Skip to main content
Operations Research

Return to Answer

Long edit to clarify the thinking here, which I hope is helpful. It feels like a lot of discussion about a simple concept, but oh well.
Source Link
alerera
alerera
  • 1.6k
  • 7
  • 13

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$ and $Q$ measures the inventory count in items after placing the order at the beginning of the cycle. If you want $K$ were to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual tocomplete the model and compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problemmodel, where a dimension is a quantity that must be measured to specify the model. UsuallyThe three fundamental dimensions are cost is(typically measured in \$), time is measured(measured in years), and inventory (measured in items). In my setup above, $TC$ is a cost variable and is comprised of two cost components, both measured in \$. So $K$ has units of \$ and provides the fixed cost incurred in an order cycle. The second term has $h$ measured in \$ per item-year multiplied by $\frac{Q^2}{2 \lambda}$ measuring average item-years of inventory in an order cycle since the average inventory $\frac{Q}{2}$ is measured in items and the cycle length (in your example)$\frac{Q}{\lambda}$ is measured in years assuming an initial inventory of $Q$ items. Following your usageIn the EOQ setup, constant order quantities makes it unnecessary to measure orders separately to specify the problem. If the unit of orders is not a dimension but rathercalled for simplicity unit order, then the unit orders per time is just an endogenous performance metric given by demand (for exampleitems per year) divided by order size (items per unit order), the countor $\frac{\lambda}{Q}$. Note that it is perfectly fine for $Q$ to be a quantity of items or a quantity of items per unit order, depending on the setting.

If you are still wanting more, it is also possible to guess wrong and assume that it is necessary to measure orders explicitly when specifying a correct EOQ model. Let's think about this. Let $K$ in this case be a cost per order, measured in \$ per unit order. We can furthermore define the variable $Q$ as measuring inventory per order, the increase in inventory obtained for every order measured in items per unit order. Let me also suggest that variable quantity $o_t$ captures the orders placed at time $t$ again measured in unit orders. Consider a time duration that begins with a time $t$ when an order is alwaysplaced. It isn't difficult to see that there should be no items in inventory at $\frac{\lambda}{Q}$)$t$ for an optimal ordering strategy. You could reformulateIf we place $o_t$ unit orders, the EOQ problem by measuring inventory level will be raised to $Q o_t$ at a cost of $K o_t$. During the time interval while the inventory again drops to zero, the total cost will be: $$TC' = K o_t + h \frac{Q o_t}{2} \frac{Q o_t}{\lambda} \; ,$$ where again both terms measure cost in "orders" instead\$ (the second term is again \$ per item-year multiplied first by items and then by years). Dividing again by the length of "items"the time interval yields: $$\frac{TC'}{t} = \frac{K \lambda}{Q} + h \frac{Q o_t}{2} \; .$$ Both terms measure cost correctly in \$ per year, but givenit should be clear that cost only increases in the setup each impliesdecision variable $o_t$ so it should be set equal to one unit order before optimizing for $Q$. In this formulation, it is true that $Q$ (and $Q^*$) is in units of items per unit order whereas in the otheroriginal formulation $Q$ simply measures the increase in inventory that results by placing the single order. Since only one unit order is placed in a reorder cycle, we strip away the need to have $K$ and $Q$ measure per unit order quantities.

I'm sorry for the long answer, but I hope it helps.

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$. If you want $K$ to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual to compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problem. Usually cost is measured in \$, time is measured in years, and inventory is measured in items (in your example). Following your usage, orders is not a dimension but rather an endogenous performance metric (for example, the count of orders per time is always $\frac{\lambda}{Q}$). You could reformulate the EOQ problem by measuring inventory in "orders" instead of "items", but given the setup each implies the other.

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$ and $Q$ measures the inventory count in items after placing the order at the beginning of the cycle. If $K$ were to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. Given $TC$ measured only in \$, we complete the model and compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ model, where a dimension is a quantity that must be measured to specify the model. The three fundamental dimensions are cost (typically measured in \$), time (measured in years), and inventory (measured in items). In my setup above, $TC$ is a cost variable and is comprised of two cost components, both measured in \$. So $K$ has units of \$ and provides the fixed cost incurred in an order cycle. The second term has $h$ measured in \$ per item-year multiplied by $\frac{Q^2}{2 \lambda}$ measuring average item-years of inventory in an order cycle since the average inventory $\frac{Q}{2}$ is measured in items and the cycle length $\frac{Q}{\lambda}$ is measured in years assuming an initial inventory of $Q$ items. In the EOQ setup, constant order quantities makes it unnecessary to measure orders separately to specify the problem. If the unit of orders is called for simplicity unit order, then the unit orders per time is just an endogenous performance metric given by demand (items per year) divided by order size (items per unit order), or $\frac{\lambda}{Q}$. Note that it is perfectly fine for $Q$ to be a quantity of items or a quantity of items per unit order, depending on the setting.

