Inventory Model - ROL 2011 - 2

8/12/2019 Inventory Model - ROL 2011 - 2

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INVENTORY MODELSfor

a Time VaryingDemand Pattern

=

Advance Research Operational


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Practical Situations

Multiechelon assembly operations almost always varyappreciably with time.

Production contract requires that certain quantities to bedelivered on specific dates.

Items having a seasonal demand pattern. (Artificialseasonality can be introduced by pricing or promotionactions)

Replacement parts for an item that is being phased out of

operation. Here the demand rate drops off with time. (Class C) Items with known trends in demand that are expected to

continue. Parts for preventive maintenance where the schedule is

accurately known.


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The Choice of Approaches

Lot sizing model with EOQ (MRP)* The Wagner Whitin (WW) Algorithm

Wagelmans Hoesel Kolen (WHK) Algorithm Silver Meal Heuristic Least Unit Cost Heuristic

* Explained in a certain lecture


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Wagner-Within Algorithm

Denote the fixed ordering cost and the holding cost in period tby K t and h t , respectively .

Specifically, if C t is the per-unit purchasing cost , then theanalysis requires that C t + h t C t+1 for all t.

It is not necessary for all the periods to be of equal length. d t is the known deterministic demand in period t. x t to represent the inventory at the beginning of period t

before the order-placement decision is made. For simplicity, assumed that lead time is zero.


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... y

t is the on-hand inventory after the order-placement decision is made

and the order is received, or equivalent to x t plus the order quantity. Herey t x t

The fixed cost is equal to Kif y t > x t ; otherwise it is 0 . Equivalently, the fixed cost as K t where

The purchasing cost is equal to the product of the unit purchasing cost Cand the order quantity y t x t .

Finally, the leftover inventory at the end of period t is y t d t , and thecorresponding holding cost is h(y t d t ).

Combining the three terms, the total cost in period t as

otherwise,0

,1 t t t

x y

)d h(y ) xC(y K t t t t t


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The cumulative cost over the whole horizon is

Since the order quantity is non-negative, y t must be greater than or equalto x t in any period t. To ensure that there is no backlogging, y t d t .

If Z 1(x1) be the optimal cost for the whole horizon given that the inventoryat the beginning of the horizon is x

1 , the formulation of the dynamic lot

sizing problem is as follows:

T

t t t t t t )}d h(y ) xC(y{K 1


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The cumulative variable purchasing cost is independent of thedecision variables, y 1 ,y 2 , . . . ,y T , The sum total of the order quantities overthe horizon is

Using the relationship, x t +1 = y t d t , substitute for y 1 with x 2 +d 1 , y 2 with x

3+d

2 , and, in general , y

t with x

t+1+d

t in the above expression .

After the substitution, the above expression becomes:

T

t t t x yC

1

)(


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Observe that all the terms on the right-hand side except x T+1 are constantand thus independent of the decision variables. The term x T+1 is theinventory remaining at the end of the horizon. This inventory is unutilized

and is wasted. Therefore, in any optimal policy, x T+1 must be equal to zero . This implies that the cumulative order quantity over the horizon is

independent of the decision variables. Since the total purchasing cost overthe horizon is equal to the product of this cumulative order quantity and aconstant C, the variable cost of order placement is also independent of the

decision variables. Therefore, it is excluded from the cost function. The revised formulation is

as follows:


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Solution Approach for Wagner Within Model


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... Assume that the inventory at the beginning of the horizon x 1 is equal to

zero. For example, with T = 5


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Wagner Within Algorithm Define F(t) to be the optimal cost from period 1 through period t when

inventory at the end of period t is zero. Also, define to be the minimum cost over periods 1 through t when the

inventory level at the end of period t is zero and period ts demand

is satisfied by an order placed in period s. Cst can be written as the sum of the minimum costs incurred over periods

1 through s1 and periods s through t when x s = 0 and The optimal cost over periods 1 through s1 is equal to F(s1). The cost incurred between periods s and t includes the fixed cost incurred

in period s and the holding costs incurred in periods s, s+1, . . . , t. T he holding cost in period s is proportional to the inventory at the end of

period s, which is equal to the sum of the demands in periods s+1, s+2, . . . , t. Similarly, the holding cost in period s+1 is proportional to the inventorycarried forward to period s+2 which is equal to the sum of the demands in

periods s+2, s+3, . . . , t .


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... Since inventory at the end of period t is assumed to be 0, no holding cost is

incurred in period t. Therefore, the total cost incurred in periods s through tis equal to:

(2.1)

an expression for C st by adding F(s1) to (2.1)

(2. (2.2)

if period t1s demand is optimally satisfied by an order placed in period v inthe t 1-period problem, then it is sufficient to consider periods v,v+1, . . . , tas periods in which to optimally place an order to satisfy period ts demandin a t- period problem.


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...

For each of these choices, the minimum cost over periods 1through t can be computed as:


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The Algorithm Step 1: Set t = 2, v = 1 and F(1) = K. Step 2: Since an order is placed in period 1, we need to determine

whether to place an order in period 1 or in period 2 to satisfy period 2sdemand. When the order is placed in period 2, the total cost is F(1)+K 2 =K

1+K

2 since no inventory is carried into period 2. When the order placed in

period 1 is for d 1+d 2 units, the total cost is

Choose the decision with the least total cost for periods 1 and 2. That is,

Set v = 2 if K 1+K 2 < K 1+hd 2. Otherwise, v remains unchanged


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... Step 3. Consider the t-period problem. Given v, the demand for period t is

satisfied by placing the order in one of the periods v, v + 1, v + 2, . . . , t.Compute using (2.2) and find

Step 4. Set t t +1. Stop if t = T +1. Otherwise, go to Step 3 .

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Inventory Model - ROL 2011 - 2