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Part III Lean and Beyond Manufacturing The application studies in part three illustrate sophisticated strategies for operating systems, typically manufacturing systems, to effectively meet customer requirements in a timely fashion while concurrently meeting operations requirements such as keeping inventory levels low and utilization of equipment and workers high. These strategies incorporate both lean techniques as well as beyond lean modeling and analysis. Before presenting the application studies in chapters 10, 11, and 12, inventory control and organization strategies are presented in chapter 9. These include both traditional and lean strategies. Chapter 10 deals with flowing the product at the pull of the customer as implemented in the pull approach. How to concurrently model the flow of both products and information is discussed. Establishing inventory levels as a part of controlling pull manufacturing operations is illustrated. Chapter 11 discusses the cellular manufacturing approach to facility layout. A typical manufacturing cell involving semi-automated machines is studied. The assignment of workers to machines is of interest along with a detailed assessment of the movement of workers within the cell. Chapter 12 shows how flexible machines could be used together for production. Flexible machines are programmable and thus can perform multiple operations on multiple types of parts. Alternative assignments of operations and part types to machines are compared. The importance of simulating complex, deterministic systems is discussed. The application studies in this and the remaining parts of the book are more challenging than those in the previous part. They are designed to be metaphors for actual or typical problems that can be addressed using simulation. The applications problems make use of the modeling and experimentation techniques from the corresponding application studies but vary significantly from them. Thus some reflection is required in accomplishing modeling, experimentation, and analysis. Questions associated with application problems provide guidance in accomplishing these activities. Chapter 9 Inventory Organization and Control 9.1 Introduction Even before a full conversion to lean manufacturing, a facility can be converted to a pull production strategy. Such a conversion is the subject of chapter 10. An understanding of the nature of inventories is pre-requisite for a conversion to pull. Thus, the organization and control of inventories is the subject of this chapter. Traditional inventory models are presented first. Next the lean idea of the control of inventories using kanbans is described. Finally, a generalization of the kanban approach called constant work in process (CONWIP) is discussed. In addition, a basic simulation model for inventories is shown. 9.2 Traditional Inventory Models 9.2.1 Trading off Number of Setups (Orders) for Inventory Consider the following situation, commonly called the economic order quantity problem. A product is produced (or purchased) to inventory periodically. Demand for the product is satisfied from inventory and is deterministic and constant in time. How many units of the product should be produced (or purchased) at a time to minimize the annual cost, assuming that all demand must be satisfied on time? This number of units is called the batch size. The analysis might proceed upon the following lines. 1. What costs are relevant? a. The production (or purchase) cost of each unit of the product is sunk, that is the same no matter how many are made at once. b. There is a fixed cost per production run (or purchase) no matter how many are made. c. There is a cost of holding a unit of product in inventory until it is sold, expressed in $/year. Holding a unit in inventory is analogous to borrowing money. An expense is incurred to produce the product. This expense cannot be repaid until the product is sold. There is an “interest charge” on the expense until it is repaid. This is the same as the holding cost. Thus, the annual holding cost per unit is often calculated as the company minimum attractive rate of return times the cost of one unit of the product. 2. What assumptions are made? a. Production is instantaneous. This may or may not be a bad assumption. If product is removed from inventory once per day and the inventory can be replenished by a scheduled production run of length one day every week or two, this assumption is fine. If production runs cannot be precisely scheduled in time due to capacity constraints or competition for production resources with other products or production runs take multiple days, this assumption may make the results obtained from the model questionable. b. Upon completion of production, the product can be placed in inventory for immediate delivery to customers. c. Each production run incurs the same fixed setup cost, regardless of size or competing activities in the production facility. d. There is no competition among products for production resources. If the production facility has sufficient capacity this may be a reasonable assumption. If not, production may not occur exactly at the time needed. The definitions of all symbols used in the economic order quantity (EOQ) model are given in Table 9-1. 