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Dealing with limiting factors

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DEALING WITH LIMITING

FACTORS

Multiple limitingfactors - linear

programming

Optimal productionplan where there is asingle limiting factor

Algebraic solutions

1 What is a limiting factor?

• Alimiting factor is a factor that prevents a company from achieving thelevel of activity that itwould like to.

Limiting factor analysis looks atusing the contribution concept toaddress the problem of scarce resources.

Scarce resources are where one or more of the manufacturing inputs(materials, labour, machine time) needed to make a product is in shortsupply.

Production can also be affected by the number of units of a product thatis likely to bedemanded in a period (the sales demand).

Illustration 1 - What is a limiting factor?

Suppose ALtd makes two products, Xand Y. Both products use thesame machine and the same raw material that are limited to 600 hoursand $800 per week respectively. Individual product details are asfollows.

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Product X per Product Y perunit unit

Machine hours 5.0 2.5

Materials $10 $5

Contribution $20 $15

Maximum weekly demand 50 units 100 units

Comment on whether machine hours and/or materials are limitingfactors

Solution

To make the maximum demand of 50 units of X and 100 units of Yrequires the following inputs:

Machine hours:

Materials:

50x5+100x2.5

50x10+100x5

= 500 hours

= $1,000

Thus there are enough machine hours available to make all units thatcould be sold but materials limit the production plan.

Test your understanding 1

Two products, Alpha and Gamma are made of Material X and requireskilled labour in the production process. The product details are asfollows:

Selling price

Variable cost

Contribution

Material X required per unit

Skilled labour time required per unit

Alpha Gamma

$ $

10.00 15.00

6.00 7.50

4.00 7.50

2 kg 4 kg

1 hour 3 hours

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The maximum demand per week is 30 units of Alpha and 10 units ofGamma. '

The company can sell all the Alphas and Gammas that it can make buthere is a restriction on the availability of both Material Xand skilled

labour There are 150 kg of material, and 45 hours of skilled labouravailable per week.

Identify the limiting factor.

2Optimal production plan where there is asingle limiting factorpontributipn perunit of limiting factor

In order to decide which products should be made in which order it isnecessary to calculate the contribution per unit of limiting factor (or scarceresource). K

Contribution per unitof limiting factor

Optimal production plan

Contribution per unit

Units of limiting factor required per unit

When limiting factors are present, contribution (and therefore profits) aremaximised when products earning the highest amount of contribution perunit of limiting factor are manufactured first. The profit-maximisingproduction mix is known as the optimal production plan.

The optimal production plan is established as follows.

Step 1 Calculate the contribution per unit of product.Step 2 Calculate the contribution per unit of scarce resource.Step 3 Rank products.

Step 4 Allocatethe scarce resource to the highest-ranking product.Step 5 Once the demand for the highest-ranking product is satisfied,move on to the next highest-ranking product and so on until the scarceresource (limiting factor) is used up.

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Illustration 2 - Optimal production plan where there is a single

Acompany is able to produce four products and is planning itsproduction mix for the following period. Relevant data is given below:

A B C D

Selling price ($) per unit 19 25 40 50

Labour cost per unit ($) 6 12 18 24

Material cost per unit ($) 9 9 15 16

Maximum demand (units) 1,000 5,000 4,000 2,000

Labour is paid $6 per hour and labour hours are limited to 12,000 hoursin the period.

Required:

Determine the optimal production plan and calculate the totalcontribution it earnsfor the company.

Solution

Selling price19

Variable costs:

Direct labour (6)

Direct material (9)


Hours perunit (labour cost/$6)

Contribution per hour $4

Rank

B

25 40

(12) (18)

(9) (15)

$2 $2.33

D

$

50

(24)

(16)

10

D

$2.50

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Remember to allocate the scarce resource (labour hours) to the highest-ranking product first (A). Once the demand for the highest-rankingproduct is satisfied, move on to the next highest-ranking product (D) andthen the next (C) until the scarce resource (labour hours) is used up.

Optimal production plan

Product Units Hours

used

Hours

left

Contribution

per unit ($)Total

contribution

($)

A 1,000 1,000 11,000 4 4,000

D 2,000 8,000 3,000 10 20,000

C 1,000 3,000 0 7 7,000

31,000


The following data relates to Products Able and Baker.

