Inventory Models in SC Environment

ByDebadyuti Das

Three levels of Inventory Decisions

Supply Chain Decisions (Strategic)- What are the potential alternatives to inventory?- How should the product be designed?

Deployment Decisions (Strategic)- What items should be carried as inventory?- In what form should they be maintained?- How much of each should be held and where?

Replenishment Decisions (Tactical/Operational)- How often should inventory status be determined?- When should a replenishment decision be made?- How large should the replenishment be?

Classification of Inventory Financial / Accounting Categories

- Raw Materials - Work in process (WIP)- Components, Semi-Finished Goods- Finished Goods

Functional Classification- Cycle Stock- Safety Stock- Pipeline Inventory- Decoupling Stock- Anticipation Inventory

Total Relevant Costs (TRC) TC = Purchase + Order + Holding + Shortage

What makes a cost relevant?

Four standard cost components- Purchase (unit value) cost

- Ordering (Set up) cost- Holding (Carrying) cost- Shortage cost

What factors influence inventory replenishment models? Demand Lead time Dependence of items Review time Number of echelons Discounts Shortages Perishability Planning horizon Number of items

Basic EOQ Model: Assumptions Demand

- Constant- Known- Continuous

Lead time- Instantaneous

Dependence of items- Independent

Review time- Continuous

Number of echelons- One

Basic EOQ Model: Assumptions Discounts

- None Shortages

- None Perishability

- None Planning horizon

- Infinite Number of items

- One

The Inventory Cycle

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Time

Basic Terminologies

D: Annual demand S: Setup or Order CostC: Cost per unith: Holding cost per year as a fraction of product

costH: Holding cost per unit per yearQ: Lot SizeT: Reorder interval

Material cost is constant and therefore is not considered in this model

Basic Terminologies

Number of orders per year = D/QAnnual material cost = CDAnnual order cost = (D/Q)SAnnual holding cost = (Q/2)H = (Q/2) hCTotal annual cost = (TC) =

CD + (D/Q)S + (Q/2) hC

Cost Minimization Goal

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al C

os

t

(optimal order quantity)

TCQH

D

QS

2

Deriving the EOQ

Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

Minimum Total Cost

The total cost curve reaches its minimum where the carrying and ordering costs are equal.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

EOQ: Important Observations*

Tradeoff between set-up costs and holding costs when determining order quantity. In fact, we order so that these costs are equal per unit time

Total Cost is not particularly sensitive to the optimal order quantity

Order Quantity 50% 80% 90% 100% 110% 120% 150% 200%

Increase in cost 25% 2.5 % 0.5 % 0 % 0.4 % 1.6 % 8.0 % 25.0 %

Example (1)

Demand, D = 12,000 computers per yeard = 1000 computers/monthUnit cost, C = $500Holding cost fraction, h = 0.2Fixed cost, S = $4,000/orderQ* = Sqrt[(2)(12000)(4000)/(0.2)(500)] = 980

computers Cycle inventory = Q/2 = 490Flow time = Q/2d = 980/(2)(1000) = 0.49 monthReorder interval, T = Q/d = 0.98 month

Example 1(continued)

Annual ordering and holding cost = = (12000/980)(4000) + (980/2)(0.2)(500) = $97,980Suppose lot size is reduced to Q=200, which would

reduce flow time:Annual ordering and holding cost = = (12000/200)(4000) + (200/2)(0.2)(500) =

$250,000

To make it economically feasible to reduce lot size, the fixed cost associated with each lot would have to be reduced

Example 2

If desired lot size = Q* = 200 units, what would S haveto be?

D = 12000 unitsC = $500h = 0.2Use EOQ equation and solve for S:S = [hC(Q*)2]/2D = [(0.2)(500)(200)2]/(2)(12000) =

$166.67

To reduce optimal lot size by a factor of k, the fixed order cost must be reduced by a factor of k2

Key Points from EOQ Model

In deciding the optimal lot size, the tradeoff is between setup (order) cost and holding cost.

If demand increases by a factor of k, it is optimal to increase batch size by a factor of Sqrt k. No of orders placed per year should increase by a factor of Sqrt k. Flow time attributed to Cycle inventory should decrease by a factor of Sqrt k.

If lot size is to be reduced, one has to reduce fixed order cost. To reduce lot size by a factor of 2, order cost has to be reduced by a factor of 4.

Single period Inventory Model Model for ordering of perishables and other

items with limited useful lives One time purchasing decision (Example:

newspapers, vegetables, fresh fruits etc.)

Shortage cost:(or the underestimated cost) generally the unrealized profits per unit (Revenue/unit minus cost/unit).

Excess cost: (or the overestimated cost) difference between purchase cost and salvage value of items left over at the end of a period.

Single period Inventory Model

Seeks to balance the costs of inventory overstock and under stock.

The goal is to identify the order quantity that will minimize the long run excess and shortage costs.

Service level is the key to determining the optimal stock level in this model.

