Chapter 7: Optimal Inventory PolicyÑA Single Variable Unconstrained Optimization Problem with Comparative Statics

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1 Chapter 7: Optimal Inventory PolicyÑA Single Variable Unconstrained Optimization Problem with Comparative Statics Background Just-In-Time Inventory (JIT) is a popular business management technique first developed in Japan. The basic idea behind the JIT concept is very simple: parts and supplies arrive just before they are needed, allowing firms to keep lower inventory stocks. Today, JIT is commonly used throughout the United States. It was not, however, immediately accepted. For years, JIT was available, but not implemented. Then, in the 1980s, the rapid and wide adoption of JIT methods fundamentally transformed the way many American industries ran their businesses. This brief history lesson leads to an obvious question: What explains the adoption of the JIT management technique in the 1980s? More specifically, why was JIT ignored, then accepted? What caused the switch? In this chapter we will build an economic model of inventory management and check out its comparative statics properties to see what changes in the external environment might have led to the decrease in optimal inventory levels associated with the JIT revolution. A Story about Inventory A Chicago electronics retailer confidently expects to sell some fixed number, Q units (say, 3,600), of televisions over the course of the next year for some predetermined price, with sales spread perfectly evenly over the year. (See below for a picture of the retailerõs expected sales pattern.) Sales of TVs (# units/month) Q 12 = 300 Total sales in the year is Q; Q/12 sold every month Time Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Obviously, the retailer must have TVs in stock, i.e. in her inventory, in order to sell them. Inventory is simply your stock of unsold goods. OneÕs inventory level is determined by two basic forces: how much is flowing out of your store (sales) compared to how much flows into your store (re-orders from your supplier). C7Lab.pdf 1

2 The perfectly competitive retailer has the most control over how much product flows into the store: that is, the size and frequency of re-orders. How much flows out of the store in terms of sales is set, we will assume, by exogenous factors such as how popular the product is, the price, etc. ÒWait just a second! Why,Ó you ask, Òis this an interesting problem? It sounds trivial. Why not order the entire yearõs supply, 3600 TVs, in one shot on January 1? That way, sheõll always have TVs in stock as people come in to buy them.ó CARRYING COSTS: The answer to that question is that holding inventory is not costless. For example, if you order 3,600 TVs on January 1, you have to have a place to store them so that you can re-stock your shelves. Storage costs money (usually storage charges are assessed based on volume), whether you rent storage space or whether you use your own (more expensive) retail space for storage when you could be using it for product displays. The larger the average size of your inventory, the more you have to pay to store it. In addition to the costs of storage, the money that you tie up in lots of inventory at the beginning of the year has an opportunity cost. Suppose that you chose to ship your inventory in two lots, 1800 units on January 1 and 1800 units on July 1, instead of all 3600 at the beginning of the year. When you postpone payment for half of the inventory for six months, you get to use that money in other waysñsay, investing it in short-term T-bills. The larger the average size of your inventory, the greater the opportunity cost of holding that inventory. This is because you have money tied up in inventory that you could have used to buy interest-bearing assets. ÒFine,Ó you say, ÒI can see that making one monster order on January 1 has high carrying costs. But then, what about the other extreme? Why not order 300 TVs the first of every month? In fact, if sales are distributed evenly down to the day, she could order 10 TVs per day (with a few days ordering only 9 TVs). Her carrying costs would be minimized!ó True enough, but hold on. Carrying costs arenõt the whole story. REORDER COSTS: Every time the retailer makes an order for TV sets, a truck from her supplier must deliver TVs to her store. There are costs associated with delivery. The more frequently you reorder, the greater your reorder costs. This gives you an idea of the inherent tension that is present in every optimization problem: you have to balance the reorder cost against the cost of carrying inventory. One order per year makes reorder costs really low, but carrying costs very high; while an order a day (or 365 orders per year), does the oppositeñvery high reorder costs with extremely low carrying costs. The point is that there is a particular number of orders and associated level of inventory that works best, that minimizes the total inventory costs. C7Lab.pdf 2