Product

Able Baker

Direct materials per $10 $30unit

,

Direct labour:

Grinding $5 per 7 hours per unit 5 hourshour per unit

Finishing $7.50 15 hours per unit 9 hoursper hour per unit

Selling price per $206.50 $168.00unit

Budgeted 1,200 units 600 unitsproduction

•

Maximum sales for 1,500 units 800 units

the period

t

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Notes:

(1) No opening or closing inventory is anticipated.

(2) The skilled labour used for the grinding processes is highlyspecialised and in short supply, although there isjustsufficient tomeet the budgeted production. However, it will not be possible toincrease the supplyfor the budget period.

Determine the optimal production plan and calculate the totalcontribution it earns for the company.

3 Multiple limiting factors - linear programmingLinear programming

As we have seen, when there is only one resource that limits the activities ofan organisation (other than sales demand), products are ranked in order ofcontribution per unit of limiting factor in order to establish the optimalproduction plan.

• When there is more than one limiting factor (apart from sales demand)the optimal production plan cannot be established by ranking products.In such situations, a technique known is linear programming is used.

Formulating a linear programming problem

Thefirst stage in solving a linear programming problem is to 'formulate' theproblem, i.e. translate the problem into a mathematical formula.

The steps involved in this stage are as follows.

Step 1 Define the unknowns, i.e. the variabJes (that need to bedetermined).

Step 2 Formulate the constraints, i.e. the limitations that must beplaced on the variables.

Step 3 Formulate the objective function (that needs to be maximised orminimised).

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Formulating the problem

The constraints are determined bythe scarce resources, forexample, if labour or materials are restricted.

There is also a constraint known as the non-negativity constraint.The non-negativity constraint fulfils the requirement of linearprogramming that there should be no negative values in a linearprogramming solution. You cannot make a negative amount of aproduct. Each variable in a linear programming problem musttherefore be greater than or equal to 0.

The objective function ofa linear programming problem mustalsobe formulated. The objective ofa linear programming problem isusually to maximise or minimise something. Most organisations willwish to maximise profit orcontribution. Sometimes organisationsmay wish to minimise costs.

Illustration 3 - Multiple factors - linear programming

A company makes two products, Xand Y, and wishes to maximiseprofit. Information on X and Y is as follows:

Material kg per unit

Labour hours per unit

Selling price per unit

Variable cost per unit


Product X Product Y

1 1

5 10

$ $

80 100

50 50

30 " 50

The company can sell anynumber of product X, butexpects themaximum annual demand for Y to be 1,500 units. Labour is limited to20,000 hours and materials to 3,000 kg per annum.

Required:

Using the information given, formulate the linear programming problem.

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Solution

Step 1: Define the unknowns, i.e. the variables that need to bedetermined

Let x=number of units of Xproduced and sold each year

Let y=number of units of Yproduced and sold each year

Step 2: Formulate the constraints, i.e. the limitations that must beplaced on the variables

Materials x +y 0

Step 3: Formulate the objective function that needs to bemaximised or minimised

The objective is to maximise contribution, C=30x +50y.


Abuilder has purchased 21,000 square metres of land on which it isplanned to build two types of dwelling, detached and town houses, withinan overall budget of $2.1 million. '

Adetached costs $35,000 to build and requires 600 square metres of

Atown house costs $60,000 to build and requires 300 square metres of

To comply with local planning regulations, not more than 40 buildingsmay be constructed on this land, but there must be at least 5of each

From past experience the builder estimates the contribution on theS ™nerh°.Ul81° bG ab°Ut $1°'000 8nd °n the town house t0 be about$6,000. Contribution is to be maximised.

Using the information given, formulate the linear programming problem.

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Determine the optimal solution to this linear programming problem usinga graphical approach. (Note: do not draw the iso-contribution line inorder to determine the optimal solution.)

5 Algebraic solutions

Using equations to solve linear programming problems

Equations can be used to determine where two lines cross.

For example, in Illustration 4, we established that the optimal solutionwas at Point C using the graphical method.

Point C represents the point at which the sales constraint intersects thelabour constraint.

Labour constraint 5x + 10y = 20,000 (1)

Materials constraint x + y = 3,000 (2)

The basic method is to eliminate one of the two unknowns between the

equations.

• This is achieved by adding or subtracting the equations.

• This process is known as solving simultaneous equations.

Illustration 5 - Algebraic solutions

Solve the following simultaneous equations.

