Alimiting factor is a factor that prevents a company from achieving the level of activity that it would like to.

Transcription

1 T Dealing with limitingfactors DEALING WITH LIMITING FACTORS Multiple limiting factors - linear programming Optimal production plan where there is a single limiting factor Algebraic solutions 1 What is a limiting factor? Alimiting factor is a factor that prevents a company from achieving the level of activity that it would like to. SJ Limiting factor analysis looks at using the contribution concept to address 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 isin short supply. Production can also be affected by the number of units of a product that is 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 the same machine and the same raw material that are limited to 600 hours and $800 per week respectively. Individual product details are as follows. 224

2 chapter 10 Product X per Product Y per unit unit Machine hours Materials $10 $5 Contribution $20 $15 Maximum weekly demand 50 units 100 units Comment on whether machine hours and/or materials are limiting factors ffi Solution To make the maximum demand of 50 units of X and 100 units of Y requires 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 that could be sold but materials limit the production plan. Test your understanding 1 Two products, Alpha and Gamma are made of Material X and require skilled labour in the production process. The product details are as follows: Alpha Gamma $ $ Selling price Variable cost Contribution Material X required per unit Skilled labour time required per unit 2 kg 4 kg 1 hour 3 hours 225

3 Dealing with limiting factors The maximum demand per week is 30 units of Alpha and 10 units of Gamma. ' The company can sell all the Alphas and Gammas that it can make but here is a restriction on the availability of both Material Xand skilled labour There are 150 kg of material, and 45 hours of skilled labour available per week. Identify the limiting factor. ;; ': 2Optimal production plan where there is asingle limiting factor pontributipn perunit of limiting factor In order to decide which products should be made in which order it is necessary to calculate the contribution per unit of limiting factor (or scarce resource). K ::- : : &&% r.'v;-; -':::, :*; ; :: :' '&:} :--.'-.---."""."v;.' Contribution per unit of limiting factor Contribution per unit Units of limiting factor required per unit Optimal production plan.-, :,. '''. :-.: mm?. v ;?; -.(, :;V; -: -:-..-. When limiting factors are present, contribution (and therefore profits) are maximised when products earning the highest amount of contribution per unit of limiting factor are manufactured first. The profit-maximising production 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 scarce resource (limiting factor) is used up. 226

4 Eg Illustration 2 - Optimal production plan where there is a single Acompany is able to produce four products and is planning its production mix for the following period. Relevant data is given below: chapter 10 A B C D Selling price ($) per unit Labour cost per unit ($) Material cost per unit ($) 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 hours in the period. Required: Determine the optimal production plan and calculate the total contribution it earnsfor the company. 3 Solution B D Selling price $ 50 Variable costs: Direct labour (6) (12) (18) (24) frtf;. Direct material Contribution per unit (9) (9) (15) (16) 10 Hours per unit (labour cost/$6) D Contribution per hour $4 Rank $2 $2.33 $2.50 [ 227

5 Dealing with limiting factors Remember to allocate the scarce resource (labour hours) to the highestranking product first (A). Once the demand for the highest-ranking product is satisfied, move on to the next highest-ranking product (D) and then 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 D 2,000 8,000 3, ,000 C 1,000 3, ,000 31,000!$r^ Test your understanding 2 The following data relates to Products Able and Baker. Product Able Baker Direct materials per $10 $30 unit, Direct labour: Grinding $5 per 7 hours per unit 5 hours per unit hour Finishing $ hours per unit 9 hours per hour per unit Selling price per $ $ unit Budgeted 1,200 units 600 units production Maximum sales for 1,500 units 800 units the period t 228

6 chapterio Notes: (1) No opening or closing inventory is anticipated. (2) The skilled labour used for the grinding processes is highly specialised and in short supply, although there isjustsufficient to meet the budgeted production. However, it will not be possible to increase the supplyfor the budget period. Determine the optimal production plan and calculate the total contribution it earns for the company. ;". ; 3 Multiple limiting factors - linear programming Linear programming ^:; v;,;\ As we have seen, when there is only one resource that limits the activities of an organisation (other than sales demand), products are ranked in order of contribution per unit of limiting factor in order to establish the optimal production 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 The first stage in solving a linear programming problem is to 'formulate' the problem, 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 be determined). Step 2 Formulate the constraints, i.e. the limitations that must be placed on the variables. Step 3 Formulate the objective function (that needs to be maximised or minimised). 229

