Inventory Models Questions and Answers

Q: The Economic Order Quantity (EOQ), primarily :

The EOQ primarily balances carrying and ordering costs.Total annual cost in the basic model is \(\dfrac{D}{Q}C_{o} + \dfrac{Q}{2}C_{h}\): ordering cost falls as the order size Q grows, while carrying cost rises. The EOQ \(Q^{*} = \sqrt{2DC_{o}/C_{h}}\) minimises the total, and at \(Q^{*}\) the two costs are exactly equal.It does not minimise set-up (ordering) cost alone, which would mean ordering as rarely as possible, and the basic EOQ model assumes no shortages, so it is not designed to reduce them. Hence 'all of the above' is wrong too.

Q: If the lead time is \(m\) periods and usage rate is \(u\), the re-order stock level is given by :

The re-order level is \(m \cdot u\).An order placed now arrives after the lead time of \(m\) periods, during which \(m \cdot u\) units are used. To avoid running out before the delivery, the order must be placed when the stock falls to exactly the lead-time demand, \(m \cdot u\) (plus any safety stock, which this simple model omits).Q/2 is the average inventory in the EOQ model, \(Q/u - m\) is a time (cycle length minus lead time), and \(m\) alone is a number of periods, not a stock level.

Q: In the model \(S = Q - ut\), a stock out will occur when :

A stock-out occurs when \(t = Q/u\).In \(S = Q - ut\), the stock starts at Q and is used at the constant rate \(u\). It runs out when \(S = 0\):\[ Q - ut = 0 \Rightarrow t = \frac{Q}{u} \]At \(t = 0\) the stock is full, \(u/Q\) inverts the ratio (it is the fraction of the order used per unit time), and \(Q = S\) holds only at \(t = 0\).

Q: Re-order level for an item is always :

The re-order level is always more than the minimum stock.The re-order level is the stock at which a new order is placed. It must cover the demand during the lead time on top of the minimum (safety) stock, so\[ \begin{aligned} &\text{re-order level} \\ &= \text{minimum stock} + \text{lead-time consumption} \end{aligned} \](in the standard formula, maximum usage times maximum lead time). With positive lead-time demand it is always above the minimum stock, so it cannot be less than or equal to it. It is usually below the average stock, since stock runs down to it from the maximum level.

Q: In the inventory model \(S = Q - u.t\) with \(Q = 1000\), \(u = 40\) and \(t = 10\), the current stock level is :

The current stock level is 600.In this model Q is the stock at the start, \(u\) the rate of usage per unit time and \(t\) the time elapsed, so the stock remaining is\[ S = Q - ut = 1000 - 40 \times 10 = 600 \]400 is the quantity used, not the stock left; 1000 ignores the usage, and 1200 adds the usage instead of subtracting it.

Q: If the demand distribution is exponential with mean \(1/\alpha\), then the optimal inventory level \(S^{*}\) is :

The optimal level is \(S^{*} = \dfrac{1}{\alpha}\log\dfrac{h + p}{h + c}\).With purchase cost \(c\), holding cost \(h\) and shortage cost \(p\), the single-period optimum satisfies \(F(S^{*}) = \frac{p - c}{p + h}\). For exponential demand, \(F(S) = 1 - e^{-\alpha S}\), so\[ \begin{aligned} e^{-\alpha S^{*}} &= 1 - \frac{p - c}{p + h} = \frac{h + c}{h + p} \\ S^{*} &= \frac{1}{\alpha}\log\frac{h + p}{h + c} \end{aligned} \]Since \(p \gt c\), the logarithm is positive, as a stock level must be. Option B has the fraction inverted, giving a negative level, and options with the factor \(\alpha\) instead of \(1/\alpha\) have the wrong units (the mean demand is \(1/\alpha\)).

Q: In inventory control, which costs can vary with order quantity ?

The answer is all of the above.In inventory models, the annual holding cost rises with the order quantity Q (average stock is about Q/2), and the annual ordering (reorder) cost falls with Q, since fewer orders are placed, about D/Q per year. The unit purchase cost can also depend on Q when suppliers offer quantity discounts for larger orders.So each of the three costs can vary with the order quantity, and the options naming only one of them are incomplete.

If orders are placed with the size determined by the economic order quantity, the re-order costs component is :

A.equal to the holding cost component
B.greater than the holding cost component
C.less than the holding cost component
D.either greater or less than the holding cost component

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: A

At the economic order quantity the re-order cost component is equal to the holding cost component.

With annual demand D, cost per order \(C_{o}\) and holding cost \(C_{h}\) per unit per year, ordering Q at a time costs

\[ TC(Q) = \frac{D}{Q}C_{o} + \frac{Q}{2}C_{h} \]

Setting \(TC'(Q) = 0\) gives \(\dfrac{DC_{o}}{Q^{2}} = \dfrac{C_{h}}{2}\), so \(\dfrac{D}{Q}C_{o} = \dfrac{Q}{2}C_{h}\): at \(Q^{*} = \sqrt{2DC_{o}/C_{h}}\) the two components are exactly equal.

Ordering more than the EOQ makes holding cost the larger component, and ordering less makes re-order cost the larger, which is why the other options describe non-optimal order sizes.

