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BUS 480 Decision Making Tools II: Operation Research Session 5 Review Materials Managers understand that working capital is necessary for the operation of a company. The decision managers need to make regarding working capital is not about whether the company should have any working capital but about the optimal amount of working capital the company should have. Inventory (together with cash, accounts receivables, accounts payables, etc.) makes up a portion of a company's working capital. In this session, we will look a specific operation research model, namely the economic order quantity (EOQ) model, commonly used to determine the optimal level of inventory a company should keep. Before we proceed with our discussions on the EOQ model, we need to first be familiar with some of the concepts associated with inventory management (or control): 1. What constitutes inventory for a company? 2. Why does a company carry inventory? 3. What affects the level of inventory a company keeps? 4. Please explain the difference between dependent demand items and independent demand items. [1] 5. Please provide a brief explanation for each of the three costs associated with carrying (and not manufacturing) inventory: Carrying cost Ordering cost Shortage cost 6. Briefly explains the two types of inventory control system: Continuous (or fixed-order quantity) system Periodical (or fixed-time) system [2] We will focus, in this session, on the continuous inventory system and how the EOQ model can be used to determine the optimal level of inventory a company should carry. 7. Please explain the objective of the EOQ model. In this session, we will look at three variations of an EOQ model, namely (i) the basic EOQ model, (ii) an EOQ model with non-instantaneous receipt of order, and (iii) an EOQ model with shortages. 8. Basic EOQ model The basic EOQ model is governed by several key assumptions: (a) the company faces a constant demand, (b) no shortage allowed, (c) the lead time is constant, and (d) the ordered quantity is received all at once (i.e. instantaneously). It is important to remember that any result (i.e. formula) generated by the EOQ model is only true if all these assumptions hold. The following graph shows how the inventory is received and depleted based on the above assumptions: Inventory Level Time [3] Before we look at the derivation of the basic EOQ model, please be aware that the following notations will be used in the formulae: Q is the quantity of inventory ordered D is the annual demand of the inventory Cc is the carrying cost per unit per year Co is the ordering cost per order With the given assumptions, we can determine the following costs with these formulae: Total carrying cost = Total ordering cost = Total inventory cost = Based on these formulae, we know that there is a direct relationship between total carrying cost and order quantity (i.e. the bigger the order quantity, the higher the carrying cost); and there is an inverse relationship between total ordering cost and order quantity (i.e. the bigger the order quantity, the lower the ordering cost). Can you intuitively explain these relationships? The relationship of these various inventory costs are illustrated in the following graph: Annual Cost ($) Total Cost Carrying Cost Ordering Cost Order Quantity, Q [4] The optimal level of order quantity (Q*) is the amount of inventory that should be ordered if the objective is to incur the lowest possible level of total inventory cost; and it can be calculated with the following formula: Optimal order quantity, Q* = Once the optimal order quantity is determined, it can be used to determine the following: Number of orders per year = Order cycle time = Example 5.1 Hayes Electronics stocks and sells a particular brand of personal computer. It costs the firm $450 each time it places an order with the manufacturer for the personal computers. The cost of carrying one PC in inventory for a year is $170. The store manager estimates that total annual demand for the computers will be 1,200 units, with a constant demand rate throughout the year. Orders are received within minutes after placement from a local warehouse maintained by the manufacturer. The store policy is never to have stockouts of the PCs. The store is open for business every day of the year except Christmas Day. Determine the following: a. Optimal order quantity per order b. Total carrying cost c. Total ordering cost d. Minimum total annual inventory costs e. The optimal number of orders per year f. The optimal time between orders (in days) In the previous example, it is assumed that when a company places an order, it will be delivered immediately. As a result, the company will wait until its inventory is completely depleted before placing a replacement order (because the order will be delivered right away and the inventory replenished). However, in most cases, there is a time gap between placing the order and receiving the order and this is known as the lead time (L). A company will have to take into consideration the lead time when placing an order because it needs to make sure to have sufficient inventory on hand (to meet the demand) while waiting for the order to arrive. The reorder point represents the level of inventory a company has on hand when it needs to place an order; and it represents the amount of inventory the company will deplete while waiting for the “replenishment” to arrive. The reorder point can be calculated as follows: Reorder point = [5] Example 5.2 The Petroco Company uses a highly toxic chemical in one of its manufacturing processes. It must have the product delivered by special cargo trucks designed for safe shipment of chemicals. As such, ordering (and delivery) costs are relatively high, at $2,600 per order. The chemical product is packaged in one-gallon plastic containers. The cost of holding the chemical in storage is $50 per gallon per year. The annual demand for the chemical, which is