5x+10y = 20,000(1)

x + y = 3,000 (2) "

Solution

Stepl

By multiplying equation (2) by 10, the coefficients of y becomeequal:

(1): 5x + 10y = 20,000

10 x (2).: 10x + 10y = 30,000. ... Equation (3)

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';£^ ; Step 2

Equation (2) when multiplied is called equation (3). You can deductequation (1) from equation (3) to eliminate y.

1' '̂.M ^ -(3) 10x-+ 10y = 30,000

(1) 5x +•10y == 20,000

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Step 3

5x 10,000

IP

• Obtain a value for x by r

v *'* "5

5x =10,000

x =10,000/5 = 2,000

Step 4

• Substitution into any of (1), (2) or (3) is possible but in this case(2)is most convenient giving:

2,000 + y = 3,000

therefore y = 1,000

• So the solution is x = 2,000 y = 1,000 as before.

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6 Further exam-style OT questions on linear programming

Many of the OT questions used throughout this chapter are longer than thoseyou would expect to see in the exam. This is to ensure that you understandthe full process of linear programming as many students consider it a

.': difficult part of the syllabus and it is essential deemed knowledge for F5. Inthis section you can practise shorter exam-style OTs.

240


Which of the following is not an assumption of linear programming?

A There are only two variables

B There must be a single objective

C The problem must be a static one

D The constraints must be linear


In a linear programming problem to determine the optimal contributionC=10x+20y, the optimal solution is given by the intersection of 5x+3y=19and 4x+y=11. The maximum profit is $ .


In a linear programming problem one constraint is that a company mustmake at least four times as manychairs as tables. If t and c representthe number of tables and chairs made respectively, what is the correctequation for this constraint?

A t = 4c

B c = 4t

C c>4t

D t>4c

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7 Chapter summary

DEAUNGWITH LIMITING

FACTORS

What is the limiting factor?

Alimiting factor is a factor that preventsacompany from achieving the level of

activity that itwould like to. It is usuallysales demand but it might be material,

labour, machine capacityand so on.

Multiple limiting factors - linearprogramming

Step 1 Define the unknowns, i.e. thevariables (that need to be declared).Step 2 Formulate the constraints, i.e.thelimitations that mustbe placed on thevariables.

Step 3 Formulatethe objectivefunction(that needs to be maximised orminimised).

Graphical solutions

Step 4 Graph the constraints andobjective function.Step 5 Determine the optimal solutionto the problem by manipulating the iso-contibution line and reading from thegraph or calculating the contributionearned at each pointof feasible area.

Optimal production plan where there is asingle limiting factor

Make products in order of contributionearned per unit of limiting factor(start

with the highest earner).

Contribution per unitof limiting factor =

Contribution per unitUnits of limiting factor required per unit

Algebraic solutions

When it is difficult to read therequiredpoints from a linear programming graph,the optimal solution to the problem canbe established (or confirmed) by solving

the relevant simultaneous equations.Simultaneous equations are used in

conjunction with the graphical method insolving linear programming problems, and

not on their own.

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BudgetingChapter learning objectives

Upon completion ofthis chapter you will be able to:

explain why organisations use budgeting, planning, control,communication, co-ordination, authorisation, motivation,evaluation

explain the stages in the budget process, including theadministrative procedures

explain, giving examples, the term 'principal budget factor' (or'limiting factor')

from data supplied, prepare budgets for sales

from data supplied, or derived, about the sales budget, preparebudgets for production

from data supplied, or derived, about the production budget,prepare budgets for material usage

from data supplied, or derived, about the materials usagebudget, prepare budgets for material purchases

from data supplied, or derived, about the production budget,prepare budgets for labour

from data supplied, or derived, about the production budget,prepare budgets for overheads

explain, and prepare from information provided: fixed, flexible,flexed budgets.

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SALES

BUDGETS

THE PURPOSES

OF

BUDGETING

BUDGETING

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BUDGET

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FIXED, FLEXIBLE AND FLEXED BUDGETS

PRODUCTION

BUDGETS

MATERIAL

BUDGETS

LABOUR

BUDGETS

1 The purposes of budgeting

Budget theory

OVERHEAD

BUDGETS

A budget is a quantitative expression of a plan of action prepared inadvance of the period to which it relates.

Budgets set out the costs and revenues that are expected to be incurred orearned in future periods.

• For example, ifyou are planning to take a holiday, you will probably havea budgeted amount that you can spend. This budget will determinewhere you go and for how long.

• Most organisations prepare budgets for the business as a whole. Thefollowing budgets may also be prepared by organisations:

- Departmental budgets.

- Functional budgets (for sales, production, expenditure and so on).

