### £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

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