QUANTITATIVE METHODS IN BUSINESS PROBLEM #5: ECONOMIC ORDER QUANTITY MODELS

Transcription

1 QUANTITATIVE METHODS IN BUSINESS PROBLEM #5: ECONOMIC ORDER QUANTITY MODELS Dr. R. K. Tibrewala Mathematics by Dr. Arnold Kleinstein Introduction All organizations keep inventory (or stock) of certain items for future use. Depending on the type of business an organization may keep inventory of raw materials, components, subassemblies, finished products or supplies. There are two main reasons for keeping inventory. The first is to take care of uncertainty or fluctuations in supply and demand. Inventory kept for this purpose is called safety stock. When we have a complete and exact knowledge of supply time (lead-time) and demand (usage), the resulting inventory models are known as deterministic. In deterministic models there is no reason to keep any safety stock and the replenishment begins when inventory is exactly zero. The second reason for keeping inventory is to take advantage of buying or producing in a large quantity. The economic benefit may result from quantity discounts, purchasing costs (or set up costs), inspection, and order processing costs. Larger order quantities (or lot sizes) reduce such costs but may increase holding costs which include the cost of money tied up in inventory, cost of space, insurance, obsolescence, etc. The economic order quantity, or EOQ models, are designed to trade off between the ordering cost (or set up cost), which is fixed regardless of the amount ordered, and the holding cost which increases as order size increases. In these models, the demand is assumed to be constant, and the lead-time assumed to be known exactly. There are many business situations where these conditions hold. For example, if a company is operating at full capacity, or on a prescheduled basis, the demand for many items will be constant. As a general rule, it has been observed that most companies keep too much inventory. Reducing inventory can, not only reduce costs, but also improve quality. We will, therefore, address the issue of how much to order (or produce) and when to order (or produce). In other words, we want to find Q, the economic order quantity, and R, the reorder point (inventory level at which we place a new order). version August 1,

2 Mathematics Relationship between Area and Multiplication Geometry gives us a way to picture some mathematical operations. For example, since the Area of rectangle = base * height The product of two numbers a*b, may be represented by the area of a rectangle whose base is a, and height b. b 6 2*6 a 2 Area of rectangle = a*b The area of two rectangles represents the sum of two multiplications: b c*d d 6 c*d 11 a c 2 7 Area of both rectangles = a*b + c*d (2*6) + (7*11) = Suppose that we are to add together the product of many different numbers. Rather than write down the numbers, let's think of it in terms of area. Area of rectangles = ab + cd + ef+ gh + hi + jk + lm version August 1,

3 Approximation Formula If these rectangles all go down evenly, as above, we can approximate the area of these rectangles by the area of triangle, QOD Q O D Area of triangle QOD = (1/2) * base * height = (1/2) * Q * D Thus, because of the correspondence between numbers and area, we can conclude: ab + cd + ef+ gh + hi + jk + lm (1/2) * Q * D Business Example: In the business problem that follows, we will see how this is applied to approximate the cost of holding inventory. The inventory cycle begins when Q items arrive. We assume the daily demand is constant, so that the inventory is reduced by the same amount each day. When the inventory becomes zero a new order arrives. Let D be the number of days in the inventory cycle. Q O D =days in cycle version August 1,

4 In the above picture, the x-axis represents days, the y-axis inventory. The area of each rectangle is the amount of inventory on hand that day. This is because base of each rectangle is 1 day. Area of each rectangle = base * height = 1 * height = height = inventory on hand that day The sum of the number of inventory items on hand each day of the inventory cycle is equal to the sum of the area of all the rectangles, which, we saw, is approximately equal to the area of the triangle QOD. Thus, Sum of the number of inventory items on hand each day of the inventory cycle = (1/2) * Q * D We will now use this to calculate the calculate the cost of holding the items in inventory. Let H = the annual holding cost per item, then H Daily holding cost per item = days in. year The holding cost for inventory over the course of the inventory cycle is the daily holding cost per item * sum of the inventory on hand each day of the cycle. Thus, Holding cost over an inventory cycle = H days in *area of rectangle year Holding cost for the inventory cycle = H days in 1 * * Q * D year 2 version August 1,

