Thursday, January 31, 2008

NIOSH Lifting Equation

'Safe Lifting' Guidelines
Lifting safely will protect your back while you lift. Before you lift an object ask yourself the following questions:
• Do you think you can lift it alone?
• Is the load too big or too awkward?
• Does the load have good handles or grips?
• Is there anything to obstruct proper lifting?
• Could the contents of the load shift while being lifted?
For safe lifting, remember to:
• Stand as close to the load as possible
• Bend at your knees NOT your waist
• Hug the load close to your body, don't hold it away from you
• Raise yourself up with the strong thigh muscles.

Low back pain risks increase when the compressive force at the L5-S1 (lumbar 5 sacral 1) disc exceeds 770 lbs.

NIOSH Lifting Equation
1981 Equation
In 1981 the National Institute of Occupational Safety and Health(NIOSH) issued a Work Practices Guide for Manual Lifting that used 770 lbs. of L5-S1 compressive force as one of the criteria for establishing an Action Limit (AL). Exceeding the action limit required implementation of administrative controls or job redesign. The AL is the weight that can safely be lifted by 75% of the female and 99% of the male population. A Maximum Permissible Limit (MPL is 3 times the action limit) was also set that was equivalent to a compressive force of 770 lbs on the lumbar spine.
The 1981 NIOSH lifting equation is as follows:
Action Limit (AL) = 90lbs. (6/H)(1-.01[V-30])(.7+3/D)(1-F/Fmax)
where:
H = horizontal location of the load forward of the midpoint between the ankles at the origin of the lift (in inches)
V = vertical location of the load at the origin of the lift (in inches)
D = vertical travel distance between the origin and the destination (in inches)
F = average frequency of lifts (lifts/minute)
Fmax = maximum frequency of lifting which can be sustained (from a NIOSH table)
The Maximum Permissible Load (MPL) = 3 (AL)
1991 Equation
In 1991 the NIOSH equation was revised to account for the effects of other variables, such as asymmetrical lifting, good or poor handles, and the total time spent lifting during the workday. Another lifting equation, based on the 1981 equation, was developed that yields a Recommended Weight Limit (RWL) as follows:

Recommended Weight Limit (RWL) = LC x HM x VM x DM x AM x FM x CM
where:
LC = load constant (51 lbs.)
HM = horizontal multiplier = 10/H
VM = vertical multiplier = (1- (0.0075 [V-30])
DM = distance multiplier = (0.82 + (1.8/D))
AM = asymmetric multiplier = (1 - (0.0032A))
FM = frequency multiplier (from a table)
CM = coupling multiplier (from a table)
A = angle of asymmetry = angular displacement of the load from the saggital plane, measured at the origin and destination of the lift
and where H,V,D and F are identical to the 1981 equation.
The RWL protects about 85% of women and 95% of men.
Ways to Protect Your Back
• Give yourself a lot of support. For stability, spread your feet at least as for apart as your shoulder width. Distribute weight evenly throughout the soles of both feetand keep your feet firmly planted, with your center of gravity in your abdominal cavity.
• Tighten your abdominal muscles.The abdominal cavity, consists of the abdominal muscles in front, the diaphragm and ribs above the pelvic floor below. Pressure in the abdomen that helps share the loads placed upon the spine.
• Bend form your knees. Alwways bend from our knees, so the legs can serve as shock absorbers. The pelvis to find its balance over the hips when the knees are slightly bent, so that weight comes first into the thighs and hips instead of the spine. Don't lift with locked knees because they tighten the hamstring muscles and lock the pelvis into an unbalanced position. Don't bend from the waist because it puts tremendous pressure on the lumbar vertebrae.
• Keep your spine in balance. Balance your shoulders and chest over the lower spine, to lessen the force placed on it.A balanced back, with its normal 3 curves, keeps the spinal muscles active so they can share the load placed on the bones, ligaments and discs.

Tuesday, January 29, 2008

Just-in-time (JIT)

