College · B.Com. (Programme) · Semester 4

DSC-4.2 — Cost Accounting

Compiled notes

DSC-4.2 — Cost Accounting

DSC-4.2 — Cost Accounting

Compiled from actual DU previous year question papers — key concepts, definitions, and exam patterns.

Cost Accounting - Complete Visual Study Guide

Course: B.Com. (Prog.) CBCS - Paper BC 4.3: Cost Accounting

Purpose: A complete, exam-focused Markdown revision guide based on the supplied course material. It is an original synthesis: use the source PDF for its full question bank and detailed numerical practice.


Contents

  1. The big picture

  2. Unit I - Introduction and Cost Concepts

  3. Unit II-A - Materials

  4. Unit II-B - Labour

  5. Unit III - Overheads and ABC

  6. Unit IV-A - Costing Methods

  7. Unit IV-B - Job, Contract and Process Costing

  8. Unit IV-C - Joint Products and By-products

  9. Unit V - Service Costing and Accounting Systems

  10. Formula bank and exam toolkit

  11. Practice checklist


1. The big picture

Cost accounting in one view

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Syllabus map

UnitWhat you must be able to doHigh-value outputs
IExplain the purpose, concepts and classifications of costDefinitions, distinctions, cost sheet
II-ABuy, store, issue and value materialsEOQ, stock levels, FIFO/LIFO/weighted average
II-BRecord and control labourIdle time, overtime, turnover, incentive plans
IIICollect, distribute and recover overheadAllocation, apportionment, absorption, ABC
IVSelect and apply an appropriate costing methodJob, contract, process, unit costing
VCost a service and reconcile accounting systemsPassenger-km, integral/non-integral accounts

The cost pyramid - a memory infographic

LayerFormulaWhat it represents
Selling PriceCost of Sales + ProfitPrice charged to customer
Cost of SalesCost of Production + Selling and Distribution OH +/- Finished Goods stock adjustmentCost of goods sold
Cost of ProductionWorks Cost + Administration OH +/- WIP adjustmentFinished output cost
Works CostPrime Cost + Factory OH +/- WIP adjustmentFactory conversion cost
Prime CostDirect Material + Direct Labour + Direct ExpensesDirectly traceable production cost

Fast rule: direct costs are traced to a cost unit. Indirect costs are collected as overhead and then allocated, apportioned and absorbed.


2. Unit I - Introduction and Cost Concepts

2.1 Meaning of cost, costing and cost accounting

TermPlain meaning
CostAmount of resources sacrificed to obtain or produce something.
CostingTechniques and processes used to ascertain cost.
Cost accountingRecording, classifying, analysing and reporting costs to support control and decisions.
Cost accountancyThe wider application of costing principles to cost ascertainment, profitability and management decision-making.
Cost unitUnit for which cost is measured, e.g., one unit, tonne, job, passenger-km.
Cost centreLocation, person, machine or department for which costs are collected.

2.2 Objectives of cost accounting

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  • Cost ascertainment: determine the cost of a product, job, process, department or service.

  • Price fixation: set a rational selling price, especially where cost-plus pricing or tenders are used.

  • Cost control: compare actual cost against standards, budgets or prior periods; investigate exceptions.

  • Profitability analysis: identify profitable products, customers, jobs or activities.

  • Decision support: compare relevant costs and benefits of alternatives.

  • Information to management: provide timely, classified data for planning and control.

2.3 Limitations

  • There is no single universal costing procedure for every business.

  • Several allocations, apportionment bases and valuation methods use judgement.

  • A cost useful for one decision may not be relevant for another.

  • It can be expensive to design, operate and continuously update a suitable system.

  • Results are estimates where assumptions are involved; cost accounting is not an exact science.