7 Dealing with limiting factors 9 Formulating the problem The constraints are determined bythe scarce resources, for example, 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 linear programming that there should be no negative values in a linear programming solution. You cannot make a negative amount of a product. Each variable in a linear programming problem must therefore be greater than or equal to 0. The objective function ofa linear programming problem must also be formulated. The objective ofa linear programming problem is usually to maximise or minimise something. Most organisations will wish to maximise profit orcontribution. Sometimes organisations may wish to minimise costs. KJ Illustration 3 - Multiple factors - linear programming A company makes two products, Xand Y, and wishes to maximise profit. 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 $ $ Contribution per unit 30 " 50 The company can sell anynumber of product X, butexpects the maximum annual demand for Y to be 1,500 units. Labour is limited to 20,000 hours and materials to 3,000 kg per annum. Required: Using the information given, formulate the linear programming problem. 230

8 ::. ^-...,.r~. _: ---^. m p Solution Step 1: Define the unknowns, i.e. the variables that need to be determined chapter 10 J-S 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 be placed on the variables mm ;; :. r KJi : %, Materials x +y <3,000 Labour hours 5x + 10y <20,000 Maximum sales y< 1,500 Non-negativity x >0, y>0 Step 3: Formulate the objective function that needs to be maximised or minimised The objective is to maximise contribution, C=30x +50y. % m.-- : 1 y 1;:' Test your understanding 3 Abuilder has purchased 21,000 square metres of land on which it is planned to build two types of dwelling, detached and town houses, within an 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 buildings may be constructed on this land, but there must be at least 5of each From past experience the builder estimates the contribution on the S 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. 231

9 chapterio Determine the optimal solution to this linear programming problem using a graphical approach. (Note: do not draw the iso-contribution line in order 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 solution was at Point C using the graphical method. Point C represents the point at which the sales constraint intersects the labour 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) ".'..:.v.-- m ' ' - - 'j. ;;:. i=v --: '4 m;, f p : : "... ;'' SeSSiS M Hi I Solution Stepl By multiplying equation (2) by 10, the coefficients of y become equal: (1): 5x + 10y = 20, x (2).: 10x + 10y = 30, Equation (3) 237

10 Dealing with limiting factors '; ^ ; Step 2 Equation (2) when multiplied is called equation (3). You can deduct equation (1) from equation (3) to eliminate y. 1'^'.M ^ - (3) 10x-+ 10y = 30,000 (1) 5x + 10y == 20,000 v 'A.* - 5x 10,000 IP Step 3 Obtain a value for x by r 5x =10,000 v *'* "5 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. fiv.^j KAPLANPUBLISHING

11 . ft-.-: -r :> : ; chapter 10 i:s.i^l.- v --.;,,_ - VW Establishing the optimal solution using simultaneous equations You may consider that the whole process of linear programming would be easier by solving the constraints as sets of simultaneous equations and not bothering with a graph. This is possible and you may get the right answer, but such a technique should be used with caution! It is not recommended that you solve sets of simultaneous equations until you have determined graphically which constraints are effective in determining the optimal solution. The solutions obtained for linear programming problems using the graphical method rely on the graphs being drawn very clearly so that accurate readings may be taken. If it is not easyto read the co-ordinates ofthe required points from a linear programming graph then the use of simultaneous equations provides a reliable check. Test your understanding 5 Alfred Ltd is preparing its plan for the coming month. It manufactures two products, the Flaktrap and the Saptrap. Details are as follows. V"'-.-,--,;."- -"..«: :-. m \ : -: : ; ift. : ' -' *«..-'=.? :'.. Amount per unit Selling price ($) Raw material (kg) Labour hours Skilled Semi-skilled Product iktrap Saptrap Price/wage rate $5/kg $3/hour 5 25 $3/hour : ",.? ',.- " ; I: i :":-".' :.-' -L'.';~- The company's variable overhead rate is $1 per labour hour (for both skilled and semi-skilled labour). The supply of skilled labour is limited to 2,000 hours per month and the supply of semi-skilled labour is limited to 2,500 hours per month. At the selling prices indicated, maximum demand for Flaktraps is expected to be 150 units per month and the maximum demand for Saptraps is expected to be 80 units per month. Use simultaneous equations to solve the linear programming problem given above. \,.y.-j....;.; - - Hi s.<r--s.-..-?;. :-:.,:. - :-.i it:'-:-,; - '-" : 239