Introduction
Just-in-time (JIT) is easy to grasp conceptually, everything happens just-in-time. For example consider my journey to work this morning, I could have left my house, just-in-time to catch a bus to the train station, just-in-time to catch the train, just-in-time to arrive at my office, just-in-time to pick up my lecture notes, just-in-time to walk into this lecture theatre to start the lecture. Conceptually there is no problem about this; however achieving it in practice is likely to be difficult!
So too in a manufacturing operation component parts could conceptually arrive just-in-time to be picked up by a worker and used. So we would at a stroke eliminate any inventory of parts, they would simply arrive just-in-time! Similarly we could produce finished goods just-in-time to be handed to a customer who wants them. So, at a conceptual extreme, JIT has no need for inventory or stock, either of raw materials or work in progress or finished goods.
Obviously any sensible person will appreciate that achieving the conceptual extreme outlined above might well be difficult, or impossible, or extremely expensive, in real-life. However that extreme does illustrate that, perhaps, we could move an existing system towards a system with more of a JIT element than it currently contains. For example, consider a manufacturing process - whilst we might not be able to have a JIT process in terms of handing finished goods to customers, so we would still need some inventory of finished goods, perhaps it might be possible to arrange raw material deliveries so that, for example, materials needed for one day's production arrive at the start of the day and are consumed during the day - effectively reducing/eliminating raw material inventory.
Adopting a JIT system is also sometimes referred to as adopting a lean production system. More about JIT can be found here, here, here and here.
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History
JIT originated in Japan. Its introduction as a recognized technique/philosophy/way of working is generally associated with the Toyota motor company, JIT being initially known as the "Toyota Production System". Note the emphasis here - JIT is very much a mindset/way of looking at a production system that is distinctly different from what (traditionally) had been done previous to its conception.
Within Toyota Taiichi Ohno is most commonly credited as the father/originator of this way of working. The beginnings of this production system are rooted in the historical situation that Toyota faced. After the Second World War the president of Toyota said "Catch up with America in three years, otherwise the automobile industry of Japan will not survive". At that time one American worker produced approximately nine times as much as a Japanese worker. Taiichi Ohno examined the American industry and found that American manufacturers made great use of economic order quantities - the traditional idea that it is best to make a "lot" or "batch" of an item (such as a particular model of car or a particular component) before switching to a new item. They also made use of economic order quantities in terms of ordering and stocking the many parts needed to assemble a car.
Ohno felt that such methods would not work in Japan - total domestic demand was low and the domestic marketplace demanded production of small quantities of many different models. Accordingly Ohno devised a new system of production based on the elimination of waste. In his system waste was eliminated by:
• just-in-time - items only move through the production system as and when they are needed
• autonomation - (spelt correctly in case you have never met the word before) - automating the production system so as to include inspection - human attention only being needed when a defect is automatically detected whereupon the system will stop and not proceed until the problem has been solved
In this system inventory (stock) is regarded as an unnecessary waste as too has to deal with defects.
Ohno regarded waste as a general term including time and resources as well as materials. He identified a number of sources of waste that he felt should be eliminated:
• overproduction - waste from producing more than is needed
• time spent waiting - waste such as that associated with a worker being idle whilst waiting for another worker to pass him an item he needs (e.g. such as may occur in a sequential line production process)
• transportation/movement - waste such as that associated with transporting/moving items around a factory
• processing time - waste such as that associated with spending more time than is necessary processing an item on a machine
• inventory - waste associated with keeping stocks
• defects - waste associated with defective items
At the time car prices in the USA where typically set using selling price = cost plus profit mark-up. However in Japan low demand meant that manufacturers faced price resistance, so if the selling price is fixed how one can increase the profit mark-up? Obviously by reducing costs and hence a large focus of the system that Toyota implemented was to do with cost reduction.
To aid in cost reduction Toyota instituted production leveling - eliminating unevenness in the flow of items. So if a component which required assembly had an associated requirement of 100 during a 25 day working month then 4 were assembled per day, one every two hours in an eight hour working day. Leveling was also applied to the flow of finished goods out of the factory and to the flow of raw materials into the factory.
Toyota changed their factory layout. Previously all machines of the same type, e.g. presses, were together in the same area of the factory. This meant that items had to be transported back and forth as they needed processing on different machines. To eliminate these transportation different machines were clustered together so items could move smoothly from one machine to another as they were processed. This meant that workers had to become skilled on more than one machine - previously workers were skilled at operating just one type of machine. Although this initially met resistance from the workforce it was eventually overcome.
Whilst we may think today that Japan has harmonious industrial relations with management and workers working together for the common good the fact is that, in the past, this has not been true. In the immediate post Second World War period, for example, Japan had one of the worse strike records in the world. Toyota had a strike in 1950 for example. In 1953 the car maker Nissan suffered a four month strike - involving a lockout and barbed wire barricades to prevent workers returning to work. That dispute ended with the formation of a company backed union, formed initially by members of the Nissan accounting department. Striking workers who joined this new union received payment for the time spent on strike, a powerful financial inventive to leave their old union during such a long dispute. The slogan of this new union was "Those who truly love their union love their company".
In order to help the workforce to adapt to what was a very different production environment Ohno introduced the analogy of teamwork in a baton relay race. As you are probably aware typically in such races four runners pass a baton between themselves and the winning team is the one that crosses the finishing line first carrying the baton and having made valid baton exchanges between runners. Within the newly rearranged factory floor workers were encouraged to think of themselves as members of a team - passing the baton (processed items) between themselves with the goal of reaching the finishing line appropriately. If one worker flagged (e.g. had an off day) then the other workers could help him, perhaps setting a machine up for him so that the team output was unaffected.
In order to have a method of controlling production (the flow of items) in this new environment Toyota introduced the Kanban. The Kanban is essentially information as to what has to be done. Within Toyota the most common form of Kanban was a rectangular piece of paper within a transparent vinyl envelope. The information listed on the paper basically tells a worker what to do - which items to collect or which items to produce. In Toyota two types of Kanban are distinguished for controlling the flow of items:
• a withdrawal Kanban - which details the items which should be withdrawn from the preceding step in the process
• a production ordering Kanban - which details the items to be produced
All movement throughout the factory is controlled by these kanbans - in addition since the kanbans specify item quantities precisely no defects can be tolerated - e.g. if a defective component is found when processing a production ordering Kanban then obviously the quantity specified on the Kanban cannot be produced. Hence the importance of autonomation (as referred to above) - the system must detect and highlight defective items so that the problem that caused the defect to occur can be resolved.
Another aspect of the Toyota Production System is the reduction of setup time. Machines and processes must be re-engineered so as to reduce the setup time required before processing of a new item can start.
Ohno has written that Toyota was only able to institute kanbans on a company wide basis in 1962, ten years after they first embarked on the introduction of their new production system. Although, obviously, as the originators of the approach Toyota had much to learn and no doubt made mistakes, this illustrates the time that can be required to successfully implement a JIT system in a large company. Moreover you can reflect on the management time/effort/cost that was consumed in the development and implementation of their JIT system.
With respect to the Western world JIT only really began to impact on manufacturing in the late 1970's and early 1980's. Even then it went under a variety of names - e.g. Hewlett Packard called it "stockless production". Such adaptation by Western industry was based on informal analysis of the systems being used in Japanese companies. Books by Japanese authors (such as Ohno himself) detailing the development of JIT in Japan were not published in the West until the late 1980's.
As an indication of the growth of interest in JIT over time the graph below shows the number of documents (such as books and conference proceedings) referring to just-in-time in the British Library, which has a very extensive collection of such documents relating to the UK. The earliest material I could find was from 1984, when there was one book published and one set of conference proceedings. The graph shows the number of documents published each year as well as the cumulative number published.