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The Economic Order Quantity (EOQ), primarily :

A.minimizes the set-up cost
B.reduces shortages
C.balances carrying and ordering costs
D.All of the above

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: C

The EOQ primarily balances carrying and ordering costs.

Total annual cost in the basic model is \(\dfrac{D}{Q}C_{o} + \dfrac{Q}{2}C_{h}\): ordering cost falls as the order size Q grows, while carrying cost rises. The EOQ \(Q^{*} = \sqrt{2DC_{o}/C_{h}}\) minimises the total, and at \(Q^{*}\) the two costs are exactly equal.

It does not minimise set-up (ordering) cost alone, which would mean ordering as rarely as possible, and the basic EOQ model assumes no shortages, so it is not designed to reduce them. Hence 'all of the above' is wrong too.

Discuss in Forum

If the lead time is \(m\) periods and usage rate is \(u\), the re-order stock level is given by :

A.Q/2
B.\(\dfrac{Q}{u} - m\)
C.\(m\)
D.\(m \cdot u\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: D

The re-order level is \(m \cdot u\).

An order placed now arrives after the lead time of \(m\) periods, during which \(m \cdot u\) units are used. To avoid running out before the delivery, the order must be placed when the stock falls to exactly the lead-time demand, \(m \cdot u\) (plus any safety stock, which this simple model omits).

Q/2 is the average inventory in the EOQ model, \(Q/u - m\) is a time (cycle length minus lead time), and \(m\) alone is a number of periods, not a stock level.

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In the model \(S = Q - ut\), a stock out will occur when :

A.\(t = 0\)
B.\(t = u/Q\)
C.\(t = Q/u\)
D.\(Q = S\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: C

A stock-out occurs when \(t = Q/u\).

In \(S = Q - ut\), the stock starts at Q and is used at the constant rate \(u\). It runs out when \(S = 0\):

\[ Q - ut = 0 \Rightarrow t = \frac{Q}{u} \]

At \(t = 0\) the stock is full, \(u/Q\) inverts the ratio (it is the fraction of the order used per unit time), and \(Q = S\) holds only at \(t = 0\).

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Re-order level for an item is always :

A.Less than minimum stock
B.More than minimum stock
C.Same as the minimum stock
D.More than the average stock

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: B

The re-order level is always more than the minimum stock.

The re-order level is the stock at which a new order is placed. It must cover the demand during the lead time on top of the minimum (safety) stock, so

\[ \begin{aligned} &\text{re-order level} \\ &= \text{minimum stock} + \text{lead-time consumption} \end{aligned} \]

(in the standard formula, maximum usage times maximum lead time). With positive lead-time demand it is always above the minimum stock, so it cannot be less than or equal to it. It is usually below the average stock, since stock runs down to it from the maximum level.

Discuss in Forum

In the inventory model \(S = Q - u.t\) with \(Q = 1000\), \(u = 40\) and \(t = 10\), the current stock level is :

A.400
B.1200
C.1000
D.600

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: D

The current stock level is 600.

In this model Q is the stock at the start, \(u\) the rate of usage per unit time and \(t\) the time elapsed, so the stock remaining is

\[ S = Q - ut = 1000 - 40 \times 10 = 600 \]

400 is the quantity used, not the stock left; 1000 ignores the usage, and 1200 adds the usage instead of subtracting it.

Discuss in Forum

If the demand distribution is exponential with mean \(1/\alpha\), then the optimal inventory level \(S^{*}\) is :

A.\(\alpha \log\dfrac{h + p}{h + c}\)
B.\(\dfrac{1}{\alpha}\log\dfrac{h + c}{h + p}\)
C.\(\dfrac{1}{\alpha}\log\dfrac{h + p}{h + c}\)
D.\(\alpha \log\dfrac{h + c}{h + p}\)

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The optimal level is \(S^{*} = \dfrac{1}{\alpha}\log\dfrac{h + p}{h + c}\).

With purchase cost \(c\), holding cost \(h\) and shortage cost \(p\), the single-period optimum satisfies \(F(S^{*}) = \frac{p - c}{p + h}\). For exponential demand, \(F(S) = 1 - e^{-\alpha S}\), so

\[ \begin{aligned} e^{-\alpha S^{*}} &= 1 - \frac{p - c}{p + h} = \frac{h + c}{h + p} \\ S^{*} &= \frac{1}{\alpha}\log\frac{h + p}{h + c} \end{aligned} \]

Since \(p \gt c\), the logarithm is positive, as a stock level must be. Option B has the fraction inverted, giving a negative level, and options with the factor \(\alpha\) instead of \(1/\alpha\) have the wrong units (the mean demand is \(1/\alpha\)).

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In inventory control, which costs can vary with order quantity ?

A.unit cost only
B.reorder cost only
C.holding cost only
D.all of the above

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

The answer is all of the above.

In inventory models, the annual holding cost rises with the order quantity Q (average stock is about Q/2), and the annual ordering (reorder) cost falls with Q, since fewer orders are placed, about D/Q per year. The unit purchase cost can also depend on Q when suppliers offer quantity discounts for larger orders.

So each of the three costs can vary with the order quantity, and the options naming only one of them are incomplete.

Discuss in Forum

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