Chapter 9 & 26 - Limiting Factor, Throughput, & Linear Programming
Limiting Factor Analysis
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In the short term the sales demand maybe in excess of current production capacity.
This could be because the output is restricted by a shortage of either of the following
resources:
o Skilled Labour
o Materials
o Equipment, or
o Space
A limiting factor (AKA Scarce resource) is any factor that is in scare supply.
o The scarce resource stops the organization from expanding its activities
further, so that there is a maximum level of activity at which the organization
can operate.
Within a short-term time period it is unlikely that constraints can be removed, and
additional resources acquired.
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Assumption – Based on marginal costing principles i.e., fixed cost is unaffected.
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Contribution/Profit Maximisation Combination – Where there is a limiting factor,
products should be ranked based on their contribution per limiting factor or scarce
resource.
o Profit is maximized when the greatest possible contribution to profit is
obtained each time the scarce or limiting factor is used.
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Process
o Determine which products / activities have the greatest contribution per
scarce resource and rank products/activities accordingly - from highest to
lowest contribution per scarce resource.
o Allocate the scarce resource based on the ranking above and work out the
optimal production plan.
o This will maximize the amount of contribution (and profit) overall.
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Qualitative Factors to consider before determining the final production
programme:
o Customer Goodwill may be lost causing a fall in future sales if the
organization cannot produce all the products in its product line.
o Inability to satisfy demand for certain products may drive customers away to
competitors.
o Relatedness of the products – e.g., are they complementary or substitute
goods? If so, lower sales for one product will affect sales of the other
products, which have not been taken into account.
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Downsides of Limiting Factor Analysis:
o Difficulty in applying this procedure if there is more than one scarce resource.
Therefore, Linear Programming would have to be used to determine the
optimal production programme.
o Only applies to those situations in which capacity constraints cannot be
removed in the short term. In the longer-term additional resources should be
acquired if the contribution from the extra capacity exceeds the cost of
acquisition.
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Contribution = Selling Price – Total Variable Cost
OR
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Contribution = Profit + Fixed Cost
Explain briefly why the concept of contribution analysis is used in situations
of scarce resources?
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This concept is based on marginal costing principles. Fixed costs are sunk or
irrelevant to the decision. Therefore, it is prudent to rank products based on
contribution per unit of scarce resource and to produce and sell products that have
not the highest contribution per unit but contribution per limiting factor. This is how
the scarce resource should be allocated. Using this concept in limiting factor
decisions will maximise profits.
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We should always prioritize the sales of our products according to their contribution
per limiting factor. Usually, the limiting factor is Demand and we can use
contribution margin per unit to prioritize. When a resource becomes the limiting
factor we do the same thing and prioritize by contribution margin per limiting factor,
(e.g. the scarce resource of labour, raw materials etc.)
Throughput Accounting
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The theory of constraints (TOC) and throughput accounting are illustrations of
where cross-functional collaboration between operations management and
accounting can help to generate improved performance.
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Goldratt and Cox (1984) advocated a new approach to production management
called Optimized Production Technology (OPT).
o OPT is based on the principle that profits are expanded by increasing
throughput of the plant.
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Throughput is a measure of profitability and is defined by the following equation:
o Throughput = Sales Revenue – Direct Material Cost
 Only Direct Materials Cost is thought to be variable whereas rest of
the costs (i.e., Direct Labour, Production Overhead, etc.) are thought
to be Fixed.
The OPT approach determines what prevents throughput being higher by
distinguishing between bottleneck and non-bottleneck resources.
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A bottleneck is a limiting factor whose capacity limits the throughput of the whole
production process.
o The aim is to identify bottlenecks and remove them or, if this is not possible,
ensure that they are fully utilized at all times.
o Non-bottleneck resources should be scheduled and operated based on
constraints within the system and should not be used to produce more than
the bottlenecks can absorb.
