Lot Sizing Lot Sizing

InventoryInventory

Types of inventoryTypes of inventory– Raw materials/purchase partsRaw materials/purchase parts– Work-in-processWork-in-process– Finished goodsFinished goods

Holding of inventory is expensiveHolding of inventory is expensive– Ties up fundsTies up funds– Requires spaceRequires space– People/system needed to trackPeople/system needed to track– What happens when the product changes?What happens when the product changes?

Inventory controlInventory control

Inventory control aimed at providing items at Inventory control aimed at providing items at the time of needthe time of need– Need to determine re-order point (inventory level Need to determine re-order point (inventory level

at which a new supply should be ordered to bring at which a new supply should be ordered to bring inventory to its desired level)inventory to its desired level)

– Attempt to provide some safety stock to Attempt to provide some safety stock to compensate for fluctuations in demand, however, compensate for fluctuations in demand, however, difficult to predict reasonable safety stock due to: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 Lack of ability to determine specific timing of future demand

(timing)(timing) Leads to:Leads to:

– Inventory imbalanceInventory imbalance– Stockouts (shortages)Stockouts (shortages)– High overall inventoryHigh overall inventory

Approaches to inventory Approaches to inventory controlcontrol

Work without inventory or reduce to minimum Work without inventory or reduce to minimum (JIT)(JIT)

Utilize an inventory management system, MRP Utilize an inventory management system, MRP System (Materials Requirements Planning System (Materials Requirements Planning System)System)

Economic order quantityEconomic order quantity

MRPMRP

Example of MRP logicExample of MRP logic

It takes 1 unit of B and 2 units of C to assemble It takes 1 unit of B and 2 units of C to assemble product Aproduct A

The gross requirements for A: 200 due in week 4 and The gross requirements for A: 200 due in week 4 and 250 due in week 5250 due in week 5

At the beginning of week 1At the beginning of week 1

Determine the production schedules for A, B and CDetermine the production schedules for A, B and C

IDIDOn On

HandHandSafety Safety StockStock

AllocateAllocatedd

Lead Lead TimeTime

Level Level CodeCode

AA 100100 5050 2020 11 00

BB 150150 7070 3030 22 11

CC 8080 4040 1515 11 11

EOQ modelEOQ model

Simplest and most fundamental of all Simplest and most fundamental of all inventory modelsinventory models

Describes the important trade-off between Describes the important trade-off between fixed order cost and holding costsfixed order cost and holding costs

EOQ model assumptionsEOQ model assumptions

Demand rate is known and is a constant Demand rate is known and is a constant λλ units per unit timeunits per unit time– Unit of time may be days, weeks, months, …, as Unit of time may be days, weeks, months, …, as

long as all relevant variables are expressed in the long as all relevant variables are expressed in the same unitssame units

– Default unit of time is a yearDefault unit of time is a year Shortages are not permittedShortages are not permitted There is no order lead timeThere is no order lead time The costs includeThe costs include

– Setup costs at K per order placedSetup costs at K per order placed– Proportional order cost at c per unit orderedProportional order cost at c per unit ordered– Holding cost at h per unit held per unit timeHolding cost at h per unit held per unit time

Changes in stock levels over Changes in stock levels over timetime

EOQ model developmentEOQ model development

Objective is to choose Q to minimize the Objective is to choose Q to minimize the average cost per unit of time (typically average cost per unit of time (typically assumed to be a year)assumed to be a year)

Components of the average annual costComponents of the average annual cost– In each cycle, the total fixed order cost (K) plus the In each cycle, the total fixed order cost (K) plus the

proportional order cost (c∙Q) is K + c∙Qproportional order cost (c∙Q) is K + c∙Q– To obtain the order cost per unit time, divide by the To obtain the order cost per unit time, divide by the

cycle length T → (K + c∙Q)/Tcycle length T → (K + c∙Q)/T– The average inventory level during one cycle is Q/2 The average inventory level during one cycle is Q/2

→ because all cycles are identical, the average → because all cycles are identical, the average inventory level over a time horizon composed of inventory level over a time horizon composed of many cycles is also Q/2many cycles is also Q/2

– Holding cost (carrying cost) per unit time is h∙Q/2Holding cost (carrying cost) per unit time is h∙Q/2

Average annual costAverage annual cost

Average annual cost, G(Q)Average annual cost, G(Q)

As Q units are consumed each cycle at a rate As Q units are consumed each cycle at a rate of of λλ, it follows that T = Q/, it follows that T = Q/λλ

Three terms above represent setup cost, Three terms above represent setup cost, purchase cost, and holding costpurchase cost, and holding cost

2)(

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Determine EOQDetermine EOQ

Find Q to minimize G(Q)Find Q to minimize G(Q) Set the derivative equal to zeroSet the derivative equal to zero

Proportional order cost incurred per unit time Proportional order cost incurred per unit time ((λλ∙c) is independent of Q and is generally ∙c) is independent of Q and is generally ignored when computing average costsignored when computing average costs

c does affect the value of EOQ indirectly, as h c does affect the value of EOQ indirectly, as h appears in the formula and h = I∙c, I = interest appears in the formula and h = I∙c, I = interest rate (I is typically an annual interest rate)rate (I is typically an annual interest rate)

h

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Determine the minimum of the Determine the minimum of the average annual costaverage annual cost

EOQ exampleEOQ example

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

Another EOQ exampleAnother EOQ example

The Ohio State University location of The Ohio State University location of McDonald’s uses 120 six-ounce paper cups per McDonald’s uses 120 six-ounce paper cups per day. McDonald’s plans to be open 360 days day. McDonald’s plans to be open 360 days per year. The cups cost $.10 per dozen; per year. The cups cost $.10 per dozen; ordering costs are $5 per order; and carrying ordering costs are $5 per order; and carrying costs are 50 percent of the item cost (since costs are 50 percent of the item cost (since space is at a premium).space is at a premium).(a) Find the economic order quantity if delivery is (a) Find the economic order quantity if delivery is

instantaneous (work the problem in units of dozens instantaneous (work the problem in units of dozens of cups).of cups).

(b) Currently, cups are ordered every thirty days. (b) Currently, cups are ordered every thirty days. Relate current order quantity, optimal order quantity, Relate current order quantity, optimal order quantity, current total costs, and optimal total costs. What do current total costs, and optimal total costs. What do you recommend?you recommend?