One often reads nowadays that JIT involves employee participation, involving workers so as to gain from their knowledge and experience. Such participation is meant to ensure that workers feel involved with the system and make suggestions for improvements, cooperate in changes, etc. Personally I am not convinced that this aspect of JIT, as it is interpreted nowadays, played any part in its initial development. Certainly Ohno, writing in 1978 long before the appearance in the West of material related to JIT, in 8 pages of single spaced A4 paper outlining the Toyota Production System makes little mention of this aspect. My best guess, from my reading of the subject, is that JIT started out as a top-down, centrally organized and imposed production system. Whilst it may later have come to take on a "human-face" with connotations of worker involvement and participation I personally doubt it started out that way.
Toyota still describes itself as using the Toyota Production System for car manufacture, e.g. here in relationship to a manufacturing plant in the USA.
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Japanese terms
There are a number of Japanese terms (words) associated with JIT that you may encounter. I have listed some below for you:
• Andon - trouble lights which immediately signal to the production line that there is a problem to be resolved (typically the line is stopped until the problem is resolved)
• Jikoda - autonomation - enabling machines to be autonomous and able to automatically detect defects
• Muda - waste
• Mura - unevenness
• Muri - excess
• Poka-yoke - "foolproof" machines and methods so as to prevent production mistakes
• Shojinka - a workforce flexible enough to cope with changes in production and using different machines
• Soikufu - thinking creatively, having inventive ideas
In the Toyota system the Andon, indicating a stoppage of the line, is hung from the factory ceiling so that it can be clearly seen by everyone. This coupled with line stoppage clearly raises the profile of the problem and encourages attention/effort to its solution so that it does not reoccur.
As an indication though of the difficulty of implementing JIT in a Western environment when General Motors instituted an Andon for line stoppage workers were simply not prepared to take responsibility for stopping the line. Hence defective items were passed though the system, rather than the Andon functioning as planned and highlighting problems and hence leading to their resolution. General Motors resolved the problem by allowing workers to indicate that they had a problem whilst the line continued to operate.
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Reconciling JIT and EOQ
It is a common misconception that JIT, a Japanese originated concept, is somehow radically different from the classical Western concept of the Economic Order Quantity (EOQ), based as it is on the most economic level of stock. I hope below to convince you that this is not so.
Recall from the notes about inventory theory the problem of deciding the appropriate amount of stock to order.
Assume
• Stock used up at a constant rate (R units per year)
• Fixed setup cost co for each order - often called the order cost
• No lead time between order and arrival of order
• Variable holding cost ch per unit per year
Then we need to decide Q, the amount to order each time, often called the batch (or lot) size.
With these assumptions the graph of stock level over time takes the form shown below.




Hence we have that:
• Annual holding cost = ch(Q/2)
where Q/2 is the average inventory level
• Annual order cost = co(R/Q)
where (R/Q) is the number of orders per year (R used, Q each order)
So total annual cost = ch(Q/2) + co(R/Q)
Total annual cost is the function that we want to minimize by choosing an appropriate value of Q.
Note here that, obviously, there is a purchase cost associated with the R units per year. However as this is just constant as R is fixed we can ignore it here.
The diagram below illustrates how these two components (annual holding cost and annual order cost) change as Q, the quantity ordered, changes. As Q increases holding cost increases but order cost decreases. Hence the total annual cost curve is as shown below - somewhere on that curve lies a value of Q that corresponds to the minimum total cost.



