MBA Operations Management Final Review Problems

Queueing Theory and Service Improvement at First Local Bank

The First Local Bank process manager is evaluating strategies to enhance customer service at its drive-in facility by targeting a reduction in waiting and transaction times. Based on a pilot study, the customer arrival rate is estimated at $\lambda = 20$ cars per hour. The facility operates with a single service window ($k = 1$) where cars line up in a single file and are processed on a First-Come-First-Served (FCFS) basis. Currently, the teller requires an average of 2 minutes to complete a single transaction. This service rate ($\mu$) is calculated as $\frac{60\text{ minutes}}{2\text{ minutes/car}} = 30$ cars per hour. The capacity utilization ($\rho$), represented by the ratio of arrival rate to total service capacity, is calculated using the formula $\rho = \frac{\lambda}{k \times \mu}$. For this system, $\rho = \frac{20}{30} = 0.667$, or approximately $66.7\%$.

Under the current single-teller configuration, the average number of cars waiting in line ($L_q$) is 1.33 cars. The overall time a customer spends in the facility ($W$), including both waiting and service time, is 6 minutes. To improve these metrics, the bank is considering leasing high-speed information-retrieval and communication equipment at a cost of $30\, \text{dollars/hour}$. This equipment would serve the entire facility and reduce the average transaction-processing time to 1 minute per customer. Under this proposed upgrade, the service rate increases to $\mu = \frac{60\text{ minutes}}{1\text{ minute/car}} = 60$ cars per hour. Assuming interarrival and processing times are exponentially distributed, the average number of cars in the facility ($L$) would drop to 0.5 cars. The overall time to enter and exit the drive-in facility ($W$) would decrease to $0.025\, \text{hours}$, which is equivalent to 1.5 minutes.

Inventory Management Systems and Economic Order Quantity (EOQ)

A continuous review inventory system for a specific item involves a weekly demand of $\text{Demand} = 64\, \text{units/week}$. Given 52 weeks per year, the annual demand ($D$) is $3,328\, \text{units/year}$. The ordering cost ($S$) is $50\, \text{dollars/order}$, and the annual holding cost ($h$) is $13\, \text{dollars/unit/year}$. The lead time ($L$) for receiving an order is 4 weeks, with a standard deviation of weekly demand ($\sigma_d$) of 12 units. The company aims for a cycle-service level of \text{97.5%}. To find the Economic Order Quantity ($EOQ$ or $Q^*$), the formula used is $\sqrt{\frac{2 \times D \times S}{h}}$. Calculating this yields $\sqrt{\frac{2 \times 3,328 \times 50}{13}} = \sqrt{25,600} = 160\, \text{units}$. The cycle inventory, which represents the average inventory held during a cycle, is $\frac{Q}{2} = \frac{160}{2} = 80\, \text{units}$. The average time between orders ($TBO$) is $\frac{Q}{D} = \frac{160\, \text{units}}{64\, \text{units/week}} = 2.5\, \text{weeks}$.

Determining safety stock ($ss$) and the reorder point ($ROP$) requires the standard deviation of demand over the lead time ($\sigma_L$). Since demand is variable, $\sigma_L = \sigma_d \times \sqrt{L} = 12 \times \sqrt{4} = 24\, \text{units}$. For a cycle-service level of \text{97.5%}, the corresponding z-score from the normal distribution table is $z = 1.96$. The safety stock is calculated as $z \times \sigma_L = 1.96 \times 24 = 47.04$ units, rounded to 48 units. The reorder point is the sum of the expected demand during lead time and the safety stock: $ROP = (L \times \text{average weekly demand}) + ss = (4 \times 64) + 48 = 256 + 48 = 304\, \text{units}$.