2.4 Financial, cost and management accounting

BasisFinancial accountingCost accountingManagement accounting
Primary usersExternal usersInternal operating managementSenior management
Main focusOverall profit, financial positionProduct/service/department costDecisions, plans and performance
Time focusHistorical, periodicHistorical plus current controlFuture-oriented as well as historical
Statutory requirementGenerally compulsoryUsually voluntaryVoluntary
DetailAggregatedDetailed by cost unit/centreTailored to the decision

2.5 Cost classification

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ClassificationMeaningExample
Direct materialMaterial identifiable with a unit of outputTimber in a table
Direct labourWages traceable to a unit/job/processWages of a job worker
Direct expenseExpense directly attributable to a unitRoyalty per unit, special design fee
Indirect costCannot be economically traced to one unitFactory rent, supervisor salary
Fixed costTotal cost remains constant within a relevant rangeBuilding rent
Variable costTotal cost changes with activityDirect material
Semi-variable costContains fixed and variable elementsPower bill with minimum charge
Sunk costPast cost that cannot be changedBook value of old equipment
Opportunity costBenefit forgone by choosing one alternativeRent lost by using own building
Marginal costAdditional cost of one more unitUsually variable cost of extra output
Conversion costDirect labour + factory overheadCost of converting material into output

2.6 Role of a cost accountant

PlanningControlReportingDecision support
Budget cost and set standardsCompare actual with standardPrepare cost sheets and reportsMake/buy, pricing and product mix
Design records and proceduresInvestigate deviationsReport profitabilityEvaluate alternatives
Estimate tenders/quotationsReduce waste and idle resourcesSupply management informationAdvise on relevant costs

2.7 Installing a costing system

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Use a system that is simple, economical, flexible, timely and aligned to the organisation's product, production method and management needs.

2.8 Cost sheet format

text
Direct materials consumed                         xxx
Direct wages                                      xxx
Direct expenses                                   xxx
                                                   ---
Prime cost                                        xxx
Add: Factory / works overhead                     xxx
                                                   ---
Gross works cost                                  xxx
Add/Less: WIP adjustment                          xxx
                                                   ---
Works cost                                        xxx
Add: Administration overhead                      xxx
                                                   ---
Cost of production                                xxx
Add/Less: Finished goods adjustment               xxx
Add: Selling and distribution overhead            xxx
                                                   ---
Cost of sales                                     xxx
Add: Profit                                        xxx
                                                   ---
Sales / selling price                             xxx

2.9 Mini example - cost sheet logic

Suppose direct material is 60,000, direct wages 40,000, direct expenses 5,000, factory overhead 25,000, administration overhead 10,000, selling overhead 15,000, and desired profit is 20% of sales.

StepAmount
Prime cost = 60,000 + 40,000 + 5,000105,000
Works cost = 105,000 + 25,000130,000
Cost of production = 130,000 + 10,000140,000
Cost of sales = 140,000 + 15,000155,000
Selling price = Cost of sales / 0.80193,750

If profit is 20% of selling price, cost is 80% of sales. Divide cost by 0.80. Do not add 20% of cost.


3. Unit II-A - Materials

3.1 Material control: purpose

Material control ensures the right material is available in the right quantity, quality, time and price, while keeping stock investment and losses under control.

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3.2 Key documents

DocumentPrepared/used byMain purpose
Purchase requisitionUser departmentAuthorises need for material
Request for quotationPurchase departmentInvites supplier prices and terms
Purchase orderPurchase departmentFormal order to supplier
Goods received note (GRN)Receiving/storesRecords quantity and condition received
Bin cardStorekeeperQuantity record at storage location
Stores ledgerCost/accounts departmentQuantity and value record
Material requisition noteProduction departmentAuthorises issue to a job/process
Material return noteProduction/storesRecords unused material returned
Material transfer noteDepartment to departmentRecords internal transfer

3.3 Stock levels

LevelFormula
Re-order levelMaximum usage x maximum lead time
Minimum levelRe-order level - (normal usage x normal lead time)
Maximum levelRe-order level + EOQ - (minimum usage x minimum lead time)
Danger levelAverage usage x emergency lead time
Average stock levelMinimum stock + 1/2 EOQ, or (minimum + maximum) / 2

3.4 EOQ - Economic Order Quantity

EOQ balances ordering cost against carrying cost.

EOQ=2DSH EOQ = \sqrt{\frac{2DS}{H}}

Where:

  • D = annual demand in units

  • S = ordering cost per order

  • H = annual holding/carrying cost per unit

At EOQ, the total of ordering and carrying cost is at or near its lowest point. The first falls as order size rises; the second rises as order size rises.

