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Inventory Models


Therefore, in the model inputs we report, h, the annual cost per dollar value of ... In Model 2. the firm designs an optimal order per cycle as well as optimal ... – PowerPoint PPT presentation

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Title: Inventory Models

Inventory Models
  • 3 Models with Deterministic Demand
  • Model 1 Regular EOQ Model
  • Model 2 EOQ with Backorder
  • Model 3 EPQ with Production
  • 2 Models with Stochastic Demand
  • Model 5 Models with Discrete Probability
  • Model 6 Model with Continuous Probability

Deterministic ModelsModel 1 The EOQ Model
  • In the deterministic models under study, the firm
    faces a known linear demand.
  • As an example of deterministic models, we will
    demonstrate in detail Model 1, the basic EOQ
    model, In this model the firm orders the product.
    The goal of the firm is to determine Q, the
    Economic Order Quantity that minimizes the firms
    total cost.

  • Determining the economic order quantity uniquely
    determines the cycles length T.
  • The determination depends on the relative cost
    of making an order relative to the cost of
    holding item in inventory

Types of costs
  • The firm has three types of costs
  • Procurement Cost
  • Ordering Cost
  • Holding cost

Inputs of the EOQ Model
  • A Annual number of items demanded (Annual
  • k Fixed cost per order
  • c Unit cost of procuring an item
  • h Annual cost per dollar value of . holding
    items in inventory.

Planning Period and holding items dollars in
  • Costs are computed for a planning period.
  • In the textbook, this planning period is a year.
  • Length of use Cycle ,T, (Time between orders) is
    then specified in yearly terms. Conversion to
    days would require multiplying the result by the
    number of days in a year.

Holding Cost
  • It is important to notice, the primary component
    in the holding cost is the cost of holding the
    monetary value of the item in inventory.
    Typically this cost accounts for about 80 of the
    holding cost.
  • The holding cost of the monetary value held in
    inventory is evaluated based on the opportunity
    cost for investment of the value held in
    inventory or based on the cost of borrowed funds
    needed to hold the item.
  • For example, a dollar invested may yield 10
    return on investment or may be required borrowed
    fund at 10 interest cost. In these examples,
    the cost is .10 per dollar.

  • Other costs associated with holding are also
    therefore prorated per dollar value.
  • Therefore, in the model inputs we report, h, the
    annual cost per dollar value of holding the item
    in inventory. Hence, the corresponding holding
    cost for the item is hc.

Cost Computation is easy.
  • For every Cost Type the cost is,
  • Cost per unit of measurement times the Number of

Units of Measurement and Number of Units for Type
of Cost
  • Type of cost UOM
    Number of Units
  • Procurement One procured item
    Annual demand
  • Holding One item held
  • in inventory for a year
    Average Inventory
  • Ordering One made order
    Number of orders in a

Costs Per Unit
  • Type of Cost Cost Per Unit
  • Procurment c
  • Holing hc
  • Ordering k

Annual Number of Units per Type of Cost
  • Cost Type Number of Units Formula
  • Procurement Annual Demand A
  • Holding Average Number of Q/2
    units in Inventory
  • Ordering Annual number of A/Q

Some Outputs
  • The ordered (and use) Quantity per cycle,
  • Number of Orders, A/Q
  • Length of order and use Cycle, Q/A
  • Maximum Inventory level, Q
  • Average Inventory level, Q/2
  • Annual Holding cost per item, hc

Total Cost
  • Total annual Cost is the sum of procurement cost
    ordering costs and holding cost.
  • cA k(A/Q) hc(Q/2).
  • Since, the procurement cost cA is fixed, and does
    not affect optimization. The Relevant Cost for
    decision is then,
  • TC(Q) k(A/Q) hc(Q/2).

Model 2 Inventory with Backorder.
  • In Model 2. the firm designs an optimal order
    per cycle as well as optimal waiting list.
  • As soon as the order of size Q arrives, the firm
    supplies the waiting lists demand. The stock,
    S, of the remaining units, is left as inventory
    to serve next customers not on the waiting list.
  • The stock S is depleted till it disappears. The
    firm is then beginning to collect orders on a
    waiting list, till the optimal level of the
    waiting list, Q-S, is reached. At which point a
    new order of size Q arrives.

