at

Rs.1) 6

102

Contribution 16

During the 4th

coming accounting period the availability of labour hrs will be restricted to

2880 hrs.Material availability is limited to 3440 units . the machine has the capacity to

produce 2760 units . The marketing manager expects the maximum sales potential for y

is 420 units with respect to Product z there is no sales Limitations

Required

Formulate Linear Programing Model and solve by simplex and interpret the final /matrix

and also from the final /matrix find shadow price or opportunity cost

Solution

Maximise C = 14y+16z subject to

8y + 4Z < 3440 ( Material Constraint)

6Y + 8Z < 2880 ( Labour Constraint)

4Y + 6Z < 420 ( maximum and minimum sales limitation)

Z > 0

First Matrix

Quantity Y Z

S1 = 3440 -8 -4 (1) Material Constraint

S2 = 2880 -6 -8 (2) Labour constraint

S3 = 2760 -4 -6 (3) Machine hours constraint

S4 = 420 -1 0 (4) Sales Constraint

C = 0 +14 +16 (5) Contribution

Note that the quantity column in the matrix indicates the resources available or the slack

that is not taken up when production is zero. For example, the S1 row of the matrix

indicates that 3440 units of materials are available when production is zero. Column Y

indicates that 8 units of materials, 6 labour hours and 4 machine hours are required to

produce 1 unit of product Y, and this will reduce the potential sales of y by 1. You will

also see from column Y that the production of 1 unit of Y will yield Rs.14 contribution.

Similar reasoning applies to column Z. Note that the entry in the contribution row (i.e.

The C row) for the quantity column is zero because this first matrix is based on nil

production, which gives a contribution of zero.

Second Matrix

Quantity Y S2

S1 = 2000 -5 +1/2 (1) Material Constraint

Z = 360 -3/4 -1/8 (2)

S3 = 600 +1/2 +3/4 (3) Machine hours Constraint

S4 = 420 -1 0 (4) Sales Constraint

C = 5760 +2 -2 (5)

The substitution process obtained for the second matrix has become more complex, but

the logical basis still remains. For example, the quantity column of the second matrix

indicates that 2000 units of materials are unused,360 units of Z are to be made, 600

machine hours are still unused and sales of product Y can still be increased by another

420 units before the sales limitation is reached. The contribution row indicates that a

contribution of Rs. 5760 will be obtained from the production and sale of 360 units of

product Z. Column Y indicates that production of 1unit of product Y uses up 5 units of

the stock of materials,but, because no labour hours are available , ¾ units of product Z

must be released. This will release 3 units of materials (3/4 x 4), 6 labour hours ( ¾ x 8)

And 4 ½ machine hours ( ¾ x 6). From this substitution process we now have 8 units of

materials ( 5 units + 3 units ), 6 labour hours and 4 ½ machine hours.

From the standard cost details one unit of Y requires 8 units of materials, 6 labour hours

and 4 machine hours. This substitution process thus provides necessary resources for

producing 1 unit of product Y, as well as providing an additional half an hour of machine

capacity. This is because production of 1 unit of product Y requires that production of

item Z be reduced by ¾ units, which releases 4 ½ machine hours. However product Y

requires only 4 machine hours, so production of 1unit of Y will increase the available

machine capacity by half an hour. This agrees with the entry in column Y of the second

matrix for machine capacity . Column Y also indicates that production of 1 unit of Y

reduces the potential sales of product Y (S4) by 1 unit.

The optimum solution is achieved when the contribution row contains only negative or

zero values. Because row C contains a positive item, our current solution can be

improved by choosing the product with the highest positive contribution. Thus we should

choose to manufacture product Y, since this is the only positive item in the contribution

row. The second matrix indicates that the contribution can be increased by Rs. 2 by

substituting 1 unit of Y for ¾ units of Z. We therefore obtain an additional contribution

of Rs.14 from Y but lose at Rs.12 from Z ( ¾ x 16) by this substitution process. The

overall result is an increased contribution of Rs.2 by adopting this substitution process.