Sensitivity analysis shows that if the ordering cost ($S$) increases by $50\%$ to $75\, \text{dollars}$, the new $EOQ$ becomes $\sqrt{\frac{2 \times 3,328 \times 75}{13}} \approx 196\, \text{units}$. This results in a new cycle inventory of $98\, \text{units}$, representing a $22.5\%$ increase ($\frac{98 - 80}{80}$). Importantly, changes in ordering cost do not affect safety stock or the reorder point. Conversely, if lead time increases from 4 to 9 weeks, the $EOQ$ remains 160 units, but the new safety stock increases because $\sigma_L = 12 \times \sqrt{9} = 36$. The new safety stock becomes $1.96 \times 36 \approx 71\, \text{units}$, and the new reorder point would be $(9 \times 64) + 71 = 647\, \text{units}$.

Single-Period Inventory Models (Newsvendor Problem)

Swell Productions uses the Newsvendor model to determine the optimal order quantity for poodle skirts at a one-day event. The skirts are purchased for $c = 40\, \text{dollars}$ and sold for $p = 75\, \text{dollars}$. Unsold skirts have a salvage value (refund) of $s = 33\, \text{dollars}$. Forecasted demand is normally distributed with $\mu = 400$ and $\sigma = 100$. In a scenario where unsatisfied demand is lost, the cost of understocking ($C_u$) is $(p - c) = 75 - 40 = 35\, \text{dollars/unit}$, and the cost of overstocking ($C_o$) is $(c - s) = 40 - 33 = 7\, \text{dollars/unit}$. The critical ratio (optimal service level) is $\frac{C_u}{C_u + C_o} = \frac{35}{35 + 7} = 0.833$. The z-score for this probability is 0.97, leading to an optimal order quantity of $400 + (0.97 \times 100) = 497\, \text{units}$.

If unsatisfied customers are willing to be backlogged and satisfied via a rush order, the costs change. The rush order costs $56\, \text{dollars}$ per unit instead of $40\, \text{dollars}$. The cost of understocking ($C_u$) becomes the difference between the rush price and the original price: $(56 - 40) = 16\, \text{dollars/unit}$. The cost of overstocking ($C_o$) remains $7\, \text{dollars/unit}$. The new critical ratio is $\frac{16}{16 + 7} = 0.696$. The corresponding z-score is 0.51, resulting in an optimal order quantity of $400 + (0.51 \times 100) = 451\, \text{units}$.

McCormick Hardware applies a similar discrete demand analysis for riding lawn mowers. Mowers cost $300\, \text{dollars}$, sell for $425\, \text{dollars}$, and surplus units are sold for $250\, \text{dollars}$. Here, $C_u = 425 - 300 = 125\, \text{dollars/unit}$ and $C_o = 300 - 250 = 50\, \text{dollars/unit}$. The critical ratio is $\frac{125}{125 + 50} = 0.714$. Using the discrete probability distribution for demand (Cumulative probabilities: $D=0: 0.1, D=1: 0.25, D=2: 0.55, D=3: 0.75$), the store must order 3 mowers to ensure the cumulative probability reaches at least $71.4\%$.

Questions & Discussion

1. Discuss the role of cycle inventory in the supply chain. Answer: The primary role of cycle inventory is to allow different stages in the supply chain to purchase product in lot sizes that minimize the sum of the material, ordering, and holding cost. If a manager were considering the holding cost alone, he or she would reduce the lot size and cycle inventory. Economies of scale in purchasing and ordering, however, motivate a manager to increase the lot size and cycle inventory. A manager must make the trade-off that minimizes the total cost when making the lot sizing decision. Ideally, cycle inventory decisions should be made considering the total cost across the entire supply chain. In practice, however, each stage often makes its cycle inventory decisions independently. This practice increases the level of cycle inventory as well as the total cost in the supply chain. Cycle inventory exists in a supply chain because different stages exploit economies of scale to lower total cost. The costs considered include material cost, fixed ordering cost, and holding cost. The supply chain operation phase operates on a weekly or daily time horizon and deals with decisions concerning individual customer orders.