constant over time, is 2,000 gallons per year. The lead time from time of order placement until receipt is 10 days. The company operates 310 working days per year. Compute the following: a. Optimal order quantity b. Total minimum inventory cost c. Reorder point 9. EOQ model with non-instantaneous receipt of order In the basic EOQ model, it is assumed that the entire amount of inventory ordered will be received all at once. However, in some scenarios, the inventory ordered is received gradually over time. The following graph illustrates the scenario when the inventory is depleted and replenished gradually over time (at different rates): Inventory Level Time Since the inventory will be depleted and replenished simultaneously (at different rates), we will need to adjust the formulae developed for the basic EOQ model for this new scenario. The only exception is the total ordering cost because it is based on the number of orders placed a year. We will use p to represent the production rate (i.e. the daily rate the order is received or replenished) and d to represent the daily rate the inventory is demanded or depleted. Total carrying cost = Total ordering cost = Total inventory cost = [6] The optimal level of order quantity (Q*) that will incur the lowest possible level of total inventory cost when there is non-instantaneous receipt is calculated with the following formula: Optimal order quantity, Q* = Similarly, once the optimal order quantity is determined, it can be used to determine the following: Number of production runs = Length of production run = Keep in mind that the number of production runs is similar to the number of orders in a basic EOQ model, and the length of production run is similar to the order cycle time in a basic EOQ model. Example 5.3 The Simple Simon Bakery produces fruit pies for freezing and subsequent sale. The bakery, which operates 5 days a week, 52 weeks a year, can produce pies at the rate of 64 pies per day. The bakery sets up the pie production operation and produces until a predetermined number (Q) has been produced. When not producing pies, the bakery uses its personnel and facilities for producing other bakery items. The setup cost for a production run of fruit pies is $500. The cost of holding pies in storage is $5 per pie per year. The annual demand for frozen fruit pies, which is constant over time, is 5,000 pies. Determine the following: a. The optimal production run quantity (Q*) b. Total annual inventory costs c. The optimal number of production runs per year d. The optimal cycle time e. The run length in working days Example 5.4 The Pacific Lumber Company and Mill processes 10,000 logs annually, operating 250 days per year. Immediately upon receiving an order, the logging company's supplier begins delivery to the lumber mill at a rate of 60 logs per day. The lumber mill has determined that the ordering cost is $1,600 per order and the cost of carrying logs in inventory before they are processed is $15 per log per on an annual basis. Determine the following: a. The optimal order size b. The total inventory cost associated with the optimal order quantity c. The number of operating days between orders d. The number of operating days required to receive an order [7] 10. EOQ model with shortages We can also modify the basic EOQ model to allow for shortages, and any demand not met can be backordered and shipped to customers later (and the inventory is restocked). However, the company will encounter additional shortage cost, for example, due to lost sales and lost customer goodwill. The following graph illustrates how the inventory is received and depleted in an EOQ model when shortages are allowed (where t1 is the length of time when inventory is on hand and t2 is the length of time when there is a shortage): Inventory Level t2 Q t1 Time Shortage, S We will need a formula to calculate the additional shortage cost (where Cs is the annual per unit cost of shortages). In addition, we also need to modify the formulae of the total carrying cost and the total inventory cost (but not the total ordering cost) from the basic EOQ model to adjust for shortages as follow: Total carrying cost = Total ordering cost = Total shortage cost = Total inventory cost = [8] The relationship of these various inventory costs are illustrated in the following graph: Annual Cost ($) Total Cost Carrying Cost Ordering Cost Shortage Cost Order Quantity, Q The optimal level of order quantity (Q*) and shortages (S*) that will incur the lowest possible level of total inventory cost when shortages are allowed is calculated with the following formulae: Optimal order quantity, Q* = Optimal shortages, S* = In addition, we can use the results of Q* and S* to calculate the following: Number of orders per year = Order cycle time (in days) = Length of time with inventory on hand, t1 (in years) = Length of time when there is a shortage, t2 (in years) = [9] Example 5.5 Videoworld is a discount store that sells color televisions. The annual demand for color television sets is 400. The cost per order from the manufacturer is $650. The carrying cost is $45 per set each year. The store has an inventory policy that allows shortages. The shortage cost per set is estimated at $60. Determine the following: a. The optimal order size b. Maximum shortage level c. Minimum total annual inventory cost Example 5.6 The University Bookstore at Tech stocks the required textbooks for Management Science 2405. The demand for the text is 1,200 copies per year. The cost of placing an order is $350, while the annual carrying cost is $2.75 per book. If a student requests the book and it is not in stock the student will likely go to the privately owned Tech Bookstore. It is likely that the student will not buy books at the University Bookstore in the future; thus the shortage cost to the University Bookstore is estimated to be $45 per book. Determine the following: a. The optimal order size b. The maximum shortage level c. The total inventory cost [10]