- Income statements (in order to determine the expected futureprofits).

- Cash budgets (in order to determine future cash flows).

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Purposes of budgeting

The main aims of budgeting are as follows:

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Planning for the future - in line with the objectives of the organisation.

' Controlling costs - by comparing the plan of the budget with the actualI •- results and investigating significant differences between the two.£s§K

• Co-ordination of the different activities of the business by ensuring that1HI managers are working towards the same common goal (as stated in thejjjj budget).

• Communication-budgets communicate the targets of theorganisation to individual managers.

• Motivation - budgets can motivate managers by encouraging them tobeat targets or budgets set at the beginning of the budget period.

§|§| Bonuses are often based on 'beating budgets'. Budgets, if badly set,can also demotivate employees.

• Evaluation - the performance of managers is often judged by lookingIIIp at how well the manager has performed 'against budget'.gfpl • Authorisation - budgets act as a form of authorisation of expenditure.

2 The stages in budget preparation

How are budgets prepared?

IISBefore any budgets can be prepared, the long-term objectives of anorganisation must be defined so that the budgets prepared are workingtowards the goals of the business.

Once this has been done, the budget committee can be formed, the budgetmanual can be produced and the limiting factor can be identified.

Budget committee is formed - a typical budget committee is madeup of the chief executive, budget officer (management accountant) anddepartmental or functional heads (sales manager, purchasing manager,production manager and so on). The budget committee is responsiblefor communicating policy guidelines to the people who prepare thebudgets and for setting and approving budgets.

Budgeting

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If sales is the principal budget factor, then the sales budgetmust beproduced first.

Final steps in the budget process - oncethe budget relating to thelimiting factor has been produced then the managers responsible for theother budgets can produce them. The entire budget preparation processmay take several weeks or months to complete. Thefinal stages are asfollows.

- 1 Initial budgets are prepared. Budget managers may sometimestry to build in an element of budget slack - this is a deliberate over-estimation of costs or under-estimation of revenues which canmake it easier for managers to achieve their targets.

- 2 Initial budgets are reviewed and integrated into the completebudget system.

- 3After any necessary adjustments are made to initial budgets, theyare accepted and the master budget is prepared (budgeted incomestatement, balance sheetand cash flow). This master budget isthen shown to top management for final approval.

- 4 Budgets are reviewed regularly. Comparisons between budgetsand actual results are carried out and any differences arising areknown as variances.

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Budget preparation

The preparation of budgets is illustrated as follows.

Illustration 1 - The stages in budget preparation

RAW MATERIALS

SELLING AND

DISTRIBUTION

EXPENSES BUDGET

SALES BUDGET

^«

PRODUCTION BUDGET

» 1 i

LABOUR FACTORY OVERHEAD) •COST OF GOODS

SOLD BUDGET

MASTER: BUDGET

BUDGETED INCOME

STATEMENT

CASH BUDGET

BUDGETED BALANCE SHEET

GENERAL AND

ADMINISTRATION

EXPENSES BUDGET

.%.....

CAPITAL

EXPENDITURE

BUDGET

The diagram shown above is based on sales being the principalbudget factor. This is why the sales budget is shown in Step 1.

Remember that if labour were the principal budget factor, then thelabour budget would be produced first and this would determine theproduction budget. .

Once the production budget has been determined then theremaining functional budgets can be prepared.

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Budgeting

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Continuous budgets

Continuous budget - this type of budget is prepared a year (orbudget period) ahead and is updated regularly by adding a further .accounting period (month, quarter) when the first accounting period hasexpired. If the budget period is a year, then itwill always reflect thebudget for a year inadvance. Continuous budgets are also known asrolling budgets.

3 Sales budgets

Budget preparation - functional budgets

A functional budget is a budget of income and/or expenditure which appliesto a particular function. The main functional budgets that you need to be ableto prepare are as follows:

sales budget

production budget

raw material usage budget

raw material purchases budget

labour budget

overheads budget.

Sales budgets

We shall begin our preparation of functional budgets by looking at salesbudgets. Sales budgets are fairly straightforward to prepare as the followingillustration will demonstrate.

Illustration 2 - Sales budgets

A company makes two products - PS and TG. Sales for next year arebudgeted to be 5,000 units of PS and 1,000 units of TG. Planned sellingprices are $95 and $130 per unit respectively.

Required:

Prepare the sales budget for the next year.

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limiting factor analysis looks atusing the contribution...

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