3.5 ABC analysis - selective control infographic

CategoryTypical share of itemsTypical share of annual usage valueControl level
A10%70%Very strict; senior review; frequent monitoring
B20%20%Normal control
C70%10%Simple, economical controls

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ABC principle: rank items by annual consumption value:

Annual usage value=Unit cost×Annual quantity used \text{Annual usage value} = \text{Unit cost} \times \text{Annual quantity used}

3.6 Pricing material issues

MethodRuleEffect when prices are rising
FIFOEarliest receipts issued firstLower issue cost; higher closing stock
LIFOLatest receipts issued firstHigher issue cost; lower closing stock
Simple averageAverage of unit prices, ignoring quantitiesEasy but less representative
Weighted averageTotal value available / total quantity availableSmooths price fluctuations
Replacement priceIssue at current replacement priceUseful for current-cost decisions
Standard pricePre-set price; difference is a varianceUseful for control

FIFO, LIFO and weighted average example

Receipts: 100 units at 10 and 100 units at 14. Issue: 120 units.

MethodIssue valuationClosing stock valuation
FIFO100 x 10 + 20 x 14 = 1,28080 x 14 = 1,120
LIFO100 x 14 + 20 x 10 = 1,60080 x 10 = 800
Weighted averageAverage = 2,400 / 200 = 12; issue = 120 x 12 = 1,44080 x 12 = 960

3.7 Stores ledger workflow

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3.8 Material losses

LossMeaningAccounting treatment
Normal lossExpected, unavoidable lossCost is absorbed by good output; scrap value reduces material cost
Abnormal lossAvoidable or unusual lossCharge separately to costing P&L / loss account
ScrapSmall residue with low sale valueCredit scrap proceeds to relevant overhead/process
DefectivesUnits repairable at extra costCharge rectification according to cause
SpoilageUnits not repairableTreat as normal or abnormal based on cause

4. Unit II-B - Labour

4.1 Labour cost control system

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ActivityQuestion answeredCommon record
Time keepingWas the employee present?Attendance register, clock card, biometric log
Time bookingWhich job/process used the employee's time?Job card, time sheet, daily time record
PayrollHow much gross/net pay is due?Wage sheet, pay slip
Labour analysisWhere should labour cost be charged?Labour distribution sheet

4.2 Time keeping vs time booking

BasisTime keepingTime booking
FocusAttendanceTime spent on each job/process
Main objectivePayroll and punctuality controlAccurate job/process costing
Typical sourceGate/biometric/clock cardJob ticket/time sheet

4.3 Methods of remuneration

MethodWage calculationBest suited for
Time rateHours worked x hourly rateQuality-sensitive work; output hard to measure
Piece rateUnits produced x rate per unitMeasurable, repetitive output
Differential piece rateDifferent rate above/below standardStrong output incentive
Halsey planTime wages + worker share of time saved x rateShared saving approach
Rowan planTime wages + (time saved/time allowed) x time taken x rateIncentive with a built-in limit

Let T be time allowed, t actual time, and R wage rate.

Incentive planFormula
Halsey (50% share)tR + 50% x (T - t)R
RowantR + ((T - t) / T) x tR

4.4 Idle time

Idle time is time paid for but not used in productive work.

TypeCausesTreatment
Normal idle timeSet-up, normal waiting, minor machine adjustment, tea breakInclude in factory overhead or inflate labour rate
Abnormal idle timeBreakdown, strike, power failure, avoidable delayCharge to costing P&L / abnormal loss

4.5 Overtime

Cause of overtimeTreatment of premium
Specific customer/job requestCharge to that job
General production requirementTreat as factory overhead
Abnormal reasonCharge to costing P&L

4.6 Labour turnover

Labour turnover means change in the workforce over a period.

Average employees=Opening employees+Closing employees2 \text{Average employees} = \frac{\text{Opening employees} + \text{Closing employees}}{2}
MeasureFormula
Separation rateSeparations / average employees x 100
Replacement rateReplacements / average employees x 100
Flux rate(Separations + replacements) / average employees x 100
Avoidable causesUnavoidable causes
Low wages, poor conditions, poor supervision, lack of training, job dissatisfactionRetirement, illness, accident, death, personal/family reasons, relocation

Cost of high turnover: recruitment, training, lost output, lower quality, more supervision, overtime and higher scrap.

4.7 Fringe benefits

Fringe benefits are employment-related benefits in addition to basic wages, such as employer retirement contributions, medical support, paid leave, canteen subsidy, insurance and welfare facilities.

Treatment: charge directly when clearly attributable to a job; otherwise include in labour/factory overhead and recover systematically.