We can calculate exactly which value of Q corresponds to the minimum total cost by differentiating total cost with respect to Q and equating to zero.
d(total cost)/dQ = ch/2 - coR/Q² = 0 for minimization
which gives Q² = 2coR/ch
Hence the best value of Q (the amount to order = amount stocked) is given by
• Q =(2Rco/ch)0.5
and this is known as the Economic Order Quantity (EOQ)
To get the total annual cost associated with the EOQ we have from before that total annual cost = ch(Q/2) + co(R/Q) so putting Q =(2Rco/ch)0.5 into this we get that the total annual cost is given by
ch((2Rco/ch)0.5/2) + co(R/(2Rco/ch)0.5) = (Rcoch/2)0.5 + (Rcoch/2)0.5 = (2Rcoch)0.5
Hence total annual cost is (2Rcoch)0.5 which means that when ordering the optimal (EOQ) quantity we have that total cost is proportional to the square root of any of the factors (R, co and ch) involved.
The Economic Order Quantity is (by definition) the order quantity that minimizes total annual cost and hence (on cost grounds) should always be the quantity that we order.
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What then of JIT with its philosophy of (essentially) very small orders/levels of stock (i.e. Q=1). Is there not a contradiction?
If fact there need not be. This is because in JIT we notice that we need not take co and/or ch as fixed. In particular if we can reduce the cost of ordering co then the EOQ reduces. For example, if we were to reduce co by a factor of 4 we would reduce total cost by a factor of 2 (note the EOQ would change as well, being halved). This, in fact, is one of the ideas behind JIT to reduce (continuously) co and ch so as to drive down total cost.
Hence if, for example, we were to build close links with our suppliers so as to reduce ordering cost dramatically it becomes, just by a straightforward application of the EOQ formula, much more attractive to have small order quantities. In the limit if co is zero, i.e. ordering is free, then we order each and every unit as we need it (remember here our simple EOQ model assumes a zero lead time, i.e. orders received as soon as they are placed).
Note too from the formula (2Rcoch)0.5 for the total annual cost associated with the EOQ reducing co also reduces cost.
In summary then in order to reconcile JIT and EOQ we do not take co and/or ch as fixed but seek (continuously) to reduce them, thereby reducing the EOQ thereby simultaneously reducing the total annual cost.
Obviously seeking ways of reducing co and/or ch takes management time (thereby incurring cost) and we may reach a point of diminishing returns, i.e. it may not be worth the management effort (cost) required to reduce total annual cost further.





________________________________________General Motors
An example of the use of JIT in General Motors is given below.
General Motors (GM) in the USA has (approximately) 1700 suppliers who ship to 31 assembly plants scattered throughout the continental USA. These shipments total about 30 million metric tons per day and GM spends about 1,000 million dollars a year in transport costs on these shipments (1990 figures).
JIT implies frequent, small, shipments. When GM moved to JIT there were simply too many (lightly loaded) trucks attempting to deliver to each assembly plant. GM's solution to this problem was to introduce consolidation centers at which full truckloads were consolidated from supplier deliveries.
This obviously involved deciding how many consolidation centers to have, where they should be, their size (capacity) and which suppliers should ship to which consolidation centers (suppliers can also still ship direct to assembly plants).
As of 1990 some 20% by weight of shipments go through consolidation centers and about 98% of suppliers ship at least one item through a consolidation centre.
All this has been achieved without sacrificing the benefits of JIT.
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JIT outline points
Originated in Japan
Often said Japanese industry works - just-in-time, Western industry works - just-in-case
JIT is also known as stockless production or lean production
JIT is a suitable production system when:
• have steady production of clearly defined standard products
• a reasonable number of units made
• a high value product
• have flexible working practices and a disciplined workforce
• short setup times on machines
• quality can be assured, e.g. zero defects either though good working practices or though a cost penalty
Kanban
• a signal or message or communication, e.g. wave hands, shout, send a card, electronic
• used to control the flow of items though the production process
It is often said that:
• Materials Requirements Planning (MRP) = a 'Push' system
• JIT = a 'Pull' system
I believe that this is an incorrect analysis - MRP is a system based on fulfilling predicted usage in a set time period.
JIT is a system based on actual usage - parts of the production system are "linked" together via kanbans as the system runs
It is this linkage that is the distinguishing difference between MRP and JIT - JIT is a dynamic linked system, MRP is not
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JIT philosophy
• elimination of waste in its many forms
• belief that ordering/holding costs can be reduced
• continuous improvement, always striving to improve
Elements of JIT
• regular meetings of the workforce (e.g. daily/weekly)
• discuss work practices, confront and solve problems
• an emphasis on consultation and cooperation (i.e. involving the workforce) rather than confrontation
• modify machinery, e.g. to reduce setup time
• reduce buffer stock
• expose problems, rather than have them covered up
• reveal bad practices
• take away the "security blanket" of stock
JIT need not be applied to all stages of the process. For example we could keep large stocks of raw material but operate our production process internally in a JIT fashion (hence eliminating work-in-progress stocks).
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Classic JIT diagram
The classic JIT diagram is as below. There the company (the boat) floats on a sea of inventory, lurking beneath the sea are the rocks, the problems that are hidden by the sea of inventory.
|
--|--
|
---------------
\ /
========\ Company /============ Sea of inventory
\---------/
x
xxx xxxx
xxxxx xxxxxx Rocks - the problems hidden
xxxxxxxxxxxxxxxx by the sea of inventory
If we reduce the inventory level then the rocks become exposed, as below.
|
--|--
|
--------------- x
\ / xxx xxxx
========\ Company /====xxxxx===xxxxxx========
\---------/ xxxxxxxxxxxxxxxx
Now the company can see the rocks (problems) and hopefully solve them before it runs aground!
One plan to expose the problems is simply to:
• make a large amount of finished goods stock to keep the customers supplied
• try running the production system with less inventory to expose problems
• revert to the original levels of inventory until you have had time to fix the problems you exposed
• repeat the above - hence continuous improvement
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Benefits
The benefits of JIT are:
• better quality products
• quality the responsibility of every worker, not just quality control inspectors
• reduced scrap and rework
• reduced cycle times
• lower setup times
• smoother production flow
• less inventory, of raw materials, work-in-progress and finished goods
• cost savings
• higher productivity
• higher worker participation
• more skilled workforce, able and wiling to switch roles
• reduced space requirements
• improved relationships with suppliers

