Yuji Yamamoto

PPU411 – VT 2017

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Continuous review (Q) system

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A continuous review (Q) system, sometimes called a reorder point (ROP) system or fixed order-quantity system, tracks the remaining inventory of a SKU each time a withdrawal is made to determine whether it is time to reorder.

Periodic review (P) system

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A continuous review (Q) system, sometimes called a reorder point (ROP) system or fixed order-quantity system, tracks the remaining inventory of a SKU each time a withdrawal is made to determine whether it is time to reorder.

A periodic review (P) system, reviews the remaining inventory of a SKU periodically on fixed intervals and determine what quantity to reorder

Periodic review (P) system

Two decisions must be made:

• P: Length of time between reviewsCould e.g. be based on the cost trade-offs of the EOQ, average time between orders of the EOQ (= TBO).

• T: Target inventory levelMust equal to the expected demand during protection interval (P + L) plus enough safety stock to protect against demand and lead time uncertainties.

T = average demand during the protection interval+ safety stock for protection interval

! = ! ! + ! + !! !!! = ! ! + ! + !!!! ! + !!

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EXAMPLE 7P system

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Demand for a bird feeder is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units. The lead time is 2 weeks, and the business operates 52 weeks per year. EOQ is 75 units and the safety stock is 9 units for a cycle-service level of 90 percent.

What is the equivalent P system? Find P and T!Answers are to be rounded to the nearest integer.

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Q system vs. P system

Primary advantages of Q systems• Review frequency may be individualized• Fixed lot sizes can result in quantity discounts• Lower safety stocks

Primary advantages of P systems• Convenient • Orders can be combined • Only need to know IP when review is made

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Hybrid systems

Optional replenishment systems• Optimal review, min-max, or (s,S) system, like the P system• Reviews IP at fixed time intervals and places a variable-sized order to

cover expected needs• Ensures that a reasonable large order is placed

Base-stock system• Replenishment order is issued each time a withdrawal is made• Order quantities vary to keep the inventory position at R• Minimizes cycle inventory, but increases ordering costs• Appropriate for expensive items

Visual systems• Allows employees to place orders when inventory visibly reaches a

certain marker. (E.g. Kanban)

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Non-instantaneous replenishment

If an item is being produced internally (rather than purchased)finished units may be used or sold as soon as they are completed, without waiting until a full lot is completed!

Usually production rate, p, exceeds the demand rate, d, so there is a build-up of (p – d) units for Q/p days!

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Non-instantaneous replenishment

( ) ÷ø

öçè

æ -=-=

pdpQdp

pQImax

Maximum cycle inventory (Imax):

wherep = production rate d = demand rateQ = lot size

Average cycle inventory is no longer Q/2 (as in EOQ). Instead it is Imax /2

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Non-instantaneous replenishment

Total annual cost = Annual holding cost + Annual ordering or setup cost

=>

=> Economic production lot size:

( ) ( )SQDHIC +=

2max ( ) ( )S

QDH

pdpQC +÷÷ø

öççè

æ -=2

dpp

HDSELS

-=2

I = inventoryD = annual demandp = production rate d = demand rateQ = lot sizeH = holding costS = ordering/setup cost

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EOQ vs. ELSDo NOT mix up

Economic order size

EOQ =

Economic production lot size

ELS =dpp

HDS

-2

HDS2

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EXAMPLE 8Non-instantaneous replenishment

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Solution

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Problem to solve

Skärmurklipp gjort: 2010-11-21; 13:18

A toy manufacturer uses 48000 rubber wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 8 per day. The toy trucks are assembled uniformly over the entire year. Carrying cost is €1 per wheel a year. Setup cost for a production run of wheels is €45. The firm operates 240 days per year.Determine the:a. Optimal run sizeb. Minimum total annual cost for carrying and setupc. Cycle time for the optimum run sized. Run time

D = 48000 wheels per yearS = €45H = €1 per wheel per yearp = 800 wheels per dayd = 48000 wheels per 240 days, or 200 wheels per day

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Solution

Skärmurklipp gjort: 2010-11-21; 13:18

√ 2DSH √ p

p - d √ 2x48000x451 √ 800

800 - 200= = 2400 wheelsa: ELS =

b: Cmin = Carrying cost + Setup cost = Imax

2( ) H + ( ) SD

ELS

Imax = (p – d) = (800 – 200) = 1800 wheelsELSp

2400800

Cmin = 18002( ) 1 + ( ) 45 = 900 + 900 = €1800

480002400

c: Cycle time = ELSd( ) = = 12 days

2400200

d: Run time = ELSp( ) = = 3 days

2400800

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Quantity discounts

Price incentives to purchase large quantities, create pressure to maintain a large inventory. Hence, a new approach is needed to find the best lot size - one that balances the advantages of lower prices for purchased materials and fewer orders (which are benefits of large order quantities) against the disadvantage of the increased cost of holding more inventory.

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Quantity discounts

Total annual cost =Annual holding cost + Annual ordering or setup cost + Annual cost of materials

whereQ = lot sizeH = holding costD = annual demandd = demand rateS = ordering and/or setup costP = price/unit

( ) ( ) PDSQDHQC ++=

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Quantity discountsStep 1.Beginning with lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its price. Each subsequent EOQ is smaller than the previous one, because P, and thus H, gets larger and because the larger H is in the denominator of the EOQ formula.

Step 2.If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.

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EXAMPLE 9Quantity discounts

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One period decisions

One of the dilemmas facing many retailers is how to handle seasonal goods. Often, they cannot be sold at full markup the next year because of changes in styles. This type of situation is often called the newsboy problem.

Step 1: List different demand levels and probabilities.

Step 2: Develop a payoff table that shows the profit for each purchase quantity, Q, at each assumed demand level, D.Each row represents a different order quantity and each column represents a different demand.The payoff depends on whether all units are sold at the regular profit margin which results in two possible cases.

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One period decisions

Case 1.

Case 2.

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One period decisions

Step 3: Calculate the expected payoff of each Q by using the expected value decision rule: For a specific Q, first multiply each payoff by its demand probability, and then add the products.

Step 4: Choose the order quantity Q with the highest expected payoff.

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EXAMPLE 10One period decisions

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Supply exceeds demand => Case 2

Demand exceeds supply => Case 1

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Relevant book chapters

• Chapter: Managing inventories

• Supplement C: Special inventory models

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Questions?yuji.yamamoto@mdh.se

Next lecture on Tuesday 2017-05-09

Workshop

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