3 Setting Up the Optimal Inventory Problem Objective Function. Firms are profit-maximizers. The Chicago retailer in the example above is assumed to be a perfectly competitive firm so she knows what her annual revenues will be (annual unit sales times price), so long as she has the TVs in stock when people come in to buy them. What she has to do to maximize profits, then is to minimize inventory costs. As far as inventory costs go, the wholesale cost of the TVs is exogenous for the retailer; but she DOES have control over other aspects of her inventory costs. Composition of Inventory Costs. Inventory costs have TWO main parts. The first part is the cost of actually buying the stuff from the wholesaler and having it shipped. We will call this part the Reorder Cost/year. This in turn depends on two things: how much you buy in total over the year, and how many shipments you choose to deliver the goods over the year. Before we can write out an equation to express this part of inventory costs, we need to develop some notation. If we let Q stand for annual unit sales (in our example, it was 3,600 TVs per year), and b for the price the retailer has to pay for a TV from the wholesaler, then the cost of the yearõs stock is b*q. If the shipping charge is a dollars per shipment, then shipping cost is a*number of shipments/year. Summarizing the first part of inventory costs, we have: ReorderCost/year = a*number of shipments/year + b*totalunitsales/year In the equation above, only the number of shipments/year is controlled by the retailerõs inventory manager. WeÕre assuming that total unit sales/year is given and known. Reorder cost is the first part of inventory costs. The second main part of inventory costs is a catch-all category, which includes (among other things) the interest on the money you borrow buying the goods in inventory. This amount represents the opportunity cost of the inventory. In addition, as we discussed above, there is the cost of storing the inventory. C7Lab.pdf 3

4 We will call the various expenses associated with inventory Carrying Cost/year. It is reasonable to assume that the total amount of the carrying cost is proportional to the average size of the inventory. We will let k represent this constant of proportionality so that carrying costs are: CarryingCost/year = k*average Size of Inventory/year The average size of inventory per year is the average number of TVs in inventory throughout the year. For example, suppose that you choose one shipment of your given total unit sales of 3,600 TVs. On January 1, you have an inventory of 3,600 TVs. As you sell the TVs, your inventory continuously falls. Exactly halfway through the year, you will have 1800 TVs in inventory. On the morning of December 31, you will have 9 or 10 TVs in inventory (which you sell that day). Half of the year you will have more than 1800 TVs in stock; half the year you will have less than 1800 TVs in stock. Thus, your average inventory in the year is 1800 or 3600/ High inventory in Jan Inventory by Day Average Size of Inventory over the year is 1800 TVs Low inventory in Dec Day Obviously, the inventory manager looks to have little inventory on hand so that average size of inventory is small which makes carrying costs small. The tension in the problem, however, is that the only way to reduce the average size of inventory is by ordering more frequentlyñwhich is expensive! We can combine Reorder and Carrying Costs in order to write the objective function in the following way: Total Inventory Cost/yr = Reorder Cost/yr + Carrying Cost/yr = a*# of shipments/yr + b*totalunitsales/yr + k*avgsizeinventory/yr The retailer wants to MINIMIZE total inventory costs per year. C7Lab.pdf 4

5 Exogenous/Endogenous Variables. Now that we have determined the objective function, we need to decide which variables are choice variables and which are exogenous variables. Take a look at the variables in the objective function: = a*# of shipments/yr + b*totalunitsales/yr + k*avgsizeinventory/yr Clearly, a (shipping charge in $/shipment), b (wholesale cost per TV in $/TV), and k (carrying cost charge in $/unit of inventory) are constants. These values are given to the retailer by her supplier and current market conditions. Take two minutes here to THINK about what sorts of real world market conditions might affect a (shipping charge), b (wholesale cost per TV) and k (carrying cost charge). Your ability to connect the real world to the problem is CRUCIAL. Total Unit Sales/yr (or TVs sold) per year (Q) is also a given in this model. In our concrete example, we assumed it was 3,600 TVs. The remaining variables from the above equations are Average size of inventory and number of shipments. These two variables are chosen by the retailer and, therefore, are the endogenous variables in this problem. ÒHold on here!ó you object. ÒI thought we were still on single-variable optimization problems. In the objective function there are two choice variables, average size of inventory and number of shipments. IÕm not ready to tackle anything with two choice variables in it yet!ó YouÕre quite right. What weõre going to do is similar to the strategy adopted in the lifeguard problem. Instead of choosing distance on sand and distance in water, we collapsed the problem into one variable because distance in water depended in a strict way on distance on sand. Something similar, but not exactly the same, is going to happen between average size of inventory and number of shipments. Making the Problem a Single Variable Unconstrained Optimization Problem: We begin by noting that Average Size of Inventory and Number of Shipments are related. For a fixed amount of total unit sales and, thus, total TVs shipped by the supplier, the greater the number of shipments in a year, the smaller must be the average size of inventory. Instead of receiving the total amount on January 1, if you receive many small shipments, youõll have a smaller average size of inventory. C7Lab.pdf 5

