In the UK, householders pay their electricity supply companies for the units that they use. Those supply companies then pay the producers and distribution companies. Our local distribution company sends out newsletters to householders, even though we are at one step removed from one another financially. However, there is a direct link when the power goes down, either by accident or by design. So the newsletter is relevant (are there other businesses which communicate with the customers of their customers?)

The latest issue included the following:

In customer service, we answered over 99% of inbound calls within 20 seconds - with an average speed of answer of under 2 seconds. This is quite an achievement given that it involved answering over 1.1 million calls.

So, congratulations to Western Power Distribution. If you think about the OR involved with their phone answering system, the achievement really is substantial. First of all, which average of which measurement are they using that is less than 2 seconds? Given the skewness of the distribution of waiting times for phones to be answered in general, I would suspect that the average is either the median or the mode (where the time to answer has been sliced into units of seconds or fractions of seconds). Using the mean seems a little unlikely, given that there are still 1% of calls taking over 20 seconds. And do they measure the time to get through to a human voice or a computer choice? Secondly, and this is impressive, the calls to the company will not come at a constant rate; the rate will vary with time of day, day of week, by month, and - most important - will have peaks at times of poor weather. So, Western Power has managed to find a way of handling calls at its call centres which cope with the vagaries of the British climate, and seasonal variations.

Other companies have something to learn; my recent experience of calling 24-hour call centres at weekends and holidays has been somewhat less satisfactory - Exeter City Council, no response after 15 mins; Devon & Cornwall Police, no response after 8mins for a non-emergency call.

## Friday, 22 February 2013

## Tuesday, 19 February 2013

### £1.43 (one pound, forty three pence)

Few operational research projects involve the OR people actually deciding the retail price of an item, although there are times when OR people interact with the marketing department and prices are bound to be discussed at times. So, for OR people who work in consumer retailing, a little knowledge about the psychology of pricing is useful, coupled with the willingness to ask the penetrating question "Why?". (I say "Penetrating" but other writers have said that the characteristic of good OR questions is that they are the ones that nobody else is brave enough to ask.)

Recently, the Canadian mint decided that the one cent coin must be phased out; it costs more to make a one cent coin than it is worth. (The same thing happened in the UK, several times over history. When "silver" coins actually contained silver, the mint continued to make them even though the price of silver gradually increased. Then they changed to an alloy. Bullion dealers collected old coins for their scrap value. More recently, the same happened with "copper" coins, and the value of copper - so now "copper" coins are steel electrocoated with copper. But I digress.) A consequence of the demise of the one cent coin is assumed to be that items cannot be priced to end in odd cents - prices must end in 0 (zero) or 5 (five), allowing the 5 cent coin to become the lowest denomination of use. Other countries have already done this - in the Netherlands, despite being in the Euro-zone, there are no Dutch 1cent and 2cent coins.

However, I deliberately used the word "assumed", because there are places where the smallest coin has a face value of 5, but prices can be any integer value. Buyers have their total bill rounded up or down to the nearest multiple of 5, if they are paying by cash, or pay the total value if they are paying by credit or debit card. The demise of a coin does not lead to the demise of that unit of currency. It could, but it need not. In the UK, fuel for cars is advertised at a price which ends in a fraction of a penny. (We paid 143.9 pence per litre of diesel for our car today, even though the pump records the price to a multiple of one penny.)

So here are roles for OR. If the country decides that all prices should be multiples of 5, what will be the consequences of changing the prices of low value retail goods? Psychology enters this. It is well known that consumers can be duped into thinking that a price ending in 99 is much better than the price one penny dearer, ending in 00. Leading digits also matter. A survey showed that consumers thought that the difference between £20.99 and £25.99 was less than the difference between £19.99 and £24.99 - because the consumers looked at the leading digit. There are many 99p stores in the UK, as well as £1 stores, and in several cases, the two stores exist happily alongside each other in the High Street. So what model should a country adopt for its currency? And within that, what model should a retailer adopt? Room for some OR models.