9-1 Table 9-1: Definition of Symbols for the Economic Order Quantity Model Term Annual demand rate (D) Unit production cost (c) Fixed cost per batch (A) Inventory cost per unit per year (h) Batch size (Q) Orders per year (F) Time between orders Cost per year Definition Units demanded per year Production cost per unit Cost of setting up to produce or purchase one batch h = i * c where i is the corporate interest rate Optimal value computed using the inventory model D/Q 1/F = Q/D Run (order) setup cost + inventory cost = A * F + h * Q/2 The cost components of the model are the annual inventory cost and the annual cost of setting up production runs. The annual inventory cost is the average number of units in inventory times the inventory cost per unit per year. Since demand is constant, inventory declines at a constant rate from its maximum level, the batch size Q, to 0. Thus, the average inventory level is simply Q/2. This idea is shown in Figure 9-1. The number of production runs (orders) per year is the demand divided by the batch size. Thus the total cost per year is given by equation 9-1. Y Q   h * Q 2  A* D (9-1) Q Finding the optimal value of Q is accomplished by taking the derivative with respect to Q, setting it equal to 0 and solving for Q. This yields equation 9-2. 9-2 Q *  2* A*D  h A * (9-2) 2* D h Notice that the optimal batch size Q depends on the square root of the ratio of the fixed cost per batch, A, to the inventory holding cost, h. Thus, the cost of a batch trades off with the inventory holding cost in determining the batch size. Other quantities of interest are the number of orders per year (F) and the time between orders (T). F T *  D /Q *  1/ F * *  Q (9-3) * (9-4) /D It is important to note that: Mathematical models help reveal tradeoffs between competing system components or parameters and help resolve them. Even if values are not available for all model parameters, mathematical models are valuable because they give insight into the nature of tradeoffs. For example in equation 9- 2, as the holding cost increases the batch size decreases and more orders are made per year. This makes sense, since an increase in inventory cost per unit should lead to a smaller average inventory. As the fixed cost per batch increases, batch size increases and fewer orders are made per year. This makes sense since an increase in the cost fixed cost per batch results in fewer batches. Suppose cost information is unknown and cannot be determined. What can be done in this application? One approach is to construct a graph of the average inventory level versus the number of production runs (orders) per year. An example graph is shown in Figure 9-2. The optimal tradeoff point is in the “elbow” of the curve. To the right of the elbow, increasing the number of production runs (orders) does little to lower the average inventory. To the left of the elbow, increasing the average inventory does little to reduce the number of production runs (orders). In Figure 9-2, an average inventory of about 20 to 40 units leads to about 40 to 75 production runs a year. This suggests that optimal batch size can be changed within a reasonably wide range without changing the optimal cost very much. This can be very important as batch sizes may be for practical purposes restricted to a certain set of values, such as multiples of 12, as order placement could be restricted to weekly or monthly. Example. Perform an inventory versus batch size analysis on the following situation. Demand for medical racks is 4000 racks per year. The production cost of a single rack is $250 with a production run setup cost of $500. The rate of return used by the company is 20%. Production runs can be made once per week, once every two weeks, or once every four weeks. The optimal batch size (number of units per production run) is given by equation 9-2: Q *  2* A*D h  2 * 500 * 4000  283 250 * 20 % 9-3 Inventory vs Production Run Tradeoff 180 Average inventory 160 140 120 100 80 60 40 20 0 0 50 100 150 200 250 # of Runs Figure 9- 2: Inventory versus Production Run Tradeoff Graph The number of production runs per year and the time between production runs is given by equations 9-3 and 4: T  D /Q * F *  1/ F * *  14 . 1  Q / D  3 . 7 weeks * The optimal cost is given by equation 9-1:   Y Q * h* Q * D  A* 2 * Q  250 * 20 % * 283  500 * 14 . 