 E.g. Two machines – X & Y. X can produce 500 units and Y produces
600 units then Machine X is a bottle neck and only 500 units should
be produced as an additional 100 units will just result in increase in
WIP and no increase in sales revenue.
o The OPT philosophy therefore advocates that non-bottleneck resources
should not be utilized to 100 per cent of their capacity, since this would
merely result in an increase in inventory. There is no point in a nonbottleneck activity supplying more than the bottleneck activity can consume.
This would merely result in an increase in WIP inventories and no increase in
sales volume.
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The theory of constraints (TOC) is described by Goldratt and Cox (1992) as the
process of maximizing operating profit when faced with bottleneck and nonbottleneck operations. The process involves five steps:
1.
2.
3.
4.
Identify the system’s bottlenecks;
Decide how to exploit the bottlenecks;
Subordinate everything else to the decision in step 2;
Elevate (remove) the system’s bottlenecks;
5. If, in the previous steps a bottleneck has been broken, go back to step 1 and
repeat the process for the new bottleneck.
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The TOC is a process of continuous improvement to clear the throughput chain of all
constraints. Thus, step 4 involves taking action to remove (that is elevate) the
constraint.
o This might involve replacing a bottleneck machine with a faster one,
increasing the bottleneck efficiency or changing the design of the product to
reduce the processing time required by the activity.
o When a bottleneck activity has been elevated and replaced by a new
bottleneck, it is necessary to return to step 1 and repeat the process.
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To apply TOC ideas Goldratt and Cox advocate the use of three key measures:
1. Throughput contribution - which is the rate at which the system generates profit
through sales. It is defined as sales less direct materials.
2. Investments (inventory) - which is the sum of inventories, research and
development costs and the costs of equipment and buildings.
3. Other operational expenses (also known as total factory cost and conversion
costs), - which include all operating costs (other than direct materials) incurred
to earn throughput contribution.
o The TOC aims to increase throughput contribution while simultaneously
reducing inventory and operational expenses.
o However, the scope for reducing the other operational expenses is limited
since they must be maintained at some minimum level for production to take
place at all. In other words, other operational expenses are assumed to be
fixed costs.
o The duo advocated a throughput orientation whereby increasing throughput
must be given first priority, decreasing inventories second and decreasing
operational expenses last.
o The TOC adopts a short-run time horizon and treats all operating expenses
(including direct labour but excluding direct materials) as fixed, thus
implying that variable costing should be used for decision-making, profit
measurement and inventory valuation.
Throughput Accounting ratio
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Galloway and Waldron (1988) advocated an approach called throughput accounting
to apply the TOC philosophy.
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To ascertain the optimal use of the bottleneck activity, they rank the products
according to a measure they have devised called the throughput accounting (TA)
ratio. The define the TA ratio as:
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𝑻𝒉𝒓𝒐𝒖𝒈𝒉𝒑𝒖𝒕 𝒂𝒄𝒄𝒐𝒖𝒏𝒕𝒊𝒏𝒈 𝒓𝒂𝒕𝒊𝒐 [𝑻𝑷𝑨𝑹] =
Where
𝑇ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 [𝑅𝑒𝑡𝑢𝑟𝑛] 𝑝𝑒𝑟 𝑓𝑎𝑐𝑡𝑜𝑟𝑦 ℎ𝑟
𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑓𝑎𝑐𝑡𝑜𝑟𝑦 ℎ𝑟
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𝑻𝒉𝒓𝒐𝒖𝒈𝒉𝒑𝒖𝒕 𝒄𝒐𝒏𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏 [𝑹𝒆𝒕𝒖𝒓𝒏] 𝒑𝒆𝒓 𝒇𝒂𝒄𝒕𝒐𝒓𝒚 𝒉𝒓 =
𝑇ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
𝑃𝑟𝑜𝑑𝑢𝑐𝑡 ′ 𝑠 𝑡𝑖𝑚𝑒 𝑜𝑛 𝑏𝑜𝑡𝑡𝑙𝑒𝑛𝑒𝑐𝑘 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
o Throughput per unit – Sales price – direct materials cost
𝑇𝑜𝑡𝑎𝑙 𝐹𝑎𝑐𝑡𝑜𝑟𝑦 𝐶𝑜𝑠𝑡 (𝑖.𝑒.𝑜𝑡ℎ𝑒𝑟 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠)
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𝑪𝒐𝒔𝒕 𝒑𝒆𝒓 𝒇𝒂𝒄𝒕𝒐𝒓𝒚 𝒉𝒓 =
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For multi products, return per factory hr differs due to different throughput
contribution per unit but cost per factory hr remains the same.
