Loading…

cost

Melika Hashemi

Updated May 30, 2022

Transcript

logistics systems design & analysis

cost

Introduction

presentres:

Farideh Golkarami

Fatemeh Natanzian

Sedigheh Hashemi

professor:

Dr. Roghanian

production route tracing

carried (handled) from the production area to a storage area
held in this area with other items
loaded into a transportation vehicle
transported to the destination
unloaded, handled, and held for consumption at the destination

initial remarks

Holding cost:

"rent" costs and "waiting" costs

rent costs:

the rent for the space, machinery needed

to store the items in place, plus any maintenance costs

waiting costs:

the opportunity cost of the capital tied up in storage, any value lost while waiting, etc.

*the rent costs remain fixed

summary

Holding cost

"n" represents the amount of time spent by that item between the corresponding stations.
"tm" represents the transportation time

Holding cost

rent item

This is the cost of the space and facilities needed to hold the maximum accumulation

The proportionality factor will depend on the size of the items, their storage requirements, and the prevailing rents for space

If the facilities are owned (and not leased), then the purchase cost should increase roughly linearly with size

rent item

cr is the proportionality constant

rent cost/year = cr (maximum accumulation)

if the demand is constant:

rent cost/item = cr (max accumulation)/ D'= crH1

D': the production and consumption rate

waiting cost

also called inventory cost, is the cost associated with delay to the items.

waiting cost/year= ci (total wait per yea)

waiting cost/item=ci ( average wait/item)

*ci would represent the "value of time".

Transportation cost

one-origin/one-destination situation
total cost per year=sum costs of each individual shipment
Published rates increase roughly linearly with shipment size
"v" is the shipment size
"cf" is a fixed cost per shipment
"cv" is the rate at which the variable cost per shipment increases size.

Transportation cost

transportation cost

The total cost only depends on the number of shipments, regardless of what they contain and when they happen, and the total number of items shipped.
The cost per item, thus, decreases with the average shipment size

EoS arise because all the items in a shipment share the fixed cost, "cf".
the variable cost should not influence shipping decisions
"cv" depends on distance; and for problems with many origins and destinations, the distance traveled is not fixed

Relationship to Headway

The cost of transportation depends on the dispatching headway.

The transportation cost decreases with the average headway,unlike holding costs which increased with the maximum headway.

Transportation cost is independent of the specific headway.

Relationship to Distance

cf and cv depend mainly on distance
The cost for "n" shipments totaling "V" items, when the origin and destination are "d" distance units apart, can be broken up in four terms as follows.

"cs" , is the cost attributable to each trip.It is the fixed cost of stopping.
"cd" , is the cost attributable to each incremental vehicle-mile.
"c's" , represents the added cost of carrying an extra item.

Relationship to Distance

The fourth constant is the cost attributable to each incremental item-mile.
The marginal wear and tear and operating cost per mile for each extra item carried.
several destinations:

if we redefine "cs" to be the fixed cost per stop, then the cost of making "n" shipments is :

Relationship to Distance

The number of vehicle-years needed per year if the demand for vehicles is not seasonal.

The sum of overhead and driver wages is proportional to the total vehicle-time for the "n" shipments.

Other vehicle operating costs should be proportional to the total number of moving vehicle-hours

Total time and the time in motion are linear functions of the vehicle-miles traveled "nd"

Relationship to Distance

the average cost per item is obtained:

As a function of the average headway, the costs per item and per unit time are

Relationship to Distance

Irregular schedules may require slightly larger cost coefficients if the shipper exclusively uses its own private fleet.

the in-vehicle time of a typical item, tm, is also a linear function of distance, d, and number of stops, ns.

Relationship to Size; Capacity Restrictions

single origin and single destination situation

vmax , for a firm that owns its own vehicles

Whenever the shipment size reaches and exceeds a multiple of vmax a new vehicle needs to be dispatched with a resulting jump in cost.

the transportation cost per shipment function:

Relationship to Size; Capacity Restrictions

consider the linear part of "ft" between 0 and vmax , as shipments larger than vmax are not

economical.

We are assuming here that headway are regular

Relationship to Size; Capacity Restrictions

The optimal shipment size is the value of v for which the vertical separation between the two curves of Fig. 2.4 is minimum

one can ignore the behavior of the transportation curve for v > vmax

If one remembers to abide by the constraint v=<vmax

problem: solution: optimal shipment size

where

Relationship to Size; Capacity Restrictions

"lot size" or "economic order quantity (EOQ)" model of the inventory control

Relationship to Size: Multiple Transportation Modes

This qualification was made because if shipment size varies by a large amount, it may be cost effective to change transportation modes.

some shipping modes, such as mail, exhibit a low fixed cost per shipment and a high cost per item, others may be the opposite.

the best mode depends on the shipment size; as it grows, one tends to favor the modes with lower variable cost and higher fixed cost.

