Multi-Item Joint ReplenishmentMulti-Item Joint Replenishment

ADVANTAGES OF JOINT ADVANTAGES OF JOINT REPLENISHMENTREPLENISHMENT

• Savings on Unit Purchase Costs

If many items are bought from the same vendor, a quantity discount if the total order is > breakpoint qty is realized

When a order is placed to a vendor, certain fixed cost is incurred but adding a line item to order incurs a smaller fixed cost

Vendor - imposed minimum qty like 5000, 10000 units etc

It doesn’t make sense to buy a single item to avail discount, so it is necessary to coordinate several items

ADVANTAGES OF ADVANTAGES OF COORDINATIONCOORDINATION

• Savings on Unit Transportation Costs

Grouping items to achieve a carload or container load on a ship

Pool the items till the container is full to avail the discount on full container load

• Savings on Ordering Costs

In case fixed cost of ordering is high, then it makes sense to order several of these at once

In case of coordinated production, mfg set-up cost is high

ADVANTAGES OF COORDINATIONADVANTAGES OF COORDINATION

• Ease of Scheduling

Scheduling of buyer time, receiving and inspection and etc would be easy

AssumptionsAssumptions

Demand rate of each item is constant and deterministic

Replenishment qty need not be an integer

The unit variable cost doesn’t depend on qty (no discounts)

Replenishment lead time is zero

No shortages allowed

The entire order quantity is delivered at the same time

Few NotationsFew NotationsS = major Setup cost

si = minor setup cost i.e. Item dependent marginal cost of placing an order

A$ = annual rupee value of all items in the group ordered

A$i = annual rupee value of item i in the group ordered

Ci = unit cost of an item i

Di = annual demand of item i in number of units

Q$ = total rupee value of all items ordered during a cycle

Q$i = rupee value of item i ordered during a cycle

Total Relevant CostTotal Relevant CostNumber of orders N = D/Q = A$ / Q$ = a$ / Q$i

Total Relevant Costs = Ordering cost + Inventory carrying cost

Ordering cost involves a major setup ‘S’ (ordering) cost + si minor setup ‘si’ (ordering) cost for any item i. So, Total cost of placing an order for a group of items will be [S + ∑si] i = 1....n.

Inventory carrying cost would be I*(Q$/2)

So, TRC = Ordering cost + Inventory Carrying cost = N. [S + ∑si] + I*(Q$/2) = A$/Q$ * [S + ∑si] + I*(Q$/2)

Differentiate TRC w.r.t Qv and equate it with ‘0’

Total Relevant CostTotal Relevant Cost

Q$ = SQRT[(2.[S + ∑si].A$)/I]

Q$i = Q$ * (a$i / A$)

Qi = Q$i / Ci

Iowa Abe Sporting Goods Company order five different tennis racquets from a major distributor. The annual demand, cost and other data are shown in the table below. The major setup (order) cost with group of items is Rs.75 per order and the annual inventory carrying % ‘r’ is 15% of the cost of items. Find EOQ in rupee value ‘Qivi’ and units of all items ‘Q’.

Total Relevant CostTotal Relevant Cost

Item No.

Annual Demand Dv

Unit Cost

vsi

Order Size

Rupees (Qivi)

Qi

1 5000 5 52 4000 8 53 10000 10 84 18000 12 85 1000 20 10

Total 38000 36

Find Number of orders per year N, Time between orders ‘T’, Annual Cost of Ordering (Setup) and Average Annual Inventory Carrying Costs.

The Decision RuleThe Decision RuleTEOQ = √2S/hD = √2S/h*A$

S/A$ - An item with high S and low A$ value will have high TEOQ meaning less no. of replenishments than an item with low S and high A$ (in which case TEOQ is small)

Inventory policy is to consider the use of time interval (T) between replenishments of a family and a set of mi’s, the number of T intervals that replenishment quantity of item i will last

M15 = 4 means that 15th item should be included every 4th replenishment of the family, with a replenishment quantity sufficient to last a time interval of duration 4T

BROWN’S ALGORITHMBROWN’S ALGORITHM

Calculate T

Then calculate ‘mi’ = 1/T * SQRT [(2 * si)/(I * a$i)]

•Then calculate revised value of T using:

T = SQRT [ (2 * [S + (∑si/mi)] / (I * ∑mi.a$i)]

Q$i = mi . a$i. T

SILVER’S ALGORITHMSILVER’S ALGORITHM

Find minimum [si/a$i] and let that be m1 = 1 and let this be item j

Then calculate ‘mi’ = SQRT [(si/a$i) * (a$j / (S + sj))]

•Then calculate revised value of T using:

T = SQRT [ (2 * [S + (∑si/n)] / (I * ∑mi . a$i)]

Q$i = mi . a$i . T