One of the elementary examples of dynamic programming from many university courses is to determine the minimum number of coins that are needed to give change. I offer you a coin of value 100 to pay for an item costing 53 - what is the minimum number of coins needed? The formulation should be straightforward, and the answer depends on the coins that are used in that land.

Puzzles have been created which turn the problem around. Suppose that you are introducing new coinage for a country, and want to ensure that every sum of money up to say 100 can be paid using no more than 4 coins; what units should there be in your currency? (The problem is also written in terms of postage stamps, since postage does not offer scope for receiving change in cases of overpayment.) The answers turn out to be rather silly units for the coins, which is probably why such questions are left to books of mathematical puzzles, and not implemented by banks and mints.

So, why the title to this blog? It is the answer to a maximisation problem. So it is OR. What is the largest amount in current UK coins (with faceless than one pound) that does not have a subset equal to one pound? So, it is the mathematical programming problem:

maximise X1+2X2+5X5+10X10+20X20+50X50

where Xj is the number of coins of denomination j

Xj integer

and

Y1+2Y2+5Y5+10Y10+20Y20+50Y50 <> 100 for all 0<=Yj<=Xj,

Yj integer

Recently, the Canadian mint decided that the one cent coin must be phased out; it costs more to make a one cent coin than it is worth. (The same thing happened in the UK, several times over history. When "silver" coins actually contained silver, the mint continued to make them even though the price of silver gradually increased. Then they changed to an alloy. Bullion dealers collected old coins for their scrap value. More recently, the same happened with "copper" coins, and the value of copper - so now "copper" coins are steel electrocoated with copper. But I digress.) A consequence of the demise of the one cent coin is assumed to be that items cannot be priced to end in odd cents - prices must end in 0 (zero) or 5 (five), allowing the 5 cent coin to become the lowest denomination of use. Other countries have already done this - in the Netherlands, despite being in the Euro-zone, there are no Dutch 1cent and 2cent coins.

However, I deliberately used the word "assumed", because there are places where the smallest coin has a face value of 5, but prices can be any integer value. Buyers have their total bill rounded up or down to the nearest multiple of 5, if they are paying by cash, or pay the total value if they are paying by credit or debit card. The demise of a coin does not lead to the demise of that unit of currency. It could, but it need not. In the UK, fuel for cars is advertised at a price which ends in a fraction of a penny. (We paid 143.9 pence per litre of diesel for our car today, even though the pump records the price to a multiple of one penny.)

So here are roles for OR. If the country decides that all prices should be multiples of 5, what will be the consequences of changing the prices of low value retail goods? Psychology enters this. It is well known that consumers can be duped into thinking that a price ending in 99 is much better than the price one penny dearer, ending in 00. Leading digits also matter. A survey showed that consumers thought that the difference between £20.99 and £25.99 was less than the difference between £19.99 and £24.99 - because the consumers looked at the leading digit. There are many 99p stores in the UK, as well as £1 stores, and in several cases, the two stores exist happily alongside each other in the High Street. So what model should a country adopt for its currency? And within that, what model should a retailer adopt? Room for some OR models.

One of the elementary examples of dynamic programming from many university courses is to determine the minimum number of coins that are needed to give change. I offer you a coin of value 100 to pay for an item costing 53 - what is the minimum number of coins needed? The formulation should be straightforward, and the answer depends on the coins that are used in that land.

Puzzles have been created which turn the problem around. Suppose that you are introducing new coinage for a country, and want to ensure that every sum of money up to say 100 can be paid using no more than 4 coins; what units should there be in your currency? (The problem is also written in terms of postage stamps, since postage does not offer scope for receiving change in cases of overpayment.) The answers turn out to be rather silly units for the coins, which is probably why such questions are left to books of mathematical puzzles, and not implemented by banks and mints.