If you are still wanting more, it is also possible to guess wrong and assume that it is necessary to measure orders explicitly when specifying a correct EOQ model. Let's think about this. Let $K$ in this case be a cost per order, measured in \$ per unit order. We can furthermore define the variable $Q$ as measuring inventory per order, the increase in inventory obtained for every order measured in items per unit order. Let me also suggest that variable quantity $o_t$ captures the orders placed at time $t$ again measured in unit orders. Consider a time duration that begins with a time $t$ when an order is placed. It isn't difficult to see that there should be no items in inventory at $t$ for an optimal ordering strategy. If we place $o_t$ unit orders, the inventory level will be raised to $Q o_t$ at a cost of $K o_t$. During the time interval while the inventory again drops to zero, the total cost will be: $$TC' = K o_t + h \frac{Q o_t}{2} \frac{Q o_t}{\lambda} \; ,$$ where again both terms measure cost in \$ (the second term is again \$ per item-year multiplied first by items and then by years). Dividing again by the length of the time interval yields: $$\frac{TC'}{t} = \frac{K \lambda}{Q} + h \frac{Q o_t}{2} \; .$$ Both terms measure cost correctly in \$ per year, but it should be clear that cost only increases in the decision variable $o_t$ so it should be set equal to one unit order before optimizing for $Q$. In this formulation, it is true that $Q$ (and $Q^*$) is in units of items per unit order whereas in the original formulation $Q$ simply measures the increase in inventory that results by placing the single order. Since only one unit order is placed in a reorder cycle, we strip away the need to have $K$ and $Q$ measure per unit order quantities.

I'm sorry for the long answer, but I hope it helps.

added 2 characters in body
Source Link
alerera
alerera
  • 1.6k
  • 7
  • 13

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$. If you want $K$ to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual to compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problem. Usually cost is measured in \$, time is measured in years, and inventory is measured in items (in your example). Following your usage, an orderorders is not a dimension but rather an endogenous performance metric (for example, the count of orders per time is always $\frac{\lambda}{Q}$). You could reformulate the EOQ problem by measuring inventory in orders"orders" instead of items"items", but given the setup each implies the other.

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$. If you want $K$ to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual to compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problem. Usually cost is measured in \$, time is measured in years, and inventory is measured in items (in your example). Following your usage, an order is not a dimension but rather an endogenous performance metric (for example, the count of orders per time is always $\frac{\lambda}{Q}$). You could reformulate the EOQ problem by measuring inventory in orders instead of items, but given the setup each implies the other.

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$. If you want $K$ to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual to compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problem. Usually cost is measured in \$, time is measured in years, and inventory is measured in items (in your example). Following your usage, orders is not a dimension but rather an endogenous performance metric (for example, the count of orders per time is always $\frac{\lambda}{Q}$). You could reformulate the EOQ problem by measuring inventory in "orders" instead of "items", but given the setup each implies the other.

Source Link
alerera
alerera
  • 1.6k
  • 7
  • 13

In the EOQ setting, the total cost incurred during one order cycle is: $$TC = K + \frac{hQ^2}{2 \lambda} \;\;,$$ where the units of $K$ must be only \$. If you want $K$ to have units of \$ per order (or per cycle), then the second term must also have those units otherwise you are adding apples to oranges. If you want $K$ to have original units of \$ per order, then you could think that since there is one order per order/cycle that it is multiplied by another variable $o=1$ in units of orders, so thus $Ko = K$ and still has units of \$. Given $TC$ measured only in \$, we proceed as usual to compute $\frac{TC}{t}$ by dividing by the cycle length of $\frac{Q}{\lambda}$ and then find $Q^*$ to minimize cost per time in the right units of items.

One possibly important idea is that there are only three fundamental dimensions in the EOQ problem. Usually cost is measured in \$, time is measured in years, and inventory is measured in items (in your example). Following your usage, an order is not a dimension but rather an endogenous performance metric (for example, the count of orders per time is always $\frac{\lambda}{Q}$). You could reformulate the EOQ problem by measuring inventory in orders instead of items, but given the setup each implies the other.