1  7075  7071  14146 2 Applying the constraint on the time between production runs yields the following. T F Q  4 weeks ' '  52 weeks / 4 weeks '  4000 / F Y (Q )  h * ' Q 2 '  13  308 '  A* D Q '  250 * 20 % * 308  500 * 13  7700  6500  14200 2 Note that when the optimal value of Q is used the inventory cost and the setup cost of production runs are approximately equal. When the constrained value is used, the inventory cost increases since batch sizes are larger but the setup cost decreases since fewer production runs are made. The total cost is about the same. 9-4 9.2.2 Trading Off Customer Service Level for Inventory Ideally, no inventory would be necessary. Goods would be produced to customer order and delivered to the customer in a timely fashion. However, this is not always possible. Wendy’s can cook your hamburger to order but a Christmas tree cannot be grown to the exact size required while the customer waits on the lot. In addition, how many items customers demand and when these demands will occur is not known in advance and is subject variation Keeping inventory helps satisfy customer demand on-time in light of the conditions described in the preceding paragraph. The service level is defined as the percent of the customer demand that is met on time. Consider the problem of deciding how many Christmas trees to purchase for a Christmas tree lot. Only one order can be placed. The trees may be delivered before the lot opens for business. How many Christmas trees should be ordered if demand is a normally distributed random variable with known mean and standard deviation? There is a trade-off between: 1. 2. Having unsold trees that are not even good for firewood. Having no trees to sell to a customer who would have bought a tree at a profit for the lot. Relevant quantities are defined in Table 9-2. Table 9-2; Definition of Symbols for Service Level – Inventory Trade-off Models Term cs co SL Q   zp Definition Cost of a stock out, for example not having a Christmas tree when a customer wants one. Cost of an overage, for example having left over Christmas trees Service level Batch size or number of units to order Mean demand Standard deviation of demand Percent point of the standard normal distribution: P(Z  zp) = p. In Excel this is given by NORMSINV(p) Then it can be shown that the following equation holds: SL  cs cs  co  1 (9-5) 1  co / cs This equation states that the cost-optimal service level depends on the ratio of the cost of a stock out and the cost of an overage. In the Christmas tree example, the cost of an overage is the cost of a Christmas tree. The cost of a stock out is the profit made on selling a tree. Suppose the cost of Christmas tree to the lot is $15 and the tree is sold for $50 (there’s the Christmas spirit for you). This implies that the cost of a stock out is $50 - $15 = $35. The cost-optimal service level is given by equation 9-5. SL  cs cs  co  35 35  15  70 % 9-5 If demand is normally distributed, the optimal number of units to order is given by the general equation: Q *     * z SL (9-6) Thus, the optimal number of Christmas trees to purchase if demand is normally distributed with mean 100 and standard deviation 20 is Q *  100  20 * z 0 . 70  100  20 * 0 . 524  111 There are numerous similar situations to which the same logic can be applied. For example, consider a store that sells a particular popular electronics product. The product is re-supplied via a delivery truck periodically. In this application, the overage cost is equal to the inventory holding cost that can be computed from the cost of the product and the company interest rate as was done in the EOQ model. The shortage cost could be computed as the unit profit on the sale of the product. However, the manager of the store feels that if the product is out of stock, the customer may go elsewhere for all their shopping needs and never come back. Thus, a pre-specified service level, usually in the range 90% to 99% is required. What is the implied shortage cost? This is given in general terms by equation 9- 7. cs  co * SL (9-7) 1  SL Notice that this is equation is highly non-linear with respect to the service level. Suppose deliveries are made weekly, the overage cost (inventory holding cost) is $1/per week, and that a manager specifies the service level to be 90%. What is the implied cost of a stock out? From equation 97, this cost is computed as follows: cs  co * SL 1  SL  $1 * 90 % 1  90 %  $9 Note that if the service level is 99%, the cost of a stock out is $99. 9.3 Inventory Models for Lean Manufacturing In a lean manufacturing setting, the service level is most often an operating parameter specified by management. Inventory is kept to co-ordinate production and shipping, to guard against variation in demand, and to guard against variation in production. The latter could be due to variation in supplier shipping times, variation in production times, production downtimes and any other cause that makes the completion of production on time uncertain. A very important idea is that the target inventory level needed to achieve a specified service level is a function of the variance in the process that adds items to the inventory, production, as well as the process the removes items from the inventory, customer demand. If there is no variation in these processes, then there is no need for inventory. Furthermore, the less the variation, the less inventory is needed. Variation could be random, such as the number of units demanded per day by customers, or structural: product A is produced on Monday and Wednesday and product B is produced on Tuesday and Thursday but there is customer demand for each product each day. 9-6 We will confine our discussion to the following situation of interest. Product is shipped to the customer early in the morning from inventory and is replaced by a