Lot Sizing

Inventory Types of inventory Types of inventory –Raw materials/purchase parts –Work-in-process –Finished goods Holding of inventory is expensive Holding of inventory is expensive –Ties up funds –Requires space –People/system needed to track –What happens when the product changes?

Inventory control Inventory control aimed at providing items at the time of need Inventory control aimed at providing items at the time of need –Need to determine re-order point (inventory level at which a new supply should be ordered to bring inventory to its desired level) –Attempt to provide some safety stock to compensate for fluctuations in demand, however, difficult to predict reasonable safety stock due to: False assumptions about observed demand (quantity) False assumptions about observed demand (quantity) Lack of ability to determine specific timing of future demand (timing) Lack of ability to determine specific timing of future demand (timing) Leads to: Leads to: –Inventory imbalance –Stockouts (shortages) –High overall inventory

Approaches to inventory control Work without inventory or reduce to minimum (JIT) Work without inventory or reduce to minimum (JIT) Utilize an inventory management system, MRP System (Materials Requirements Planning System) Utilize an inventory management system, MRP System (Materials Requirements Planning System) Economic order quantity Economic order quantity

MRP

Example of MRP logic It takes 1 unit of B and 2 units of C to assemble product A It takes 1 unit of B and 2 units of C to assemble product A The gross requirements for A: 200 due in week 4 and 250 due in week 5 The gross requirements for A: 200 due in week 4 and 250 due in week 5 At the beginning of week 1 At the beginning of week 1 Determine the production schedules for A, B and C Determine the production schedules for A, B and C ID On Hand Safety Stock Allocated Lead Time Level Code A B C

EOQ model Simplest and most fundamental of all inventory models Simplest and most fundamental of all inventory models Describes the important trade-off between fixed order cost and holding costs Describes the important trade-off between fixed order cost and holding costs

EOQ model assumptions Demand rate is known and is a constant λ units per unit time Demand rate is known and is a constant λ units per unit time –Unit of time may be days, weeks, months, …, as long as all relevant variables are expressed in the same units –Default unit of time is a year Shortages are not permitted Shortages are not permitted There is no order lead time There is no order lead time The costs include The costs include –Setup costs at K per order placed –Proportional order cost at c per unit ordered –Holding cost at h per unit held per unit time

Changes in stock levels over time

EOQ model development Objective is to choose Q to minimize the average cost per unit of time (typically assumed to be a year) Objective is to choose Q to minimize the average cost per unit of time (typically assumed to be a year) Components of the average annual cost Components of the average annual cost –In each cycle, the total fixed order cost (K) plus the proportional order cost (c∙Q) is K + c∙Q –To obtain the order cost per unit time, divide by the cycle length T → (K + c∙Q)/T –The average inventory level during one cycle is Q/2 → because all cycles are identical, the average inventory level over a time horizon composed of many cycles is also Q/2 –Holding cost (carrying cost) per unit time is h∙Q/2

Average annual cost Average annual cost, G(Q) Average annual cost, G(Q) As Q units are consumed each cycle at a rate of λ, it follows that T = Q/λ As Q units are consumed each cycle at a rate of λ, it follows that T = Q/λ Three terms above represent setup cost, purchase cost, and holding cost Three terms above represent setup cost, purchase cost, and holding cost

Determine EOQ Find Q to minimize G(Q) Find Q to minimize G(Q) Set the derivative equal to zero Set the derivative equal to zero Proportional order cost incurred per unit time (λ∙c) is independent of Q and is generally ignored when computing average costs Proportional order cost incurred per unit time (λ∙c) is independent of Q and is generally ignored when computing average costs c does affect the value of EOQ indirectly, as h appears in the formula and h = I∙c, I = interest rate (I is typically an annual interest rate) c does affect the value of EOQ indirectly, as h appears in the formula and h = I∙c, I = interest rate (I is typically an annual interest rate)

Determine the minimum of the average annual cost

EOQ example Number 2 pencils at the bookstore are sold at a fairly steady rate of 60 per week. The pencils cost the bookstore 2 cents each and sell for 15 cents each. It costs the bookstore $12 to initiate an order, and holding costs are based on an annual interest rate of 25 percent. Determine the optimal number of pencils for the bookstore to purchase and the time between placement of orders. On average, what are the yearly holding and setup costs for this item? Number 2 pencils at the bookstore are sold at a fairly steady rate of 60 per week. The pencils cost the bookstore 2 cents each and sell for 15 cents each. It costs the bookstore $12 to initiate an order, and holding costs are based on an annual interest rate of 25 percent. Determine the optimal number of pencils for the bookstore to purchase and the time between placement of orders. On average, what are the yearly holding and setup costs for this item?

Another EOQ example The Ohio State University location of McDonald’s uses 120 six-ounce paper cups per day. McDonald’s plans to be open 360 days per year. The cups cost $.10 per dozen; ordering costs are $5 per order; and carrying costs are 50 percent of the item cost (since space is at a premium). The Ohio State University location of McDonald’s uses 120 six-ounce paper cups per day. McDonald’s plans to be open 360 days per year. The cups cost $.10 per dozen; ordering costs are $5 per order; and carrying costs are 50 percent of the item cost (since space is at a premium). (a) Find the economic order quantity if delivery is instantaneous (work the problem in units of dozens of cups). (b) Currently, cups are ordered every thirty days. Relate current order quantity, optimal order quantity, current total costs, and optimal total costs. What do you recommend?