Relationship to Size: Multiple Transportation Modes

Fig. 2.5 displays the transportation cost that results if one ships everything by the cheapest mode – the lower envelope of the three cost curves.

The lower envelope is optimal.

If the individual modal component curves are merely subadditive, e.g., they exhibit jumps as in Fig. 2.3, then the lower envelope is not necessarily optimal, or subadditive.

Relationship to Size: Multiple Transportation Modes

Relationship to Size:

Multiple Transportation Modes

shippers do not change the vehicle fleet often

the appropriate (linear) component curve should be used to evaluate transportation cost

Relationship to Size: Multiple Transportation Modes

Handling costs

Handling

cost

Include loading individual items onto a "container", moving the container to the transportation vehicle threshold, and reversing these operations at the destination.
We examine here the cost of handling a shipment of size, "v".
If the items are handled individually, the handling cost per shipment should be proportional to v , so that.

If the items are small, it is not economical to move them individually "handling vehicles"

Handling costs

the handling cost should have a similar form as the transportation function since items are being transported within a compound.
If the batch is smaller than one pallet the cost of handling it should therefore be:

The constant "c'f" represents the (fixed) cost of moving the pallet regardless of what it contains, including the forklift driver's wages, plus the forklift's depreciation and operating cost.
The constant "c'v" captures the cost, accounting for both labor and capital, of loading one item on the pallet

Handling costs

Handling cost

If "v" is larger than the maximum number of items that fit on a pallet, "v'max", then the handling cost function per shipment, "fh(v)" will still be a scaled down version of the transportation function.

we could compare the cost of moving items individually and moving them in pallets. But if more than one item fits on a pallet, it will usually be cheaper to move them in pallets

Handling cost

Motion cost

depicts the sum of transportation plus handling costs for v'max << vmax . The function, fm = ft + fh , is still subadditive and increasing.

Motion cost

Note that to within an error of c'f , the motion cost per shipment, fm(v) , can be approximated by line PQ of the figure, which is a lower bound.

If v < v'max it is better to use: c"f = cf + c'f and c"v = cv + c'v
The variable motion cost per item, c"v .
The variable transportation and handling costs per item, cv and c'v
It includes each item's prorated share of the fixed cost per pallet, c'f /v'max

The Lot Size Trade-Off with Handling Costs

Note from the figure that if the waiting cost curve is pushed upwards, the first point of contact is either v < v'max (if the waiting cost curve is very steep), or else it is likely to be an integer multiple of v'max .

The lower bound from Eq. 2.7 is exact when v is an integer multiple of v'max , one could use it instead of the exact (scalloped) curve while restricting "v" to be a multiple of v'max.

The Lot Size Trade-Off with Handling Costs

we saw already that variable costs do not influence the optimal shipment size.

if shipment size is restricted to be an integer multiple of v'max , the optimal shipment size is independent of handling costs.

If the optimal shipment size, v* , is greater than one pallet,that allowing "v" to differ from a multiple of a pallet cannot improve things appreciably.

In the most favorable case the cost savings can be shown to be about one tenth of cf/v'max , with much smaller savings in other cases.

The Lot Size Trade-Off with Handling Costs

If the optimal shipment size, v* , is greater than one pallet, we see from Fig. 2.8 that allowing "v" to differ from a multiple of a pallet cannot improve things appreciably.

If, on the other hand, v* is smaller than one pallet, then handling costs should be considered.

If c'f >> cf , then the optimal shipment size may be noticeably larger than if handling costs had been ignored.

The Lot Size Trade-Off with Handling Costs

If economic shipment sizes are likely to be larger than a pallet:

ignore handling costs in the decision;

If shipment sizes are smaller than a pallet:

then include the fixed cost of handling a pallet as part of the fixed cost per shipment and select the shipment size which is the minimum of problem "EOQ" with A = ch/D' and B = c"f .

The Lot Size Trade-Off with Handling Costs

the cost of moving and filling boxes can be ignored if the optimal shipment size is larger than a box; otherwise, the fixed cost per shipment, c"f , should include the fixed cost of moving one box including opening and closing it, but not the cost of filling it. With a properly defined c"f , the optimal shipment size should still follow from the solution of the "EOQ" problem:

where:

Stochastic effect

The production and demand rates

(and the travel time perhaps as well) may vary over time in a predictable manner, and also unpredictably.

Unpredictable variations require additional inventories, and may also increase transportation cost.

the destination requests deliveries so that its inventory level can sustain at all times the demand that is anticipated.

"safety stock", reorder "trigger points"

Stochastic

effect

Stochastic Effects Using Public Carriers

The demand and travel time uncertainty should influence

neither the frequency of dispatching nor the average lot size.

