How to calculate EOQ in this problem?

Question

I am trying to solve a problem that requires I find the EOQ. This problem is from the book (translation from greek) Operational Research: Theory, Algorithms and Applications by authors Coletsos John & Stogiannis Dimitris.

A publishing house puts out a variety of books. One of the books with the most stable sales is the book “Master Chef”, which is on its 4th edition. The publishing house estimates that the next edition of the book will have circa 25,000 sales per year.

The past years, the publishing house printed and bound the stock of a year of this book. However, recently, the big carrying cost is putting pressure on the managers who are reconsidering their policy. In particular, the publishing house is thinking about binding half the amount of copies that are printed and keeping the rest unbound. When needed, they’ll have the opportunity to bind the rest of the copies. The costs are presented in the following table (the transportation cost and the other costs that are not associated with this decision are omitted). The carrying cost is 25% of the value of the book per € per year.

(a) What’s the EOQ if the printing and binding of the book happen at the same time? What’s the yearly total cost?

(b) What’s the EOQ if only half the books are bound after printing? What’s the yearly TC?

The table that is provided is:

                Printing (in€)        Binding (in€)
Setup cost          7,500                  1,500

Variable cost         6                      3
per unit
(I think they refer to carrying cost)

I am unsure about some things and how to solve the problem. I started by writing down the given information and I for sure have the demand $D=25,000$. In order to calculate the EOQ and the TC I need the following: $K$ (i’ve also seen it symbolized with $S$), which is the cost of the order, $K_c$ which is the cost to store the books. Then, I can find the EOQ from its formula $\sqrt\frac{2\cdot K\cdot D}{K_c}$ and $TC = \frac{D}{EOQ}\cdot K+ \frac{EOQ}{2}\cdot K_c$.

I think for (a) $K=7,500+1,500 =9,000$. Then, $K_c=25$%$\cdot Value$, but that’s where I get stuck. I don’t really understand how the second row of the table comes in hand in the carrying cost.

Like, if I go to calculate in (a) with these I get: $EOQ =\sqrt{\frac{2\cdot 9,000 \cdot 25,000}{0.25\cdot Value}}=14,142.14$ $TC=\frac{25,000}{14142.14}\cdot 9,000+\frac{14142.14}{2}\cdot 0.25\cdot 9=31,819.81$

Can someone help me to understand how to utilise the information of the problem?

Edit: I am going to accept @Steven01123581321 's answer as the correct one, because it's the closest to the correct one and the main reason I managed to get through this problem unscathed.

For (a) Steven had it completely correct and nicely explained.

As for (b), the thought process is this: The publishing house is going to make "orders" in batches, printing all of the books in the batch and binding half of them. But before making another order, they'll obviously just bind the half that are left before printing more books. Therefore, the cost per order is going to be:

• $7,500$ for the printing of the batch
• $1,500$ for the first half's binding and
• $1,500$ for the second half's binding later.

That brings the setup cost per order to a sum of $K=10,500$ euros. As for the rest, Steven got that covered, with the carrying cost being $K_c=7.5\cdot 0.25$ euros per copy.

Answer 1

I find the wordings pretty confusing, but the way I see it:

for (a)

If binding & printing both happen, then for every ordered batch, you incur the costs $7500 + 1500 = 9000$. So ordering $10$ times, would incur this costs $10$ times. So when ordering a batch of $2000$, your total order costs would be $\frac{25000\times 9000}{2000} = 112500$. When ordering a batch of $2000$ each time, we know from the underlying demand model that the average inventory will be $\frac{2000}{2} = 1000$. For each unit, when binding and printing happen both, to me it looks like the total cost per unit is $9$ (and you pay $0.25$ over this value as yearly inventory costs), which would lead to a yearly inventory cost of $\frac{2000}{2} \times 0.25 \times 9 = 2250$.

Applying the EOQ formula with this knowledge, gives

$$EOQ = \sqrt{\frac{2\times 25000 \times 9000}{0.25\times 9}} \approx 14142$$

with a total cost of:

$$\frac{25000}{14142} \times 9000 + \frac{14142}{2} \times 0.25 \times 9 = 31819.81$$

for (b)

Because of the underlying model of the EOQ, I believe the authors are implying that there is a fixed order and unit cost as well, when half of the copies get printed but not bound.

So when you order $2000$ copies each batch and half of the copies are printed but not bounded, then the order cost is still $9000$.

edit: the order cost is $10500$ following the logic of @Tia in the answer. I'll adjust my answer this cost to be complete.

When looking at yearly inventory costs: if you order $2000$ pcs each times, your average inventory will be $1000$ and the cost per unit will be $\frac{500 \times 6 + 500 \times 9}{1000} = 7.5$. This thus always leads to a unit cost of $\frac{6 + 9}{2} = 7.5$

Using this in the EOQ, I get

$$EOQ = \sqrt{\frac{2\times 25000 \times 10500}{0.25\times 7.5}} \approx 16733$$

with a total cost of:

$$\frac{25000}{16733} \times 10500+ \frac{16733}{2} \times 0.25 \times 7.5 = 31374.75$$

We could also have calculated the change in the EOQ directly:

$\sqrt{\frac{9}{7.5}}\times \sqrt{\frac{10500}{9000}} = 1.1832$ and multiplying $14142$ with $1.1832$ leads to $\approx 16733$.

Answer 2

You are correct about the demand and setup cost. The presence of "Value" in the numerator of the EOQ formula is an error. $K$ is the order setup cost (which is why you have seen it as $S$), which is independent of the value of the order. As for "Value" in the denominator, I would take it to mean the variable cost of one unit (which would depend on whether it was bound or not).