The procedure is then repeated to formulate the third matrix. Column Y of

The second matrix indicates that we should use 5 units of materials and release ¾ units of

Z to obtain an additional unit of Y, but there are limitations in adopting this plan. The

unused materials are 2000 units, and each unit of y will require 5 units, giving a

maximum production of 400 units of Y. We have 360 units of Z allocated to production,

and each unit of Y requires us to release ¾ units of Z. A maximum production of 480

units of Y (360/ ¾ ) can therefore be obtained from this substitution process. There is no

limitaion on machine hours, since the second matrix indicates that the substitution

process increases machine hours by half an hour for each unit of Y produced. The sales

limitaion of Y indicates that a maximum of 420 units of Y can be produced. The

following is the summary of the limitations in producing product Y:

S1 (materials) = 400 units (2000 / 5)

Z ( substitution of product Z ) = 480 units ( 360/ ¾ )

S4 ( maximum sales of Y) = 420 units ( 420/1)

In other words, we merely divide the negative items in column Y into the quantity

column. The first limitation we reach is 400 units, and this indicates the maximum

production of Y because of the impact of the material constraint.

Third Matrix

Quantity S1 S2

Y = 400 -1/5 +1/10 (1)

Z = 60 +3/20 -1/5 (2)

S3 = 800 -1/10 +4/5 (3)

S4 = 20 +1/5 -1/10 (4)

C = 6560 -2/5 -1 4/5 (5)

The contribution row (equation 5) contains only negative items, which signifies that the

optimal solution has been reached. The quantity column for any products listed on the left

hand side of the matrix indicates the number of units of the product that should be

manufactured when the optimum solution is reached. 400 units of Y and 60 units of Z

Should therefore be produced, giving a total contribution of Rs.6560. This aggress with

the results we obtained using the graphical method. When an equation appears for slack

variable, this indicates that unused resources exist. The third matrix therefore indicates

that the optimal plan will result in 800 unused machine hours ( S3) and an unused sales

potential of 20 units for product Y (S4). The fact that there is no equation for S1 and S2

means that these are the inputs that are fully utilized and that limit further increases in

output and profit.

Interpreting the Final Matrix

The S1 column (materials) of the third matrix indicates that the materials are fully

utilized. (Whenever resources appear as column headings in the final matrix, this

indicates that they are fully utilized.) So, to obtain a unit of materials, the column for S1

Indicates that we must alter the optimum production programme by increasing production

of product Z by 3/20 of a unit and decreasing production of product Y by 1/5 of a unit.

If we increase production of product Z by 3/20 of a unit the more machine hours will be

required, leading to the available capacity being reduced by 9/10 of an hour. Each unit of

Product Z requires 6 machine hours, so 3/20 of a unit will require 9/10 of an hour (3/20 x

6). Deceasing production of product Y by 1/5 unit will release 4/5 of a machine hour,

given that 1 unit of product Y requires 4 machine hours. The overall effect of this process

is to reduce the available machine capacity by 1/10 of a machine hour.

The S1 column indicates that to release 1 unit of materials from the optimum production

programme we should increase the output of product Z by 3/20, and decrease product Y

by 1/5 of a unit. This substitution process will lead to the unused machine capacity being

reduced by 1/10of a machine hour, an increase in the unfulfilled sales demand of product

Y (S4) by 1/5 of a unit and a reduction in contribution of rs.2/5. All this information is

obtained from column S1 of the third matrix.

Opportunity Cost:

The contribution row of the final matrix contains some vital information for the

accountant. The figures in this row represent opportunity costs (also known as shadow

prices) for the scarce factors of materials and labour. For example the reduction in

contribution from the loss of 1 unit of materials is Rs.2/5 (Rs.0.40) and from the loss of

one labour hour is rs.1 4/5 (Rs.1.80). Our earlier studies have indicated that this

information is vital for decision-making, and we shall use this information again shortly

to establish the relevant costs of the resources.

S3 S4 S1 S2

Machine Sales of Materials Labour Contribution (Rs.)

Capacity Y

Increase

Product Z

By 3/20 of

A unit -9/10(3/20x6) ------ -3/5(3/20x4) -1 1/5(3/20x8)

+2 2/5(3/20x6)

Decrease

Product Y

By 1/5 of

A unit +4/5(1/5x4) +1/5 +1 3/5(1/5x8) +1 1/5(1/5x6) -

2 4/5(1/5x14)

Net Effect -1/10 +1/5 +1 Nil -2/5