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Interpretation
o TPAR > 1 – suggests that throughput contribution exceeds operating costs
so the product should make a profit. Priority should be given to the products
generating the best ratios.
o TPAR < 1 – suggests that throughput is insufficient to cover operating costs.
𝑇𝑜𝑡𝑎𝑙 𝑏𝑜𝑡𝑡𝑙𝑒𝑛𝑒𝑐𝑘 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒 𝑡𝑖𝑚𝑒 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒
Multi-product Decisions
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Throughput accounting may be applied to a multi-product decision making problem
in exactly the same way as was described for limiting factor analysis
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Given that fixed costs are unaffected by the production decision in the short run, the
approach should be to maximise the throughput contribution earned.
Process:
1.
2.
3.
4.
5.
Identify the bottleneck constraint.
Calculate the throughput per unit for each product.
Calculate the throughput per unit of the bottleneck resource for each product.
Rank the products in order of the throughout per unit of the bottleneck resource.
Allocate resources using this ranking
Improving TPAR
1. Increase the sales price for each unit sold, to increase the throughput per unit
2. Reduce material costs per unit (e.g., by changing materials or switching suppliers)
3. Reduce total operating expenses, to reduce the cost per factory hour
4. Improve the productivity of the bottleneck, e.g., the assembly workforce or the
bottleneck machine, thus reducing the time required to make each unit of product.
Throughput Return per factory hour would increase and therefore the TPAR would
increase.
Differences between Throughput and Contribution
1. Contribution treats direct materials, direct labour and variable overheads as variable
costs, whereas through-put accounting assumes that only direct materials represent
variable costs.
2. Throughput accounting is thus more short-term oriented and assumes that direct
labour and variable overheads cannot be avoided within a very short-term period. In
contrast, contribution assumes that the short term represents a longer period than
that assumed with throughput accounting and thus classifies direct labour and
variable overheads as variable costs that vary with output in the longer term.
Linear Programming
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Linear Programming is a technique for solving certain types of decision problems
where more than one scarce resource exists.
o It is applied to those problems which require the optimisation of some
criterion, for example maximising profit / contribution or minimising costs,
but where the actions that can be taken are restricted by limitations, such as
the amount of finance available
o Decision problems consisting of 2 products only will be dealt with
o 2 or more products – simplex method will be used (outside the scope of the
module)
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The objective function refers to the quantification of an objective, and usually takes
the form of maximizing profits or minimizing costs.
Assumptions
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Contribution per unit for each product and the utilization of resources per unit are
the same whatever quantity of output is produced and sold within the output
range being considered
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It must also be assumed that units produced, and resources allocated are infinitely
divisible. This means that an optimal plan that suggests we should produce 94.38
units is possible. However, it will be necessary to interpret the plan as a production
of 94 units.
Graphical Method
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The graphical method of linear programming can be used when there are just two
products. The steps involved are as follows:
1. Formulate the problem
a. Define Variables
b. Specify the objective function (Maximising or minimising)
c. Establish Constraints
d. Construct Objective Function
2. Draw the constraints on the graph
3. Establish the feasible region for optimal solution
4. Determine the optimal solution
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To prevent nonsensical results, we must include a non-negativity requirement or
constraint, which is a statement that all variables in the problem must be equal to or
greater than zero.
Optimal Solution
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Graphical Method
o Plot the Contribution Function inside the feasible region by choosing a random total
contribution value.
o Extend that contribution line until it intersects with the last corner of the feasible
region boundary (Corner in the feasible region that intersects with the highest
contribution line).
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Algebraically
o Find coordinates of the corner points of the feasible region
o Work out the total contribution at each corner and select the point which
results in the highest contribution.