A common ordering strategy uses a trigger point v0:

whenever the inventory on hand plus the number of items on back order equals v0 , a shipment of size v is requested. The reorder headway for this strategy vary because the demand varies, but the shipment sizes remain constant.

Stochastic Effects Using Public Carriers

demand arrival process can be approximated by a diffusion process with rate D' (items per unit time) and index of dispersion (items)

lead time, (the time between order placement and receiving) has mean and standard deviation . (The lead time should be close to the average transportation time, tm , if the origin can keep up with the requests)

Stochastic Effects Using Public Carriers

If one desires to avoid stock-outs, the trigger point, v0 , should be large

enough to ensure that no stock-out occurs immediately before the arrival of an order.

(i) ordered, (ii) received, and (iii) consumed at the destination.

This condition can be expressed probabilistically if we recognize that,

conditional on the lead time, T , PQ is normal with mean D'T and variance

D'T . The unconditional first two moments of PQ are thus:

Stochastic Effects Using Public Carriers

If the trigger point, v0 , is chosen several standard deviations greater than stock-outs will be rare. The precise value of v0 is not important for our analysis (it is a function of D', , , and nothing else); what is important is that, as the figure clearly indicates, the contribution of v toward the maximum and average accumulation is insensitive to v0

Stochastic Effects Using Public Carriers

In order to choose the optimal v , the motion and inventory costs must be balanced.

the motion costs with and without stochastic effects are the same because the same number of shipments are sent in both cases (D/v shipments per unit time), but the holding costs are larger with stochastic phenomena. Th maximum number

of items present at the destination will certainly occur after the arrival of an order. As shown in the figure, for a typical order, this number is:

which is largest when PQ is as small as possible:

Stochastic Effects Using Public Carriers

except for an additive constant, the holding costs are as in the deterministic case; the optimal shipment size remains the same.

Stochastic Effects Using Public Carriers

The irregular way in which orders are placed will undoubtedly raise inventory and production costs at the origin.

It should not influence shipping decisions.

inventories at the origin would then tend to grow with time

A simple strategy would stop production whenever the inventory at the origin (after a shipment) reaches a critical value, v1 , and would resume it (also after a shipment) when the inventory dips below another value, v2

Stochastic Effects Using Public Carriers

On a scale large compared with v , this curve shares the statistical properties of the demand curve which do not depend on v. Therefore, the optimal production decisions (i.e., the choices of v1, v2 , and D'p) do not depend on v.

the inventory (maximum and average) can be decomposed into a portion that is proportional to v (represented by the shaded area in Fig. 2.10) and independent of the production strategy.

Thus, the extra production and inventory costs arising at both the origin and the destination due to the unpredictability of demand are largely independent of v . They can be ignored when determining the optimal shipment size.

Stochastic Effects Using Two Shipping Modes

so far that stock-outs are avoided by holding inventories large enough to absorb fluctuations in demand and in the transportation lead time.

In some instances, if a second, much more expensive, shipping mode is available for expediting shipments, the total costs may be reduced by expediting small shipments at critical times. In these instances the optimal lot size v is also the result of an EOQ trade-off, although the trigger point decision is no longer independent of the shipment size decision.

Stochastic Effects Using Two Shipping Modes

Most of the time the expedite mode lies in wait, and the system operates as if the primary mode was the only mode (see Fig. 2.9). The trigger point v0 , however, does not have to be chosen as conservatively as before, because when a stock-out is imminent enough items can be sent by the premium mode to avoid it.

If, as is commonly the case, the time between reorders is large compared with the primary mode's lead time (i.e., so that when the trigger point, v0 , is reached there aren't any unfilled orders) then the probability that some items have to be expedited in the time between ordering and receiving a lot (of size v ) does not depend on v.

Stochastic Effects Using Two Shipping Modes

Assuming that the cost per item expedited is a constant, ce , we find that the expected expediting cost per regular shipment is: cef(v0) . The moving cost per item is as a result:

Stochastic Effects Using Two Shipping Modes

The maximum inventory still occurs when PQ is as small as possible, and remains: v + v0 - PQ ; the total cost per item is thus

Stochastic Effects Using Two Shipping Modes

For a given v0 , if we think of the expected amount shipped by both modes with every regular shipment, v1 = [v + f(v0)] as the "lot size," the equation is still of the EOQ form (2.8a), where the fixed moving cost has been increased to include the expected cost of expediting, (ce - cv)f(v0):

Stochastic Effects Using Two Shipping Modes

Unlike in the previous case, though, the trigger point v0 should not be chosen independently of v. If v is large so that shipments are infrequent, expediting a significant amount of freight with the average shipment only increases the moving costs marginally. But if v is small, the penalty for expediting is paid more often; it may be more efficient to increase v0