Single-Period Inventory Model

sold be unit will y that theProbabilit

estimatedunder demand ofunit per Cost C

estimatedover demand ofunit per Cost C

:Where

u

o

P

uo

u

CC

CSL

uo

u

CC

CSL

Single Period Inventory Model Example

Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell just before the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?

Cu = $10 and Co = $5; SL = $10 / ($10 + $5) = .667

Z.667 = .432therefore we need 2,400 + .432(350) = 2,551 shirts

Quantity Discount Model

Why quantity discounts? Coordination in the supply chain Price discrimination to maximize supplier profits

Lot size based Volume based How should buyer react?

All unit Quantity Discount Model

Pricing schedule has specified quantity break points q0, q1, …, qr, where q0 = 0

If an order is placed that is at least as large as qi but smaller than qi+1, then each unit has an average unit cost of Ci

The unit cost generally decreases as the quantity increases, i.e., C0>C1>…>Cr

The objective for the company (a retailer in our example) is to decide on a lot size that will minimize the sum of material, order, and holding costs

Total Cost with Constant Carrying Costs

OC

EOQ Quantity

To

tal C

os

t

TCa

TCc

TCbDecreasing Price

CC a,b,c

Total Cost with Carrying Costs expressed as % of unit price

OC

EOQ Quantity

To

tal C

os

t

TCa

TCc

TCbDecreasing Price

CCc

CCa

CCb

Case 1: All unit Quantity Discount Procedure

Compute the common minimum point Identify the feasible minimum point If the feasible minimum point is on the lowest price

range, that is the optimal order quantity. If the feasible minimum point is on any other range,

compute the total cost for the minimum point and for the price breaks of all lower unit costs.

Compare the total costs, the quantity that yields the lowest total cost is the optimal order quantity.

Case 2: All unit Quantity Discount Procedure

Step 1: Calculate the EOQ for the lowest price. If it is feasible (i.e., this order quantity is in the range for that price), then stop. This is the optimal lot size. Calculate TC for this lot size.

Step 2: If the EOQ is not feasible, calculate the TC for this price and the smallest quantity for that price.

Step 3: Calculate the EOQ for the next lowest price. If it is feasible, stop and calculate the TC for that quantity and price.

Step 4: Compare the TC for Steps 2 and 3. Choose the quantity corresponding to the lowest TC.

Step 5: If the EOQ in Step 3 is not feasible, repeat Steps 2, 3, and 4 until a feasible EOQ is found.

All-Unit Quantity Discounts: Example

Cost/Unit

$3$2.96

$2.92

Order Quantity

5,000 10,000

Order Quantity

5,000 10,000

Total Material Cost

All-Unit Quantity Discount: Example

Order quantity Unit Price0-5000 $3.005001-10000 $2.96Over 10000 $2.92

q0 = 0, q1 = 5000, q2 = 10000C0 = $3.00, C1 = $2.96, C2 = $2.92D = 120000 units/year, S = $100/lot, h = 0.2

All-Unit Quantity Discount: Example

Step 1: Calculate Q2* = Sqrt[(2DS)/hC2] = Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410Not feasible (6410 < 10001)Calculate TC2 using C2 = $2.92 and q2 = 10001TC2 = (120000/10001)(100)+(10001/2)(0.2)

(2.92)+(120000)(2.92)= $354,520

All-Unit Quantity Discount: Example

Step 2: Calculate Q1* = Sqrt[(2DS)/hC1]=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367Feasible (5000<6367<10000) StopTC1 = (120000/6367)(100)+(6367/2)(0.2)

(2.96)+(120000)(2.96) = $358,969TC2 < TC1 The optimal order quantity Q*

is q2 = 10001

Coordination forCommodity Products

D = 120,000 bottles/year SR = $100, hR = 0.2, CR = $3 SS = $250, hS = 0.2, CS = $2

Retailer’s optimal lot size = 6,324 bottlesRetailer cost = $3,795; Supplier cost =

$6,009Supply chain cost = $9,804

Coordination forCommodity Products

What can the supplier do to decrease supply chain costs? Coordinated lot size: 9,165; Retailer cost = $4,059;

Supplier cost = $5,106; Supply chain cost = $9,165 Effective pricing schemes

All-unit quantity discount $3 for lots below 9,165 $2.9978 for lots of 9,165 or more

Pass some fixed cost to retailer (enough that he raises order size from 6,324 to 9,165)

Quantity Discounts WhenFirm Has Market Power

No inventory related costs Demand curve

360,000 - 60,000pWhat are the optimal prices and profits in the

following situations? The two stages coordinate the pricing decision

Price = $4, Profit = $240,000, Demand = 120,000 The two stages make the pricing decision

independently Price = $5, Profit = $180,000, Demand = 60,000

Lessons from Discounting Schemes

Lot size based discounts increase lot size and cycle inventory in the supply chain

Lot size based discounts are justified to achieve coordination for commodity products

Volume based discounts with some fixed cost passed on to retailer are more effective in general Volume based discounts are better over rolling

horizon