Model 2, Inventory with Backorder continues
  • Since customers do not like to wait, there is a
    shortage penalty per item on the waiting list, p.
  • The penalty p may either be given directly, or
    imputed from a service level, L, where L is the
    proportion of demand met on time or alternatively
    the probability of providing an item from
    inventory (rather than from the waiting list).
    It is straight forward to impute p from L. (pp.
  • The total relevant cost equals the ordering cost
    holding cost shortage cost.

Model 3 Inventory with Production
  • In this model the firm produces the item for its
    own use. The firm wishes to optimize the
    quantity to be produced and used in every cycle,
    Q and correspondingly the optimal length of the
    production and use cycles. (T1 and T
  • First the firm sets up the machines, produces the
    product and uses it. (production rate B must be
    greater than use rate A.). Since the production
    is on-going, inventory is gradually built up to
    an optimal maximal level.
  • Once the maximum inventory level is reached the
    firm stops production. For the rest of the use
    cycle, the firm only uses the already produced
    inventory in stock. The process is repeated.

Model 3, Inventory with Production, Continued
  • In this model set up costs replaces ordering
    cost. Mathematically, it has the same formula,
  • The quantity Q stands for the quantity to use and
    produce in the cycle.
  • Average inventory is half the maximum inventory,
    as in Model 1. However it is lesser than half of
    Q, the quantity used in a cycle, as the inventory
    is only gradually built up..
  • Total Relevant Cost equals Set Up Cost Holding

An interesting feature of the Deterministic Models
  • In the Regular EOQ Model, at optimum,
  • Ordering Cost Holding Cost
  • In other models, similar observation is made
  • In the Inventory with Back Order Case
  • Ordering Cost Holding Cost Shortage Cost
  • In the inventory with Production case,
  • Set Up Cost Holding Cost

Models with Stochastic Demand The Newsvendor
  • In this case we are dealing with stochastic
    demand and given short cycles.
  • Unlike the deterministic models of chapter 15 we
    are not determining the cycles length There is
    some given cycle and we are provided with the
    holding cost of left over item at the end of the

Properties of the Newsvendor Problem Continued.
  • The firm faces possibilities of either shortage
    and surplus penalties
  • We wish to find Q, the optimal order quantity to
    be stocked at the beginning of a cycle, expected
    surplus and surplus cost, and expected shortage
    and Shortage costs, and total expected costs.

Inputs for Model 4 An Inventory Model with
Discrete Demand Distribution
  • C Procurement Cost
  • hE Additional cost of each item held at the end
    of the inventory cycle.
  • pS Penalty for each item short (loss of
  • pR Selling Price
  • Discrete Distribution of the quantity demand for
    the item in a cycle.

Per Unit Penalties for left over Item and for
  • The penalty for a left over item hE C, the
    surplus penalty per item, is the item cost c and
    the residual cost associated with the left over
    item. If the left over item has some residual
    value hE will be negative and would reduce the
    surplus penalty per item.
  • Shortage penalty per item, pS (pR C) . It has
    two components. The first is the good will loss,
    pS. The second is the opportunity loss
    encountered. If the firm had the item it could
    have made a profit pR C by selling it.

The Critical Share and Finding the optimal Order
  • The critical share is the share of the per item
    shortage penalty in the sum of the per units
    shortage and surplus penalties,

The critical share formula
  • The critical share is,
  • pS (pR c)
  • ____________________________
  • pS (pR c) (hEc)

The critical share and optimal Q
  • Marginal considerations, based on calculus led to
    the result that the optimal quantity is to be
    this where the cumulative probability of the
    demand equal the critical share.
  • In the discrete case we look at the level of the
    cumulative distribution just surpass the critical
    share. Mean demand for a cycle is labeled µ.

Total Expected Costs
  • Total expected cost is equal to the expected
    procurement cost the expected shortage cost
    the expected surplus cost.
  • Expected shortage cost equals the penalty per
    unit of shortage times the expected shortage,
  • Expected surplus cost equals the penalty per unit
    of surplus times the expected surplus.

Model 5. Inventory with Normally Distributed
  • In this case the mean and standard deviation of
    the demand are known and the determination of the
    ordered quantity, Q, based on the critical share
    may be very precisely determined. The rest of
    the inputs are the same as in Model 4.
  • The following computations of the expected
    shortage and surplus using the Normal Loss
    Function L(x), and the computations of the
    associated expected costs depend on the whether
    Q is smaller or larger than the mean demand µ.
    See page 630.