5. Unit III - Overheads and ABC

5.1 What are overheads?

Overheads are indirect material, indirect labour and indirect expenses that cannot be directly traced economically to a cost unit.

Functional classificationExamples
Factory/production OHIndirect wages, factory rent, power, depreciation
Administration OHOffice salary, audit fee, legal expenses
Selling OHSales salaries, advertising, showroom expense
Distribution OHPacking for delivery, warehouse, delivery vehicles

5.2 Four stages of charging overhead to a cost unit

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TermMeaningExample
AllocationEntire item charged to one cost centreSalary of one department's supervisor
ApportionmentCommon cost divided among cost centres on fair basisFactory rent by floor area
Re-apportionmentService-centre cost redistributed to production centresCanteen cost to production departments
AbsorptionProduction-centre OH charged to jobs/unitsOH per labour hour

5.3 Common bases of apportionment

OverheadSuitable basis
Rent, rates, lightingFloor area
PowerMachine hours / kWh
Canteen, welfare, personnelNumber of employees
Depreciation, insuranceAsset value
Indirect wages/supervisionDirect wages / number of workers
Stores expensesMaterial value / number of requisitions
RepairsMachine value / machine hours

5.4 Primary and secondary distribution

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  • Primary distribution: allocate/apportion all overheads among production and service departments.

  • Secondary distribution: redistribute service department overheads to production departments.

  • Absorption: recover each production department's final overhead from jobs/units.

5.5 Overhead absorption rates

MethodFormulaAppropriate when
Percentage of direct materialOH / direct material x 100Material drives overhead
Percentage of direct wagesOH / direct wages x 100Labour cost is main driver
Percentage of prime costOH / prime cost x 100Mixed direct inputs are relevant
Labour hour rateOH / labour hoursLabour intensity drives OH
Machine hour rateOH / machine hoursMachine-intensive production
Unit rateOH / units producedHomogeneous output

5.6 Under- and over-absorption

PositionMeaning
Under-absorptionActual OH > OH absorbed; too little was charged to output
Over-absorptionOH absorbed > actual OH; too much was charged to output

Treatments: use supplementary rate, transfer a small abnormal amount to costing P&L, or carry forward only when justified.

5.7 Machine hour rate

Machine Hour Rate=Machine-related standing and running costsProductive machine hours \text{Machine Hour Rate} = \frac{\text{Machine-related standing and running costs}}{\text{Productive machine hours}}

Include, as applicable: depreciation, repairs, insurance, power, lubricants, operator wages, lighting share, rent share and supervision. Exclude non-productive hours from the denominator when the question requires a productive-hour rate.

5.8 Activity Based Costing (ABC)

Traditional absorption often uses one broad base, such as labour hours. ABC traces overhead through activities and cost drivers.

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Traditional systemABC system
Usually one/few volume basesMultiple cause-and-effect cost drivers
Suitable when overhead is low and output homogeneousUseful with diverse products and high overhead
Can distort complex/low-volume productsMore accurate product/activity cost
Simpler and cheaperMore data-intensive

ABC sequence: identify activities -> create cost pools -> choose drivers -> calculate driver rate -> charge output according to driver usage.


6. Unit IV-A - Costing Methods

6.1 Choosing the costing method

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MethodCost unitBest suited for
Unit/output costingOne unit, tonne, litre, kgCement, bricks, sugar, paper
Job costingOne job/order/batchPrinting, repair work, furniture
Contract costingOne contractConstruction, roads, bridges
Process costingOne process/unitChemicals, textiles, food processing
Service/operating costingPassenger-km, tonne-km, room-dayTransport, hospitals, hotels, utilities

6.2 Unit costing

Use a cost sheet to calculate total cost and cost per unit.

Cost per unit=Total costUnits produced or service units \text{Cost per unit} = \frac{\text{Total cost}}{\text{Units produced or service units}}

Exam order: classify each expense correctly -> prepare total column -> divide by output -> add required profit basis accurately.


7. Unit IV-B - Job, Contract and Process Costing

7.1 Job costing

Each job receives a distinct job number and cost record.

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Job cost sheet: direct material + direct labour + direct expenses + absorbed factory OH + administration/selling OH as required.

7.2 Contract costing

Contract costing is a form of job costing for large, long-duration, site-based jobs.