Binding Constraints, Slack and Surplus
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The constraints that are at the optimal production point are called binding
constraints. Thus, if this constraint were to be changed slightly (in a certain
direction), this optimal solution would no longer be feasible.
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Slack occurs when the maximum availability of a resource or other constraining
factor is not used (there are more resources available than are required). Slack is the
amount of the unused resource or other constraint, where the constraint is a ‘less
than or equal to’ constraint.
o Slack occurs in non-binding constraint.
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Surplus occurs when more than a minimum requirement is used: surplus is the excess
over the minimum amount of constraint, where the constraint is a ‘more than or
equal to’ constraint.
Implications of Slack
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If the amount of slack is high, it is an indication of inefficient use of a particular
resource. If possible, the resource should be re-allocated to another part of the
business. Resources such as labour time may even be subcontracted to another
business.
If slack is low, then management should be alert to the fact that this resource could
become binding constraint and take on appropriate action, e.g., recruit additional
labour, etc.
Shadow Prices
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The shadow price (or dual price or opportunity cost) of a limiting factor is the
increase in value which would be created by having one additional unit of the limiting
factor (binding constraint).
o In terms of LP, it is the extra contribution or profit that may be earned by
relaxing a binding constraint by one unit i.e., extra contribution from
employing an extra labour hour or using an extra kg of materials.
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When the additional unit is added to the constraint equation, the constraint line
moves up in the graph.
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This optimal response from an independent marginal increase in a resource is called
the marginal rate of substitution
o I.e., increase in qty of products produced from increasing one kg of materials
or employing an extra labour hour.
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The shadow price represents the maximum premium above the basic rate that an
organisation should be willing to pay for one extra unit of a resource.
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If Shadow price for an additional kg of material is £0.40 and the cost of material per
unit was £4. Hence, the maximum amount that the organisation should be willing
to pay for an additional unit of material is £4.40. Of course, any amount below
£4.40 will result in an increase in total contribution and hence profits.
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Since shadow prices indicate the effect of one unit change in a constraint, they
provide a measure of the sensitivity of the result.
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The shadow price of a constraint that is not binding at the optimal solution is zero.
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Shadow prices are only valid for a small range before the constraint becomes nonbinding or different resources become critical.
Uses of Linear Programming
1. Calculation of Relevant Costs
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LP can assist in the calculation of relevant costs which is essential for decision
making.
When a resource is scarce, alternative uses exist that provide a contribution. An
opportunity cost is therefore incurred whenever the resource is used. The relevant
cost for a scarce resource is calculated as:
o Acquisition Cost of resource + Opportunity Cost
The Opportunity cost is the shadow price of the limiting constraint.
2. Maximum payment for additional scarce resources
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Opportunity costs provide important information in situations where a company can
obtain additional scarce resources, but only at a premium. How much should the
company be prepared to pay? For example, the company may be able to remove the
labour constraint by paying overtime.
The maximum the company can pay is the basic labour rate plus the shadow price
for an additional unit of labour. The total contribution will therefore be improved by
any additional payment below the shadow price.
3. Control
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Opportunity costs are also important for cost control. The responsibility centre
should therefore be identified not only with the acquisition cost but also with the
opportunity cost from the loss of one scarce unit of materials (shadow price).
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This process highlights the true cost of the inefficient usage of scarce resources and
encourages responsibility heads to pay special attention to the control of scarce
factors of production.
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This approach is particularly appropriate where a firm has adopted an optimized
production technology (OPT) strategy because variance arising from bottleneck
operations will be reported in terms of opportunity cost rather than acquisition cost.
4. Managing bottleneck Constraints
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When scarce resources are fully utilized, they are referred to as bottleneck
operations/resources. It is important that managers seek to increase the efficiency
and capacity of bottleneck operations.
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Capacity can be increased by working overtime on the bottleneck, subcontracting
some of the work that is undertaken by bottleneck operations, investing in
additional capacity at the bottleneck and implementing business process
improvements such as business process reengineering (BPR) and total quality
management processes.