6 The key to understanding the connection between the two choice variables, average size of inventory and number of shipments, is that a third variable, which we have not yet discussed, Size per Shipment, plays a crucial role. The definition is straightforward: Size of Shipment = Total Unit Sales/Number of Shipments If you choose many shipments (given a constant total unit sales), size per shipment obviously falls, and, although not as obvious, average size of inventory also falls. It is through size of shipment that number of shipments and average size of inventory are connected because of the following relationship : Average Size of Inventory = Size of Shipment/2 Before we turn to an explanation of the equation above, note that by choosing size of shipment, you automatically choose average size of inventory AND the number of shipments (since number of shipments = total unit sales/size of shipment). Thus, just like in the lifeguard problem, we transform a two variable into a single variable optimization problem. Unlike the lifeguard problem, however, where one variable was solved in terms of the other, weõre doing something a little different here. We are using a third variable, Size of Shipment, to which the other two are connected. The end result is the sameñwe end up with a single variable optimization problem. Demonstrating the Fact that Average Size of Inventory = Size of Shipment/2: The best way to show the relationship between average size of inventory and size of shipment is through a series of numerical examples. WeÕll use Excel to do this. Open C7Lab.xls now and read the first two sheets, Introduction and AvgInv. When you have finished, return here and continue reading. ***************************************************************************** The Single Variable Version of the Problem We hope that you are now convinced that average size of inventory is simply the size of each shipment divided by two. If we let D stand for the size per shipment (since it is the quantity of TVs delivered per shipment), then the average size of inventory is just D/2. We saw above that the number of shipments is simply Q divided by the size of each shipment. Thus, the retailer decides the quantity delivered in one shipment, D. That choice automatically determines the average size of inventory, D/2, as well as the number of shipments, Q/D. C7Lab.pdf 6

7 Since we now have expressions for number of shipments and average size of inventory in terms of size of shipment, we can rewrite the objective function in terms of a single choice variable. We can substitute Q/D for number of shipments per year and D/2 for average size of inventory per year into our objective function. That is, D = size of shipment in units is the single endogenous variable. Implementing this strategy, we take the original objective function = a*# of shipments/yr + b*totalunitsales/yr + k*avgsizeinventory/yr and substitute out # of shipments/yr and AvgSizeInventory/yr, writing the optimization problem as: mintotalinventorycosts D = aq kd bq D Mathematically, the tension we pointed out earlier is clear. Increases in D lower the first term, but increase the last term. The middle term is a fixed cost that does not respond to changes in the endogenous variable, D. A Concrete Example: Suppose: a = shipping charge/shipment = $2000 b = wholesale cost/tv = $200 k = carrying charge per year per unit of inventory = $64 Q = unit sales per year = 1000 LetÕs assume that the retailer chooses to have the 1000 TVs delivered in shipments of 125 TVs. D, the size of the shipment has been chosen to be 125. With D=125, there are 1000/125 = 8 shipments over the course of the year. D=125 makes the number of shipments equal to 8. D=125 also makes the average size of inventory equal to 62.5 TVs since 125/2 = Once D is chosen, the other two choice variables are automatically determined. Of course, once number of shipments is chosen or average size of inventory is chosen, the other two variables are also determined. We could solve this problem from a variety of angles. Having chosen D, the important question becomes, What are the total inventory costs (TIC) and are they as low as they can be? C7Lab.pdf 7