So, why the title to this blog? It is the answer to a maximisation problem. So it is OR. What is the largest amount in current UK coins (with faceless than one pound) that does not have a subset equal to one pound? So, it is the mathematical programming problem:

maximise X1+2X2+5X5+10X10+20X20+50X50

where Xj is the number of coins of denomination j

Xj integer

and

Y1+2Y2+5Y5+10Y10+20Y20+50Y50 <> 100 for all 0<=Yj<=Xj,

Yj integer

## Monday, 4 February 2013

### Winter holidays

There are times when your past comes back to haunt you. Once I was teaching a general course on the ideas and simple techniques of Operational Research and I wanted to have student exercises which reflected the wide range of problems for which OR is appropriate. So, I created a series of exercises that related to an imaginary tourist attraction in Devon. This gave opportunities for forecasting, with and without seasonality, scheduling (in particular the production of "home made cakes" in the cafeteria), stock control, advertising and marketing management, distribution, personnel planning .... I enjoyed setting these exercises ... maybe the students picked up something of the philosophy of OR as well.

Last week we went for a night away in Truro, Cornwall, about a hundred miles from Exeter. We did a few touristy things, and had some riverside or seaside walks in the two days. Our hotel had the customary racks of leaflets about local attractions, things to see and things to do. The distribution of these is now outsourced to professional companies which maintain the racks, so the latter have the problems of distribution that I had imagined years ago. Many leaflets are reprinted for each calendar year, but they schedule the preparation in readiness for the Easter holidays, when the main tourism business starts. So, if you go in January, many leaflets are out of date. Additionally, not all attractions will open in the winter.

Hence OR related decision problems for Cornish tourist attractions. Should you issue a winter brochure or not? Should you open during the winter months? How many copies of brochures do you produce? Do you distribute to the same catchment area in summer and winter?

The leaflets in the hotel reflected various responses to these problems. One attraction had a special winter leaflet, showing what was on display in December, January, February. Another had a brochure for 2012, with information about opening in January 2013 as well. Others were closed for the winter, or were only opened at weekends. And several all-year, all-weather, attractions had pooled their efforts to produce a booklet with one page per attraction. Still others had already produced their material for 2013.

So, I guess there is another student exercise there. Analyse why the attractions chose their particular solution. What data do they use? What models do the have to help their decisions? Is there cost-benefit analysis about the advertising?

As the Dane Piet Hein wrote:

Tina in Truro

Last week we went for a night away in Truro, Cornwall, about a hundred miles from Exeter. We did a few touristy things, and had some riverside or seaside walks in the two days. Our hotel had the customary racks of leaflets about local attractions, things to see and things to do. The distribution of these is now outsourced to professional companies which maintain the racks, so the latter have the problems of distribution that I had imagined years ago. Many leaflets are reprinted for each calendar year, but they schedule the preparation in readiness for the Easter holidays, when the main tourism business starts. So, if you go in January, many leaflets are out of date. Additionally, not all attractions will open in the winter.

Hence OR related decision problems for Cornish tourist attractions. Should you issue a winter brochure or not? Should you open during the winter months? How many copies of brochures do you produce? Do you distribute to the same catchment area in summer and winter?

The leaflets in the hotel reflected various responses to these problems. One attraction had a special winter leaflet, showing what was on display in December, January, February. Another had a brochure for 2012, with information about opening in January 2013 as well. Others were closed for the winter, or were only opened at weekends. And several all-year, all-weather, attractions had pooled their efforts to produce a booklet with one page per attraction. Still others had already produced their material for 2013.

So, I guess there is another student exercise there. Analyse why the attractions chose their particular solution. What data do they use? What models do the have to help their decisions? Is there cost-benefit analysis about the advertising?

As the Dane Piet Hein wrote:

*We shall have to evolve**Problem-solvers galore**For each problem they solve**Creates ten problems more*
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