2. Discuss the role of safety inventory in the supply chain and the trade-offs involved. Answer: The primary role of safety inventory is providing product availability for customers when demand and supply are uncertain. The trade-off that a supply chain manager must consider when planning safety inventory involve product availability and inventory holding costs. On one hand, raising the level of safety inventory increases product availability and thus the margin captured from customer purchases. On the other hand, raising the level of safety inventory increases inventory holding costs. This issue is particularly significant in industries where product life cycles are short and demand is very volatile. Carrying excessive inventory can help counter demand volatility but can really hurt if new products come on the market and demand for the product in inventory dries up. The inventory on hand then becomes worthless. In today’s business environment, firms experience great pressure to improve product availability while increasing product variety through customization. As a result, markets have become increasingly heterogeneous and demand for individual products is very unstable and difficult to forecast. Both the increased variety and the increased pressure for availability push firms to increase the level of safety inventory they hold. At the same time, product life cycles have shrunk. This increases the risk to firms of carrying too much inventory. Thus, a key to the success of any supply chain is to figure out ways to decrease the level of safety inventory carried without hurting the level of product availability.

3. Describe the two types of ordering policies and the impact each has on safety inventory. Answer: A replenishment policy consists of decisions regarding when to reorder and how much to reorder. These decisions determine the cycle and safety inventories along with the CSL. There are several forms that replenishment policies may take. We restrict attention to two instances: 1. Continuous review: Inventory is continuously tracked and an order for a lot size Q is placed when the inventory declines to the reorder point (ROP). The time between orders may fluctuate given variable demand. When using a continuous review policy, a manager has to account only for the uncertainty of demand during the lead time (L). 2. Periodic review: Inventory status is checked at regular periodic intervals and an order is placed to raise the inventory level to a specified threshold. In this case, the time between orders is fixed. The size of each order, however, can fluctuate given variable demand. Periodic review replenishment policies require more safety inventory than continuous review policies for the same lead time and level of product availability, because the safety inventory has to cover for demand uncertainty over the lead time and the review interval (L + R). Periodic review policies are simpler to implement for retailers because they do not require that the retailer have the capability of continuously monitoring inventory. Given that higher uncertainty must be accounted for, periodic review policies will require a higher level of safety inventory.

4. Explain the impact of aggregation on safety inventory. Answer: Aggregation reduces the standard deviation of demand only if demand across the regions being aggregated is not perfectly positively correlated. Demand for most products does not show perfect positive correlation across different geographical regions. In case demand in different geographical regions is about the same size and independent, aggregation reduces safety inventory by the square root of the number of areas aggregated. In other words, if the number of independent stocking locations decreases by a factor of n, the average safety inventory is expected to decrease by a factor of $\sqrt{n}$. There are two major disadvantages of aggregating all inventory in one location: 1. Increase in response time to customer order 2. Increase in transportation cost to customer. Both disadvantages result because the average distance between the inventory and the customer increases with aggregation. With this situation, either the customer has to travel more to reach the product or the product has to be shipped over longer distances to reach the customer. However, there are clear benefits to aggregating safety inventory.

5. What is the bullwhip effect and how does it relate to lack of coordination in the supply chain? Answer: Many firms have observed the bullwhip effect in which fluctuations in orders increase as they move up the supply chain from retailers to wholesalers to distributors to manufacturers. The bullwhip effect distorts demand information within the supply chain, with different stages having a very different estimate of what demand looks like. The result is a loss of supply chain coordination. This leads to increased inventories, poorer product availability, and a drop in profits. The bullwhip effect negatively impacts performance at every stage and thus hurts the relationships between different stages of the supply chain. There is the tendency to assign blame to other stages of the supply chain because each stage feels it is doing the best it can. The bullwhip effect thus leads to a loss of trust between different stages of the supply chain and makes any potential coordination efforts more difficult. It follows that the bullwhip effect and the resulting lack of coordination have a significant negative impact on the supply chain’s performance. The bullwhip effect moves a supply chain away from the efficient frontier by increasing cost and decreasing responsiveness. The bullwhip effect reduces the profitability of a supply chain by making it more expensive to provide a given level of product availability.