However you should be absolutely clear that implementing a JIT system is a task that cannot be undertaken lightly. It will be expensive in terms of management time and effort, both in terms of the initial implementation and in terms of the continuing effort required to run the system over time.
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Suppliers
Suppliers can be crucial to JIT success
Supplier gets:
• long-term, guaranteed, contract
• a good price
• steady demand
• minimal paperwork (e.g. use electronic means to order - such as email or Web or electronic data interchange, EDI)
In return the supplier agrees to
• quality components (e.g. zero defects)
• guaranteed delivery times
• a "partnership" with its customer
• contingency plans to cope with disruptions, common disruptions might be:
o the effect of bad weather
o a truck drivers strike blocking roads/ports
o a flu outbreak reducing the supplier's workforce
Supplier selection criteria:
• close to production plant (else potential transportation delays)
• good industrial relations ("involvement", "value", "dignity", "ownership"), no strike deals
• you believe that the supplier can met their promises with respect to the list of factors given above that that they are agreeing to
With suppliers satisfying these criteria you can reduce the total number of suppliers; indeed it seems logical so to do. If you had five suppliers meeting all these criteria why do you need five? Obviously you might decide to have more than one supplier for safety reasons. Even the best run supplier can suffer a factory fire or an earthquake, but probably no more than two or three suppliers.
As an illustration of this in 1997 Toyota was affected by a fire at a supplier of brake parts that cost the company an estimated $195 million and 70,000 units of production. The fire was at a plant that was the sole supplier of brake parts for all but two Toyota models and forced the company to shut its 18 assembly plants in Japan for a number of days. As a result Toyota embarked on a review of components that were sourced from a single supplier.
Having a single supplier may be attractive in cost terms, but one does need to balance the risk (albeit a low probability risk - perhaps a fire every 100-250 years say) against the cost savings.

Friday, January 25, 2008

Inventory control

The basic function of stock (inventory) is to insulate the production process from changes in the environment as shown below.




Note here that although we refer in this note to manufacturing, other industries also have stock e.g. the stock of money in a bank available to be distributed to customers, the stock of policemen in an area, etc).
The question then arises: how much stock should we have? It is this simple question that inventory control theory attempts to answer.
There are two extreme answers to this question:
a lot
• this ensures that we never run out
• is an easy way of managing stock
• is expensive in stock costs, cheap in management costs
none/very little
• this is known (effectively) as Just-in-Time (JIT)
• is a difficult way of managing stock
• is cheap in stock costs, expensive in management costs
We shall consider the problem of ordering raw material stock but the same basic theory can be applied to the problem of:
• deciding the finished goods stock; and
• deciding the size of a batch in a batch production process.
The costs that we need to consider so that we can decide the amount of stock to have can be divided into stock holding costs and stock ordering (and receiving) costs as below. Note here that, conventionally, management costs are ignored here.
Holding costs - associated with keeping stock over time
• storage costs
• rent/depreciation
• labour
• overheads (e.g. heating, lighting, security)
• money tied up (loss of interest, opportunity cost)
• obsolescence costs (if left with stock at end of product life)
• stock deterioration (lose money if product deteriorates whilst held)
• theft/insurance
Ordering costs - associated with ordering and receiving an order
• clerical/labour costs of processing orders
• inspection and return of poor quality products
• transport costs
• handling costs
Note here that a stockout occurs when we have insufficient stock to supply customers. Usually stockouts occur in the order lead time, the time between placing an order and the arrival of that order.
Given a stockout the order may be lost completely or the customer may choose to backorder, i.e. to be prepared to wait until we have sufficient stock to supply their order.
Note here that whilst conceptually we can see that these cost elements are relevant it can often be difficult to arrive at an appropriate numeric figure (e.g. if the stock is stored in a building used for many other purposes, how then shall we decide an appropriate allocation of heating/lighting/security costs).
To see how we can decide the stock level to adopt consider the very simple model below.
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Basic model
In this basic model we have the situation where:
• our company orders from an outside supplier;
• that outside supplier delivers to us precisely the quantity we ask for; and
• we pass that stock onto our customers (either external customers, or an internal customer within the same company (e.g. if ordering raw materials for use in the production process)).
Assume:
• Stock used up at a constant rate (R units per year)
• Fixed set-up cost co for each order - often called the order cost
• No lead time between placing an order and arrival of the order
• Variable stock holding cost ch per unit per year
Then we need to decide Q, the amount to order each time, often called the batch (or lot) size.
With these assumptions the graph of stock level over time takes the form shown below.