12 Dealing with limiting factors 6 Further exam-style OT questions on linear programming.': Many of the OT questions used throughout this chapter are longer than those you would expect to see in the exam. This is to ensure that you understand the full process of linear programming as many students consider it a difficult part of the syllabus and it is essential deemed knowledge for F5. In this section you can practise shorter exam-style OTs. Test your understanding 6 Which of the following is not an assumption of linear programming? A B C D There are only two variables There must be a single objective The problem must be a static one The constraints must be linear Test your understanding 7 In a linear programming problem to determine the optimal contribution C=10x+20y, the optimal solution is given by the intersection of 5x+3y=19 and 4x+y=11. The maximum profit is $. Test your understanding 8 In a linear programming problem one constraint is that a company must make at least fourtimes as manychairs as tables. If t and c represent the number of tables and chairs made respectively, what is the correct equation for this constraint? A B C D t = 4c c = 4t c>4t t>4c 240

13 Ml chapter 10 i : I Test your understanding 9 A linear programming problem has the following features: :ffik : :-;:'r=ft; s; ;.-:.'.-i.'j ": : 1 ~ ' "'. ' ': : 1«Objective function: Maximise C = 30x + 25y Non-negativity constraints: x 0, y > 0 Constraint 1: 4x+3y< 64 Constraint 2: 2x+5y< 60 The optimal solution is given by A x=10, y=8 B x=16, y=0 C x=0, y=12 D x=30,y=0 X-h:-** i'.---tt:':- & : :, -:* i-.', '(<?'.-: s H / il ill 1 :. J 241

14 Dealing with limiting factors 7 Chapter summary DEAUNGWITH LIMITING FACTORS What is the limiting factor? Alimiting factor is a factor that preventsa company from achieving the level of activity that itwould like to. It is usually sales demand but it might be material, labour, machine capacityand so on. Multiple limiting factors - linear programming Optimal production plan where there is a single limiting factor Step 1 Define the unknowns, i.e.the variables (that need to be declared). Step 2 Formulate the constraints, i.e.the limitations that mustbe placed on the variables. Step 3 Formulatethe objectivefunction (that needs to be maximised or minimised). Make products in order of contribution earned per unit oflimiting factor(start with the highest earner). Contribution per unitof limiting factor = Contribution per unit Units of limiting factor required per unit Graphical solutions Algebraic solutions :'.. Step 4 Graph the constraints and objective function. Step 5 Determine the optimal solution to the problem by manipulating the isocontibution line and reading from the graph or calculating the contribution earned at each point offeasible area. When itis difficult to read therequired points from a linear programming graph, the optimal solution to the problem can be established (or confirmed) by solving the relevant simultaneous equations. Simultaneous equations are used in conjunction withthe graphical method in solving linear programming problems, and not on their own. 242

15 chapter 13 Budgeting Chapter 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 the administrative 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, prepare budgets for production from data supplied, or derived, about the production budget, prepare budgets for material usage from data supplied, or derived, about the materials usage budget, 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. 311

16 "7 S# THE PURPOSES OF BUDGETING BUDGETING I THE STAGES OF BUDGET PREPARATION V 4 FIXED, FLEXIBLE AND FLEXED BUDGETS SALES BUDGETS PRODUCTION BUDGETS MATERIAL BUDGETS LABOUR BUDGETS OVERHEAD BUDGETS 1 The purposes of budgeting Budget theory A budget is a quantitative expression of a plan of action prepared in advance of the period to which it relates. Budgets set out the costs and revenues that are expected to be incurred or earned in future periods. / ;. ;.. ''?'. ";.'. -. For example, ifyou are planning to take a holiday, you will probably have a budgeted amount that you can spend. This budget will determine where you go and for how long. Most organisations prepare budgets for the business as a whole. The following 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 future profits). - Cash budgets (in order to determine future cash flows)...,-.-. ', v : ; k.":- /.'. : 312