8 At D=125, reorder costs =$2000*Number of shipments + $200*1000 = $16,000*4 + $200*1000 = $16,000 + $200,000 = $216,000 At D=125, carrying costs = $64*Average size of inventory = $64*62.5 = $4,000 Thus, at D=125, TIC = $216,000 + $4,000 = $220,000 Is this the best that the retailer can do? Or can she LOWER total inventory costs by choosing a shipment size that is larger or smaller? Which value of D leads to the smallest TIC? ThatÕs what the retailer must figure out. In choosing D, the retailer automatically chooses number of shipments and average size of inventory. Because of the importance of understanding the choice variables in this problem, we offer one more way of explaining whatõs going on. While it may seem there are three choice variables, thereõs actually only one because the three endogenous variables are strictly related to each other. WeÕve chosen to focus on the size per shipment (D), but the problem can be solved by writing the objective function in terms of number of shipments or average size of inventory as is shown below. Original Problem: mintotalinventorycosts = an + bq + kx NX, N Q D where = and X = D 2 Problem written in terms of D (size per shipment): mintotalinventorycosts = aq kd bq D D Problem written in terms of N (number of shipments in a year): mintotalinventorycosts = an + bq + kq N 2 N Problem written in terms of X (average size of inventory): mintotalinventorycosts = aq bq kx X 2X + + Of course, no matter which way the problem is written, the exact same answers are obtained. Using the version in which the problem is written in terms of D, we will solve the problem and use the solution to get values for N (number of shipments) and X (average size of inventory). C7Lab.pdf 8

9 Q & A Instructions: Two of the five questions (numbers 2 and 3) involve pencil and paper calculations; and two of the questions involve using Excel (numbers 1 and 4). Question 5 involves no calculations. Part I: Solving the Optimal Inventory Problem (1) Solve the minimize Total Inventory Cost problem in the concrete case with Q = 1,000, k = 64, a = 2000, b = 200. Do this using the Excel Solver. Instructions: Return to the file called C7Lab.xls. You should be on Question 1 in the Q&A Sheet. Click on ÒGo to JITÓ and follow the instructions given. If you are someplace else in the file, simply click on the worksheet tab called JIT. You will be given instructions for recording your answer on Screen 2 of the Q&A Sheet. (2) On a separate sheet of paper, solve the retailerõs problem ( i.e., minimize Total Inventory Cost) using calculus in the general case. (By the general case, we mean using the exogenous variables a, b, Q and k instead of specific numerical values.) Show your work neatly on paper. Work your way through the five steps in the solution recipe. Take a look at Chapter 5 readings and Suggested Answers. Prominently display your answers for D* and TIC*. Calculate N* after youõve found D*. NOTE: If you plug in the concrete values of the exogenous variables into your expression for D*, you may get a slightly different answer than the one that ExcelÕs Solver gave you. As you know, Solver will announce it has found a solution when the convergence criterion is met and this may be a little short of the exact solution. The two answers, Solver and pencil and paper, while not exactly the same, should be extremely close. Part II: Comparative Statics: (3) Find the reduced form for D* from (2) and use it to find dd*/dk Ñ the change in optimal order size given a change in carrying cost per TV. Put your answer on the paper which includes answers to Question 2. (4) Return to C7Lab.xls and draw a graph to illustrate how D* varies with a, the shipping cost. Use Excel, including both Solver and the Comparative Statics Wizard, to do this as you vary the value of ÒaÓ from 2000 to 1000 in increments of 200. (NOTE: Enter Ð 200, negative 200, in the amount of shock dialog box.) Record your answer in Screen 3 of the Q&A worksheet (as per the instructions in the file). (5) Use your answers to (3) and (4) above to suggest an explanation of the ÒJIT revolutionó Ñ that is, the observed reduction in inventory levels over the last few years. Do this in a text box in Screen 4 of the Q & A worksheet. C7Lab.pdf 9