CharacteristicContract costing implication
Work performed at siteSite account and site records are important
Long durationWork-in-progress and notional profit matter
Large valueContract account is prepared separately
Materials/plant at siteShow closing value/return/depreciation separately
Certification by architect/engineerWork certified is a key WIP figure

7.3 Contract account flow

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Key formulas

Notional profit=Value of work certified+Cost of work uncertified−Cost of work to date \text{Notional profit} = \text{Value of work certified} + \text{Cost of work uncertified} - \text{Cost of work to date}

Cost of work to date includes materials, wages, direct expenses, overheads and depreciation of plant, less materials/plant returned or on hand as appropriate.

Profit recognition: follow the rule specified by your university/text. A common practice is to transfer a conservative fraction of notional profit to P&L based on the stage of completion and cash received, retaining the balance as reserve.

7.4 Process costing

Process costing applies when production passes through continuous stages. Output from one process can become input to the next.

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Normal and abnormal loss

ItemMeaningTreatment
Normal lossExpected loss inherent to processCost is borne by good output; credit any scrap value
Abnormal lossActual loss exceeds normal lossValued at cost per good unit and transferred to abnormal loss account
Abnormal gainActual loss is lower than normal lossCredit process; debit abnormal gain account
Cost per expected good unit=Total process cost−Scrap value of normal lossInput units−Normal loss units \text{Cost per expected good unit} = \frac{\text{Total process cost} - \text{Scrap value of normal loss}}{\text{Input units} - \text{Normal loss units}}

Process account layout

Debit sideCredit side
Input units, material, labour, direct expenses, overheadNormal loss, abnormal loss, transfer to next process/finished stock, closing WIP

7.5 Work-in-progress (WIP) in process costing

When WIP is given, units can be incomplete for material, labour and overhead. Use equivalent production where needed.

Equivalent units=Actual units×Percentage completion \text{Equivalent units} = \text{Actual units} \times \text{Percentage completion}

Calculate material, labour and overhead equivalent units separately when their completion percentages differ.


8. Unit IV-C - Joint Products and By-products

8.1 Core idea

When one common process produces more than one output, costs incurred before separation are joint costs. The point at which outputs become separately identifiable is the split-off point.

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TermMeaning
Joint productProduct of significant relative sales value produced jointly
By-productProduct of relatively minor sales value arising incidentally
Joint costCost incurred before split-off that benefits all joint outputs
Separable costCost incurred after split-off for one individual product

8.2 Joint cost apportionment methods

MethodBasis
Physical unitsQuantity/weight/volume produced
Sales value at split-offRelative market value at split-off
Net realisable value (NRV)Final sales value less further processing and selling cost
Constant gross-margin NRVAllocates to achieve common gross margin percentage

8.3 Decision after split-off

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Only compare incremental revenue with incremental cost. Joint cost is already incurred and is usually irrelevant to this sell-or-process-further decision.


9. Unit V - Service Costing and Accounting Systems

9.1 Service/operating costing

Service costing finds cost per unit of service rather than cost per physical product.

ServiceCommon cost unit
Passenger transportPassenger-km
Goods transportTonne-km
HotelRoom-day / bed-day
HospitalPatient-day / bed-day
PowerkWh
CanteenMeal served

9.2 Transport costing

Passenger-km=Number of passengers×Distance travelled \text{Passenger-km} = \text{Number of passengers} \times \text{Distance travelled} Tonne-km=Tonnes carried×Distance travelled \text{Tonne-km} = \text{Tonnes carried} \times \text{Distance travelled}

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Cost categoryExamples
Standing/fixed chargesInsurance, road tax, garage rent, manager salary
Maintenance chargesRepairs, servicing, tyres, depreciation where treated by policy
Running/variable chargesFuel, oil, lubricants, trip wages/tolls

Operating cost sheet skeleton

text
Standing charges                                   xxx
Maintenance charges                                xxx
Running charges                                    xxx
                                                   ---
Total operating cost                               xxx
Service units (passenger-km / tonne-km)             xxx
Cost per service unit = total cost / service units  xxx
Add profit                                          xxx
Fare or freight                                     xxx

9.3 Integral vs non-integral accounting

BasisIntegral accountingNon-integral accounting
Books maintainedOne integrated setSeparate cost and financial ledgers
DuplicationAvoidedPossible
ReconciliationNormally unnecessaryOften necessary
Main advantageUnified, timely informationDetailed independent cost records
Other nameIntegrated accountsInterlocking accounts may be used

9.4 Reconciliation of profit

Reconcile profit when cost and financial books show different results, usually because of different stock valuation, overhead absorption, purely financial items or different depreciation/treatment of expenses.