17 - - chapter 13 Hi : Purposes of budgeting The main aims of budgeting are as follows: *:;.'!-.v.r. Planning for the future - in line with the objectives of the organisation.. ' Controlling costs - by comparing the plan of the budget with the actual I - results and investigating significant differences between the two. s K Co-ordination of the different activities of the business by ensuring that 1HI jjjj managers are working towards the same common goal (as stated in the budget). Communication-budgets communicate the targets of the organisation to individual managers. Motivation - budgets can motivate managers by encouraging them to beat 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. IIIp Evaluation - the performance of managers is often judged by looking at how well the manager has performed 'against budget'. gfpl Authorisation - budgets act as a form of authorisation of expenditure. vv-'-:-v.;.:v.1 i"!..vr.l IIS m mm 2 The stages in budget preparation How are budgets prepared? Before any budgets can be prepared, the long-term objectives of an organisation must be defined so that the budgets prepared are working towards the goals of the business. Once this has been done, the budget committee can be formed, the budget manual can be produced and the limiting factor can be identified. <n Budget committee is formed - a typical budget committee is made up of the chief executive, budget officer (management accountant) and departmental or functional heads (sales manager, purchasing manager, production manager and so on). The budget committee is responsible for communicating policy guidelines to the people who prepare the budgets and for setting and approving budgets. Budget manual is produced - an organisation's budget manual sets out instructions relating to the preparation and use of budgets. It also gives details of the responsibilities ofthose involved in the budgeting process, including an organisation chart and a list of budget holders. Limiting factor is identified - in budgeting, the limiting factor is known as the principal budget factor. Generally there will be one factor that will i limit the activityof an organisation in a given period. Itis usually sales that limit an organisation's performance, but it could be anything else, HfMS^Itt..., -ill -1 for example, the availability of special labour skills. 31-3

18 . Budgeting If sales is the principal budget factor, then the sales budget must be produced first. Final steps in the budget process - once the budget relating to the limiting factor has been produced then the managers responsible for the other budgets can produce them. The entire budget preparation process may take several weeks or months to complete. Thefinal stages are as follows. : ;. y :,' : : - 1 Initial budgets are prepared. Budget managers may sometimes try to build in an element of budget slack - this is a deliberate overestimation of costs or under-estimation of revenues which can make it easier for managers to achieve their targets. - 2 Initial budgets are reviewed and integrated into the complete budget system. - 3After any necessary adjustments are made to initial budgets, they are accepted and the master budget is prepared (budgeted income statement, balance sheetand cash flow). This master budget is then shown to top management for final approval. - 4 Budgets are reviewed regularly. Comparisons between budgets and actual results are carried out and any differences arising are known as variances. 3i4 >=*,!>? : :-;: -7-1 A..'-.y.o-. ^^7m&/-&&^^

19 r.... chapter 13 Budget preparation mm The preparation of budgets is illustrated as follows. Illustration 1 - The stages in budget preparation - - ' vi! SALES BUDGET -..- :.;,'' ; ^«PRODUCTION BUDGET» 1 i RAW MATERIALS LABOUR ) FACTORY OVERHEAD COST OF GOODS til :' v. ' SELLING AND DISTRIBUTION EXPENSES BUDGET SOLD BUDGET GENERAL AND ADMINISTRATION EXPENSES BUDGET MASTER: BUDGET BUDGETED INCOME STATEMENT '" '!" ; CASH BUDGET.%... CAPITAL EXPENDITURE BUDGET V :. :«; [-:v...;; BUDGETED BALANCE SHEET --..: -..: " ' V '..si;-. "'. : I.? ; -.* : &\$ 'J- v'.:-..'.;-> ;-.r.' '! : - '"'"ftp :m The diagram shown above is based on sales being the principal budget factor. This is why the sales budget is shown in Step 1. Remember that if labour were the principal budget factor, then the labour budget would be produced first and this would determine the production budget.. Once the production budget has been determined then the remaining functional budgets can be prepared. : ':: -' : : - 315

20 "T Budgeting Continuous budgets Continuous budget - this type of budget is prepared a year (or budget period) ahead and is updated regularly by adding a further. accounting period (month, quarter) when the first accounting period has expired. If the budget period is a year, then itwill always reflect the budget for a year inadvance. Continuous budgets are also known as rolling budgets. 3 Sales budgets Budget preparation - functional budgets I $ vm A functional budget is a budget of income and/or expenditure which applies to a particular function. The main functional budgets that you need to be able to prepare are as follows: sales budget production budget raw material usage budget raw material purchases budget labour budget m 1^ overheads budget. Sales budgets We shall begin our preparation of functional budgets by looking at sales budgets. Sales budgets are fairly straightforward to prepare as the following illustration will demonstrate. Illustration 2 - Sales budgets A company makes two products - PS and TG. Sales for next year are budgeted to be 5,000 units of PS and 1,000 units of TG. Planned selling prices are $95 and $130 per unit respectively. Required: Prepare the sales budget for the next year. 316