5 Derivatives and the Minimum of a Function Let f(x) be a function, the absolute minimum of f(x) on an interval (a,b) is the lowest point on the graph of f(x), for x in the interval I. Example: Let f(x) = 2x /x It is graphed below absolute minimum Tangent The maximum or minimum of a function on the interval (a, b) occurs where its tangent line is horizontal (i.e., has a slope 0). Since the derivative of a function evaluated at a point, is the slope of the tangent line at that point, we can find the minimum points (if they exit) by taking the derivative of the function and setting it equal to zero. Formula for the Minimum of a Function of the Form f(x) = ax + b/x + c The derivative of a function of the form f(x) = ax + b/x + c is df dx = a - b/x 2 Setting the derivative equal 0, since we are looking for a minimum, we get the equation: 0 = a - b/x 2 version August 1,

6 Which is solved as: x = b/ a (assuming x is a positive number). The point on the graph with this x-value is not necessarily a minimum. It may be a maximum, or a turning point. To insure it is a minimum, we need investigate further. However, it turns out that for all functions of the form f(x) = ax + b/x + c x = b/ a gives a minimum value. Example: Let f(x) = 2x /x The minimum of this function occurs at x = / 2 = = Compare this to the x-value of the lowest point as indicated on the graph above. version August 1,

7 Excel =SQRT To take the square root of a number, use the Excel function =SQRT(number). Example: =SQRT(16) = 4 If C1 contains the number 25, then =SQRT(C1) = 5, and =SQRT(4*C1) = 10 Graphing a Function In the EOQ inventory model, the function for total cost is of the form f(x) = ax + b/x + c To graph this, put x values in one column, and values of the function in another column. Then use chart wizard with graph type XY (scatter). Example: Let f(x) = 2x /x In the math section, we saw that the minimum of this function occurs at x = / 2 = = When graphing this function, choose values of x that surround 141 so that we can see the minimum. For example, to create the graph that follows, we choose x-values between 122 and 250. Below we indicate how they are entered into the spreadsheet. The rows are grouped. The graph shows the minimum point. version August 1,

8 absolute minimum version August 1,

9 Business Application American Technology Corporation is in the process of improving its management practices to increase productivity. American Technology Corporation keeps several thousand items in inventory and the decision about how much inventory to keep is left to the field operations people. Inventory cost reduction is one of the key components of the overall strategy. In order to estimate the potential cost reduction, one particular product out of thousands was chosen and the pertinent data appears below. Annual demand = 1600 items Purchasing (ordering) costs = $25 per order Annual hold cost = 25% (of cost) Cost of each item = $8 The first suggestion made was to use the economic order quantity models to minimize cost. The estimated current cost of ordering, plus holding, is $500 per year. How much money will American Technology Corporation save if the EOQ model is used? The financial group has come up with a plan to reduce the cost of operating capital that in turn will reduce the holding cost from 25% to 22%. How much additional saving can be made from this effort? The purchasing group has been negotiating with vendors. The vendor for this product has offered American Technology Corporation a discount of 2%, 5%, or 10% if the quantity purchased in one order is at least 400, 800, or 1600 items respectively. Which discount level should American Technology Corporation choose and how much money will it save by doing so? Finally, the management wants to know to know the effect of implementing both these savings options. version August 1,

10 Solution to the Business Problem The first step in solving the problem for American Technology Corporation is to develop an Economic Order Quantity model. Let us use Q (items/order) to represent the EOQ, P ($/order) to represent the ordering cost, C ($/item) to represent the item cost, I (%/year) to represent the holding cost rate, H ($/item/year) to represent item holding cost, and D (items/year) to represent annual demand. The EOQ model assumes three components make up the cost of inventory: Annual Holding Cost Total Cost = Annual Holding Cost + Ordering Cost + Purchase Cost The inventory cycle begins when Q items arrive. We assume the daily demand is constant, so that the inventory is reduced by the same amount each day. When the inventory becomes zero a new order arrives Q O In the math section we saw that: Holding cost for the inventory cycle = H days in D = days in cycle 1 * * Q * D year 2 To obtain the holding cost for the year, multiply the cost per cycle by the number of cycles: Number of cycles in a year = days in year D Holding cost for the year = days in year D * H days in 1 * * Q * D year 2 Holding cost for the year = 2 1 * H * Q version August 1,

11 This is exactly true if there are an integral number of cycles per year, and the average cost per year otherwise. Ordering Cost Since Q units are ordered each time, the number of orders will equal Q D each year. The cost per order, is P dollars. Thus, Annual Ordering cost = D * P in $/year. Q Purchase Cost Each item costs C dollars, and D items are ordered each year. Thus Purchase cost = C * D in $/year. version August 1,