Rendering diagram…

Common reconciling itemWhy it creates a difference
Over/under-absorption of OHCost books recover OH at pre-set rate; financial books show actual OH
Different stock valuationOpening/closing material, WIP or finished goods may differ
Interest, dividend, income taxUsually financial-account items only
Goodwill/preliminary expense write-offOften financial-account item only
Different depreciationCost and financial policies can differ

Direction check: Start with the stated profit. Add items missing from the target profit but present in the starting profit, and subtract items that have the opposite effect. Label every adjustment with the reason before entering it.


10. Formula bank and exam toolkit

10.1 Formula bank

TopicFormula
Prime costDirect material + direct labour + direct expenses
Conversion costDirect labour + factory overhead
Works costPrime cost + factory OH +/- WIP adjustment
Cost of productionWorks cost + administration OH
Cost of salesCost of production +/- finished goods adjustment + selling/distribution OH
EOQsqrt(2DS/H)
Re-order levelMaximum usage x maximum lead time
Average stockMinimum stock + 1/2 EOQ
Weighted average issue rateTotal stock value / total stock quantity
Labour turnoverSeparations/replacements/flux numerator divided by average employees x 100
OH absorption rateBudgeted OH / budgeted base
Machine hour rateMachine-related OH / productive machine hours
Process cost per good unit(Process cost - normal-loss scrap value) / expected good units
Passenger-kmPassengers x km travelled
Tonne-kmTonnes x km travelled

10.2 Profit basis traps

If the question saysUse
Profit is 25% on costSelling price = cost x 1.25
Profit is 25% on salesSelling price = cost / 0.75
Gross profit is 20% on salesCost of sales = 80% of sales
Cash received is 80% of work certifiedCash = 80% x work certified; retention = balance

10.3 Numerical-answer workflow

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10.4 Presentation rules that earn marks

  1. Start with a heading: Stores Ledger Account, Cost Sheet, Contract Account, Process A Account, etc.

  2. Write working notes before the final statement whenever a rate or allocation is derived.

  3. Use consistent units: rupees, units, hours, passenger-km or tonne-km.

  4. Show formulas, not just answers.

  5. Keep total columns and per-unit columns separate.

  6. For process questions, show units and value side by side.

  7. For reconciliation, state whether each item is added or deducted and why.


11. Practice checklist

Theory questions

  • Define cost accounting and explain objectives and limitations.

  • Distinguish financial, cost and management accounting.

  • Explain cost classification with examples.

  • Explain the role of a cost accountant and factors in installing a costing system.

  • Explain ABC analysis, its benefits and limitations.

  • Explain time keeping, time booking, idle time, overtime and labour turnover.

  • Explain allocation, apportionment, absorption and ABC.

  • Compare traditional overhead costing with ABC.

  • Explain job, contract, process and service costing.

  • Explain integral and non-integral accounting and why reconciliation is required.

Numerical questions

  • Prepare a cost sheet and calculate selling price under both profit bases.

  • Calculate EOQ and stock levels.

  • Prepare FIFO, LIFO and weighted average stores ledger.

  • Calculate labour turnover and incentive earnings.

  • Apportion overheads, re-apportion service department costs and calculate absorption rates.

  • Compute a machine hour rate and deal with under/over-absorption.

  • Prepare job cost sheet and quotation.

  • Prepare contract account, calculate notional profit and show WIP.

  • Prepare process accounts with normal loss, abnormal loss and abnormal gain.

  • Apportion joint cost and make sell-or-process-further decision.

  • Prepare an operating cost sheet and cost per passenger-km/tonne-km.

  • Reconcile profits from cost accounts to financial accounts.

Final one-page memory map

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Source and use note

This guide follows the syllabus and topics visible in COST ACCOUNTING SHIVDAS 2.pdf, including its worked-practice emphasis. It deliberately paraphrases and reorganises the material into concise revision notes and diagrams; consult the original PDF for the complete wording of its questions and full numerical solutions.

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