Consider drawing a horizontal line at Q/2 in the above diagram. If you were to draw this line then it is clear that the times when stock exceeds Q/2 are exactly balanced by the times when stock falls below Q/2. In other words we could equivalently regard the above diagram as representing a constant stock level of Q/2 over time.
Hence we have that:
• Annual holding cost = ch(Q/2)
where Q/2 is the average (constant) inventory level
• Annual order cost = co(R/Q)
where (R/Q) is the number of orders per year (R used, Q each order)
So total annual cost = ch(Q/2) + co(R/Q)
Total annual cost is the function that we want to minimise by choosing an appropriate value of Q.
Note here that, obviously, there is a purchase cost associated with the R units per year. However this is just a constant as R is fixed so we can ignore it here.
The diagram below illustrates how these two components (annual holding cost and annual order cost) change as Q, the quantity ordered, changes. As Q increases holding cost increases but order cost decreases. Hence the total annual cost curve is as shown below - somewhere on that curve lies a value of Q that corresponds to the minimum total cost.





We can calculate exactly which value of Q corresponds to the minimum total cost by differentiating total cost with respect to Q and equating to zero.
d(total cost)/dQ = ch/2 - coR/Q² = 0 for minimisation
which gives Q² = 2coR/ch
Hence the best value of Q (the amount to order = amount stocked) is given by
• Q =(2Rco/ch)0.5
and this is known as the Economic Order Quantity (EOQ)
Comments
This formula for the EOQ is believed to have been first derived in the early 1900's and so EOQ dates from the beginnings of mass production/assembly line production.
To get the total annual cost associated with the EOQ we have from before that total annual cost = ch(Q/2) + co(R/Q) so putting Q =(2Rco/ch)0.5 into this we get that the total annual cost is given by
ch((2Rco/ch)0.5/2) + co(R/(2Rco/ch)0.5) = (Rcoch/2)0.5 + (Rcoch/2)0.5 = (2Rcoch)0.5
Hence total annual cost is (2Rcoch)0.5 which means that when ordering the optimal (EOQ) quantity we have that total cost is proportional to the square root of any of the factors (R, co and ch) involved. For example, if we were to reduce co by a factor of 4 we would reduce total cost by a factor of 2 (note the EOQ would change as well). This, in fact, is the basis of Just-in-Time (JIT), to reduce (continuously) co and ch so as to drive down total cost.
To return to the issue of management costs being ignored for a moment the basic justification for this is that if we consider the total cost curve shown above, then - assuming we are not operating a policy with a very low Q (JIT) or a very high Q - we could argue that the management costs are effectively fixed for a fairly wide range of Q values. If this is so then such costs would not influence the decision as to what order quantity Q to adopt. Moreover if we wanted to adopt a more quantitative approach we would need some function that captures the relationship between the management costs we incur and our order quantity Q - estimating this function would certainly be a non-trivial task.
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Example
A retailer expects to sell about 200 units of a product per year. The storage space taken up in his premises by one unit of this product is costed at £20 per year. If the cost associated with ordering is £35 per order what is the economic order quantity given that interest rates are expected to remain close to 10% per year and the total cost of one unit is £100.
We use the EOQ formula,
EOQ = (2Rco/ch)0.5
Here R=200, co=35 and the holding cost ch is given by
ch = £20 (direct storage cost per unit per year) + £100 x 0.10 (this term the money interest lost if one unit sits in stock for one year)
i.e. ch = £30 per unit per year
Hence EOQ = (2Rco/ch)0.5 = (2 x 200 x 35/30)0.5 = 21.602
But as we must order a whole number of units we have that:
EOQ = 22
We can illustrate this calculation by reference to the diagram below which shows order cost, holding cost and total cost for this example.




With this EOQ we can calculate our total annual cost from the equation
Total annual cost = ch(Q/2) + co(R/Q)
Hence for this example we have that
Total annual cost = (30 x 22/2) + (35 x 200/22) = 330 + 318.2 = £648.2
Note: If we had used the exact Q value given by the EOQ formula (i.e. Q=21.602) we would have had that the two terms relating to annual holding cost and annual order cost would have been exactly equal to each other
i.e. holding cost = order cost at EOQ point (or, referring to the diagram above, the EOQ quantity is at the point associated with the Holding Cost curve and the Order Cost curve intersecting).
i.e. (chQ/2) = (coR/Q) so that Q = (2Rco/ch)0.5
In other words, as in fact might seem natural from the shape of the Holding Cost and Order Cost curves, the optimal order quantity coincides with the order quantity that exactly balances Holding Cost and Ordering Cost.
Note however that this result only applies to certain simple situations. It is not true (in general) that the best order quantity corresponds to the quantity where holding cost and ordering cost are in balance.
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Package solution
We can also solve this problem using the package, the input and output being shown below. Note here that the package can deal with more complicated factors than we have considered in the simple example given above.