12 Total Cost The total cost TC = Holding Cost + Ordering Cost + Purchas Cost, or, expressed as a formula: 1 D TC = * H * Q + * P + C* D 2 Q This equation is of the form f(x) = ax + b/x + c with Q replacing the variable x, and a = 2 1 *H b =D*P c = C*D In the math section, we saw that the minimum occurs when, x = b/ a or Q = ( D * P) /( H / 2) Simplifying the expression, we obtain the formula for Q, the order quantity that gives the minimum inventory cost. Q = 2DP H This value is denoted the minimum cost EOQ. version August 1,

13 Let us apply the concepts of EOQ to American Technology Corporation s case. D = 1600 items/year P = $25/order I = 25%/year C = $8/item H = C*I = 8*25 = $2/item/year Q = 2*1600* 25 = 40 x 5 = 200 items/order 2 TC = 1 D * Q * H + * P + purchase cost 2 Q = *200* 2 + *25 + purchase cost = purchase cost = $400/year + purchase cost Note that the holding cost equals the ordering cost and the savings from the current scenario is $100/year. (Currently the annual cost of ordering and holding is $500). version August 1,

14 Excel Spreadsheet Construction In order to analyze the effect of options suggested by the financial and the purchasing group; we will have to solve several problems under different conditions. Excel provides a convenient means for accomplishing this goal. To build a spreadsheet for the EOQ model, enter the heading, data, and formulas as shown below. Note that the formulas used are the same as those discussed before in this section. Required minimum Q is zero if American Technology Corporation does not get any discount from the vendor. The If function is used to determine order quantity to use. It checks the desired EOQ against the minimum required quantity and returns the higher of the two values. When you enter the formulas in your spreadsheet, you will see the values in column B as shown in the next spreadsheet and a savings of $100 over baseline is possible with the use of the EOQ model. A 1 American Technology Corporation EOQ 2 Using 25% rate 3 4 Annual Demand Ordering Cost 25 6 Holding Cost Rate Cost per item 8 8 Days per year Lead-time 6 10 Required minimum Q 0 11 Current total cost Unit holding cost =B6*B7 14 Minimum cost EOQ =SQRT(2*B4*B5/B13) 15 Order quantity to use =IF(B14>B10,B14,B10) Annual holding cost =B15*B13*1/2 18 Annual ordering cost =B4*B5/B15 19 Annual item cost =B7*B4 20 Annual total cost =B17+B18+B Savings over base case =B11-B20 23 B version August 1,

15 To determine the effect of various discount levels, prepare the headings in columns C, D, and E as shown. Copy everything from column B to Columns C, D, and E. Change the cost per item in columns C, D, and E to reflect discounted cost of each item, i.e., 8 *.98, 8 *.95 and 8 *.90. Enter minimum required Q to be 400, 800, and 1600 in columns C, D, and E. The spreadsheet shows the results. The best decision for American Technology Corporation is to accept a 5% discount and order 800 units each time from the vendor. This will save $330 per year from current costs. A B C D E 1 American Technology Corporation EOQ 2% disc. 5% disc. 10% disc. 2 Using 25% rate 3 4 Annual Demand Ordering Cost Holding Cost Rate Cost per item Days per year Lead-time Required minimum Q Current total cost Unit holding cost Minimum cost EOQ Order quantity to use Annual holding cost Annual ordering cost Annual item cost Annual total cost Savings over base case version August 1,

16 The last spreadsheet shows the effect of reducing the holding cost rate to.22 or 22%. Simply change the holding cost rate and the new solution appears as shown. American Technology Corporation can save $ per year by implementing both options and taking advantage of the 10% discount and ordering once a year. A B C D E 1 American Technology Corporation EOQ 2% disc. 5% disc. 10% disc. 2 Using 22% rate 3 4 Annual Demand Ordering Cost Holding Cost Rate Cost per item Days per year Lead-time Required minimum Q Current total cost Unit holding cost Minimum cost EOQ Order quantity to use Annual holding cost Annual ordering cost Annual item cost Annual total cost Savings over base case version August 1,

17 Additional Problems 1. Suppose that a company buys about $81,000 worth of items from a vendor each year. If ordering cost is $25 per order and holding cost rate is 20%, how many dollars worth of items should be bought in each order? 2. The vendor in the above problem will offer you a 2% discount if you buy at least $9,000 of items in each order. Should you accept this offer? If no, what is a fair counter offer? Research Problem: Suppose that American Technology Corporation can allow shortages for the product used in the business case, at a cost of $4 per unit per year. Construct a new spreadsheet model. version August 1,