production run during the day. Note that if the production run completes before the next shipment time, production can be considered to be instantaneous. In other words, as long as the production run is completed before the next shipment, how long before is not relevant. Suppose demand is constant and production is completely reliable. If demand is 100 units per day, then 100 units reside in the inventory until a shipment is made. Then the inventory is zero. The production run is for 100 units, which are placed in the inventory upon completion. This cycle is completed every day. The following discussion considers how to establish the target inventory level to meet a pre-established service level when demand is random, when production is unreliable, and when both are true. 9.3.1 Random Demand – Normally Distributed In lean manufacturing, a buffer inventory is established to protect against random variation in customer demand. Suppose daily demand is normally distributed with a mean of  units and a standard deviation of  units. Production capacity is such that the inventory can be reliably replaced each day. Management specifies a service level of SL. Consider equation 9-8, P(X  x) ≤ SL (9-8) This equation says that the probability that the random variable, X, daily demand, is less than the target inventory, the constant x, must be SL. Solving for the target inventory, x, yields equation 9-9. x =  +  * zSL (9-9) Exercise. Customer demand is normally distributed with a mean of 100 units per day and a standard deviation of 10 units. Production is completely reliable and replaces inventory every day. Determine the target inventory for service levels of 90%, 95%, 99% and 99.9%. Suppose production is reliable but can occur only every other day. The two-day demand follows a normal distribution with a mean of 2 *  units and a standard deviation of 2 *  units. The target inventory level is still SL. Consider the probability of sufficient inventory on the first of the two days. Since the amount of inventory is sufficient for two days, we will assume that the probability of having enough units in inventory on the first day to meet customer demand is very close to 1. Thus, the probability of sufficient inventory on the second day need only be enough such that the average of this quantity for the first day and the second day is SL. Thus, the probability of sufficient inventory on the second day is SL2 = 1 – [(1 - SL) * 2]. This means that the target inventory for replenishment every two days is given by equation 9-10. x2 = 2 *  + 2 * zSL2 (9-10) This approach can be generalized to n days between production, so long as n is small, a week or less. This condition will be met in lean production situations. 9-7 Exercise. Customer demand is normally distributed with a mean of 100 units per day and a standard deviation of 10 units. Production is completely reliable and replaces inventory every two days. Determine the target inventory for service levels of 90%, 95%, 99% and 99.9%. 9.3.2 Random Demand – Discrete Distributed In many lean manufacturing situations, customer demand per day is distributed among a relative small numbers of batches of units. For example, a batch of units might be a pallet or a tote. This situation can be modeled using a discrete distribution. The general form of a discrete distribution for this situation is:  pi = 1 (9-11) where i is the number of batches demanded and pi is the probability of the customer demand being exactly i batches. The value of i ranges from 1 to n, the maximum customer demand. If n is small enough, then a target inventory of n batches is not unreasonable and the service level would be 1. Suppose a target inventory of n batches is too large. Then the target inventory, x, is the smallest value of x for which equation 9-12 is true. x  p i  SL (9-12) i 1 Exercise Daily customer demand is expressed in batches as follows: (4, 20%), (5, 40%), (6, 30%), (7, 10%). Production is completely reliable and replaces inventory every day. Determine the target inventory for service levels of 90%, 95%, 99% and 99.9%. Suppose production is reliable but can occur only every other day. The two-day demand distribution is determined by convolving the one-day demand distribution with itself. Convolving has to do with considering all possible combinations of the demand on day one and the demand on day two. Demand amounts are added and probabilities are multiplied. This is shown in Table 9-3 for the example in the preceding box. Table 9-4 adds together the probabilities for the same values of the two-day demand (day one + day two demand). For example, the probability that the two day demand is exactly 9 batches is 16%, (8% + 8%) 9-8 Table 9-3: Possible Combinations of the Demand on Day One and Day Two Day One Demand Day Two Demand Day One + Day Two Demand Demand Probability Demand Probability Demand Probability 4 20% 4 20% 8 4% 5 40% 4 20% 9 8% 6 30% 4 20% 10 6% 7 10% 4 20% 11 2% 4 20% 5 40% 9 8% 5 40% 5 40% 10 16% 6 30% 5 40% 11 12% 7 10% 5 40% 12 4% 4 20% 6 30% 10 6% 5 40% 6 30% 11 12% 6 30% 6 30% 12 9% 7 10% 6 30% 13 3% 4 20% 7 10% 11 2% 5 40% 7 10% 12 4% 6 30% 7 10% 13 3% 7 10% 7 10% 14 1% . 9-9