Note the appearance here of the figure of 20,000 relating to material cost. This is calculated from using 200 units a year at a unit cost of £100 each. Strictly, this cost term should have been added to the total annual cost equation (ch(Q/2) + co(R/Q)) we gave above. We neglected it above as it was a constant term for this example and hence did not affect the calculation of the optimal value of Q. However, we will need to remember to include this term below when we come to consider quantity discounts.
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Example
Suppose, for administrative convenience, we ordered 20 and not 22 at each order - what would be our cost penalty for deviating from the EOQ value?
With a Q of 20 we look at the total annual cost
= (chQ/2) + (coR/Q)
= (30 x 20)/2 + (35 x 200/20) = 300 + 350 = £650
Hence the cost penalty for deviating from the EOQ value is £650 - £648.2 = £1.8
Note that this is, relatively, a very small penalty for deviating from the EOQ value. This is usually the case in inventory problems i.e. the total annual cost curve is flat near the EOQ so there is only a small cost penalty associated with slight deviations from the EOQ value (see the diagram above).
This is an important point. Essentially we should view the EOQ as a ballpark figure. That is it gives us a rough idea as to how many we should be ordering each time. After all our cost figures (such as cost of an order) are likely to be inaccurate. Also it is highly unlikely that we will use items at a constant rate (as the EOQ formula assumes). However, that said, the EOQ model provides a systematic and quantitative way of getting an idea as to how much we should order each time. If we deviate far from this ballpark figure then we will most likely be paying a large cost penalty.
The above cost calculation can also be done using the package - see below.




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Extensions
In order to illustrate extensions to the basic EOQ calculation we will consider the following example:
A company uses 12,000 components a year at a cost of 5 pence each. Order costs have been estimated to be £5 per order and inventory holding cost is estimated at 20% of the cost of a component per year.
Note here that this is the sort of cheap item that is a typical non-JIT item.
• What is the EOQ?
Here R=12000, co=5 and as the inventory holding cost is 20% per year the annual holding cost per unit ch = cost per unit x 20% = £0.05 x 0.2 per unit per year = 0.01.
Hence EOQ = (2Rco/ch)0.5 = (2 x 12000 x 5/0.01)0.5 = 3464
The package output for this problem is shown below.




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• If orders must be made for 1,2,3,4,6 or 12 monthly batches what order size would you recommend and when would you order?
Here we do not have an unrestricted choice of order quantity (as the EOQ formula assumes) but a restricted choice as explained below.
This is an important point - the EOQ calculation gives us a quantity to order, but often people are better at ordering on a time basis e.g. once every month.
In other words we need to move from a quantity basis to a time basis.
For example the EOQ quantity of 3464 has an order interval of (3464/12000) = 0.289 years, i.e. we order once every 52(0.289) = 15 weeks. Would you prefer to order once every 15 weeks or every 4 months? Recall here that we saw before that small deviations from the EOQ quantity lead to only small cost changes.
Hence if orders must be made for 1,2,3,4,6 or 12 monthly batches the best order size to use can be determined as follows.
Obviously when we order a batch we need only order sufficient to cover the number of components we are going to use until the next batch is ordered - if we order less than this we will run out of components and if we order more than this we will incur inventory holding costs unnecessarily. Hence for each possible batch size we automatically know the order quantity (e.g. for the 1-monthly batch the order quantity is the number of components used per month = R/12 = 12000/12 = 1000).
As we know the order quantity we can work out the total annual cost of each of the different options and choose the cheapest option.



The total annual cost (with an order quantity of Q) is given by (chQ/2) + (coR/Q) and we have the table below:
Batch size option Order quantity Q Total annual cost
Monthly 1000 65
2-monthly 2000 40
3-monthly 3000 35
4-monthly 4000 35
6-monthly 6000 40
12-monthly 12000 65
The least cost option therefore is to choose either the 3-monthly or the 4-monthly batch.
In fact we need not have examined all the options. As we knew that the EOQ was 3464 (associated with the minimum total annual cost) we have that the least cost option must be one of the two options that have order quantities nearest to 3464 (one order quantity above 3464, the other below 3464) i.e. either the 3-monthly (Q=3000) or the 4-monthly (Q=4000) options. This can be seen from the shape of the total annual cost curve shown below. The total annual cost for these two options could then be calculated to find which was the cheapest option.

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• If the supplier offers the following quantity discount structure what effect will this have on the order quantity?
• Order quantity Cost (per unit)
• 0-4,999 £0.05
• 5,000-9,999 £0.05 less 5%
• 10,000-19,999 £0.05 less 10%
20,000 and above £0.05 less 15%
For example, were we to order 6000 units we would only pay 0.95(0.05) for each and every one of the 6000 units, i.e. the discount would be given on the entire order.
Here, as mentioned above, we need to remember to add to the total annual cost equation (ch(Q/2) + co(R/Q)) a term relating to R multiplied by the unit cost, as the cost of a unit is now no longer fixed but variable (unit cost = a function f(Q) of the order quantity Q). Hence our total annual cost equation is
ch(Q/2) + co(R/Q) +R[f(Q)]
It is instructive to consider what changes in this equation as we change the order quantity Q. Obviously R and co remain unchanged, equally obviously Q and f(Q) change. So what of ch? Well it can remain constant or it can change. You need to look back to how you calculated ch. If it included money tied up then, as the unit cost f(Q) alters with Q, so too does the money tied up.
The effect of these quantity discounts (breaks in the cost structure) is to create a discontinuous total annual cost curve as shown below with the total annual cost curve for the combined discount structure being composed of parts of the total annual cost curves for each of the discount costs.




The order quantity which provides the lowest overall cost will be the lowest point on the Combined Cost Curve shown in the diagram above. We can precisely calculate this point as it corresponds to:
• either an EOQ for one of the discount curves considered separately (note that in some cases the EOQ for a particular discount curve may not lie within the range covered by that discount and hence will be infeasible);
• or one of the breakpoints between the individual discount curves on the total annual cost curve for the combined discount structure.
We merely have to work out the total annual cost for each of these types of points and choose the cheapest.
First the EOQ's:
Discount Cost ch EOQ Inventory Material Total
cost cost cost
0 0.05 0.01 3464 34.64 600 634.64
5% 0.0475 0.0095 3554 Infeasible
10% 0.045 0.009 3651 Infeasible
15% 0.0425 0.0085 3757 Infeasible
Note here that we now include material (purchase) cost in total annual cost.
The effect of the discount is to reduce the cost, and hence ch the inventory holding cost per unit per year - all other terms in the EOQ formula (R and co) remain the same. Of the EOQ's only one, the first, lies within the range covered by the discount rate.
For the breakpoints we have:
Order Cost ch Inventory Material Total
quantity cost cost cost
5,000 0.0475 0.0095 35.75 570 605.75
10,000 0.045 0.009 51 540 591
20,000 0.0425 0.0085 88 510 598
From these figures we can see that the economic order quantity associated with minimum total annual cost is 10,000 with a total annual cost of 591.
Note too here that this situation illustrates the point we made before when we considered the simple EOQ model, namely that it is not true (in general) that the best order quantity corresponds to the quantity where holding cost and ordering cost are in balance. This is because the holding cost associated with Q=10,000 is ch(Q/2) = 0.009(10000/2) = 45, whilst the ordering cost is co(R/Q) = 5(12000/10000) = 6.
This problem can also be solved using the package - the input being shown below. Note that in the package one needs to use Edit Discount Breaks to enter the discount structure.





One also needs to use Edit Discount Characteristics. Below we have specified that holding cost is also discounted as the cost of an item changes.




The output from the package is shown below. Notice in that output how the package is considering the same choices (EOQ's and breakpoints) as in our manual calculation above.



Note here the use of discount analysis is not restricted to buyers, it can also be used by a supplier to investigate the likely effects upon the orders he receives of changes in the discount structure. For example if the supplier lowers the order size at which a particular discount is received then how might this effect the orders he receives - will they become bigger/smaller, less frequent/more frequent?
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Newsvendor problem
Consider a newsvendor who stands on the street and sells an evening paper, the Evening News. How many copies should he stock?
He sells the paper to his customers for 35 (pence) a copy. He pays his supplier 20 (pence) a copy, but any unsold copies can be returned to the supplier and he gets 10 (pence) back. This is known as a salvage value. Assume that his demand for copies on any day is a Normal distribution of mean 100 and standard deviation 7.
Before we can compute the amount he should order we need to work out his shortage cost per unit - how much does he lose if a customer wants a copy and he does not have a copy available?
As a first analysis he loses his profit (= revenue - cost = 35 - 20 =15) so we can estimate his shortage cost (opportunity cost) as 15 (this ignores any loss of goodwill and any loss of future custom that might result from a shortage).
Giving this information to the package we get:





This tells us he should stock 104.7 (say 105) copies of the paper. This service level of 75% means that, on average, he will be able to completely supply his customer on 75 days in every 100, i.e. 3 days out of 4. The remainder of the time (1 day out of 4) he will experience shortages, some customers will not be able to buy a copy from him as he will have run out.
More sophisticated variants of this simple model can be used, for example, to decide how many copies of a magazine to have on a shelf in a newsagent (such as W.H.Smith).
Note that there is an important conceptual difference between this newsvendor problem and the EOQ/discount problems considered above. In those EOQ/discount problems we had a decision problem (how much to order) even though the situation was one of certainty - we knew precisely the rate at which we used items. In the newsvendor problem if we knew for certain how many customers will want a paper each day then the decision problem becomes trivial (order exactly that many). In other words:
• for the EOQ problem we had a decision problem even though there was no uncertainty
• for the newsvendor problem it was only the uncertainty that created the decision problem
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Comment
There are many extensions to the simple EOQ models we have considered - for example:
• reorder lead time - allow a lead time between placing an order and receiving it - this introduces the problem of when to reorder (typically at some stock level called the reorder level). You will see from the package input that the package allows for a reorder lead time.
• stockouts - we can allow stockouts (often called shortages) i.e. no stock currently available to meet orders. You will see from the package input that the package allows for shortage costs.
• often an order is not received all at once, for example if the order comes from another part of the same factory then items may be received as they are produced. You will see from the package input that the package allows for a replenishment or production rate.
• buffer (safety) stock - some stock kept back to be used only when necessary to prevent stockouts.

Saturday, January 19, 2008

Enterprise Problem Solver

Hi folks,
I wish to share my experience and thoughts on application of the Industrial Engineering& Management methodology for solving enterprise problems.
Industrial Engineering is has evolved over years to encompass all the streams of enterprise management from "MIND to MARKET".

Today enterprise is synonymous with not just a business organisation but any kind of activity to generate value. If you thought a event management company was an enterprise, think this every event that is organised by that organisation can be visualised as a enterprise with a exact termination period .

Where does all this thought take us to?

Well, every activity is a enterprise in its own sense and need methodologies to handle its complexities and constraints for a efficient & effective output.
This is a place where principles of Industrial Engineering & Management come in.........