4.8 Other Extensions
4.8.2 Interactions With Production
Another area where further results may be desirable involves the interac-tion of physical distribuinterac-tion with producinterac-tion schedules. This interacinterac-tion sometimes offers an opportunity for further cost reductions.
This subject was broached in Sec. 4.3.3, where it was suggested that production of (destination-specific) items should be rotated among geo-graphical customer regions every headway H. Dispatching the vehicles to a region immediately after its production run was completed greatly reduced the holding costs at the origin. It was assumed that production would be coordinated with transportation in this manner without much of a penalty.
More likely, though, there may be a set-up cost associated with each switch in production item types. In this case production costs may be re-duced by switching less frequently and holding higher inventories at the origin. An integrated solution can then be obtained by including in the lo-gistic cost function the production set-up costs, e.g., as explained below.
If no attempt is made to coordinate the production schedule with the physical distribution schedule, then the inventory at the origin of items of a certain type can be decomposed as shown in Figure 4.9 into a (shaded) component which depends on the time between setups for that item type,7 Hs , and a (dotted) component which depends on the transportation head-way, H; see Blumenfeld et al. (1985a):
c H.
+ c H origin
at item per
cost inventory
average i
i s
2
| 2
¸¹
¨ ·
©
§
7We are assuming that the number of item types is large and, therefore, the steps of the production curve are nearly vertical. Similar conclusions can be reached for few item types.
Fig. 4.9 Inventory accumulation when no attempt is made to coordinate pro-duction and distribution
The maximum accumulation also decomposes in a similar manner:
HD.
+ D H on accumulati
Maximum
|
sc
Because production costs depend on Hs and not on H , the sum of the pro-duction and logistics costs is made up of two components: (i) a propro-duction component with only production decision variables (including Hs ), and (ii) a logistic component with only logistics variables (including A and H).
Logistics and production decisions, thus, can be made independently of each other.
By selecting H to be an integer submultiple of Hs , or vice versa, it is possible to reduce the inventory time at the origin by an amount equal to the smallest of H and Hs (Figure 4.10 depicts the case with Hs = 3H ), and the maximum accumulation becomes the difference between the maximum and the minimum of HsD1 and HD1 .
152 One-to-Many Distribution
If this kind of coordination is feasible, the sum of the production and logis-tics costs no longer decomposes, and a coordinated production and distri-bution scheme should be considered. Blumenfeld et. al. (1985a) and (1986) have examined the case where each district is constrained to contain only one destination and all shipments are direct (ns = 1). They illustrated situa-tions where coordination of production and distribution is most conducive to cost savings, and provided a bound on the maximum possible benefit.
Further research may be worthwhile to relax the ns = 1 assumption and to allow more destinations than item types.
Fig. 4.10 Inventory accumulation with coordinated schedules
Throughout the chapter it was assumed that the total production rate, and not just the schedule by item type, could be adapted to the changing de-mand without penalty. In practice, though, this is rarely so, even if the items produced are generic. (It is more costly to change the quantity of items produced than the kind of items produced because to adjust the pro-duction rate one needs to hire extra labor, pay overtime or fire labor as needed – and the penalty for these actions is large; Newell, 1990, has
ex-amined the production rate adjustment process.) We conclude this chapter by showing that this seemingly strong assumption can often be relaxed.
Figure 4.11 shows how a production curve may be adapted to a gradu-ally decreasing demand; the objective is tracking the smooth envelope to the crests of the shipment curve (which varies like the demand curve) as closely as possible, without many production rate changes. We had seen in Sec. 2.5 that for a similar model, portrayed in Fig. 2.10, lot size decisions were independent of production decisions; fortunately, this is also true now. In Fig. 4.11, the inventory at the origin decomposes in two compo-nents: (i) a (shaded) component, which is due to the discreteness in the production rate changes and is independent of the shipping schedule,8 and (ii) a dotted component which is the same as if the production schedule was adjusted continuously as assumed in this chapter. Thus, costs can be divided into two components affected respectively only by production, or only by logistics decision variables.
Fig. 4.11 Production for a gradually decreasing demand
8It is the same as if the production curve was driven by the demand curve itself.
154 One-to-Many Distribution
Suggested Exercises
4.1 The maximum number of stops made by a vehicle delivering lots of size v ( v < vmax ) to identical customers is [vmax/v]- if a delivery lot cannot be split among vehicles. If vmax/v is an integer, an alternative way of expressing Eq. (4.5a) is:
^
E r v/v +kE`
N.distance
Total # 2 max
G
1/2If vmax/v is not an integer this expression is a lower bound. The bound is very tight if delivery lots can be split among vehicles in or-der to fill them. Hall (1993) explores in detail the shipment splitting issue.
(i) Show that this expression also applies if v > vmax and all the trucks are dispatched full (splitting delivery lots as needed), provided that a condition (equivalent to N >> C2) holds. Write the condition. Show as well that the expression is a (tight) lower bound if delivery lots smaller than vmax are not split.
(ii) Derive as well the generalization of Eq. (4.6b), when the condi-tion does not hold. Show that the distance expression is close to the above.
4.2 Section 4.4.1 describes how to design a one-to-many distribution system when the maximum number of stops per tour cannot exceed Cmax(x) , a quantity that depends on location. Develop an expression for Cmax(x) when, aside from vehicle capacity, the only restriction to number of stops is the maximum time allowed for a vehicle tour, tmax. (Assume that the vehicle's average moving speed and time per stop are known.)
4.3 If Eq. (4.21) is a good representation of the optimal cost for all (t, x), explain the logic behind the following expression for the average cost over R , for the time horizon (we assume that cs 0 ):
1/2
1 1/4
max
' 2
2
E
G
D E c kc +
/v c r
= item per cost average
d h
d
If D' and į do not change much across items – i.e. their standard de-viation to mean ratios (denoted here by İ and İ' ) are small compared
with one – show that:
average cost per item 2E(r)cd/vmax +
[ 2(kchcd)2tmax/{D(tmax) [N/ R ]3} ] [ 1 + İ2 + İ'2 ]
This expression reduces to the above if İ = İ' = 0 , and like the above it increases with the İ's .
[Hint: Use a well-known approximation to the expectation of a non-linear function of a random variable, based on a two-term Taylor se-ries expansion of the function.]
4.4 Assume that the demand for items by every customer in R follows a compound Poisson process with an inex of dispersion Ȗ; recall that the average number of events occurring per unit time is D'c = D'/Ȗ . Then explain why the proportion of all the customers in a subregion of R with no demand in time H is exp[-D'cH] . (Note that the con-stant H0 defined in the text is the inverse of D'c ; as the average time between successive events, it is: H0 = Ȗ/D' = Ȗį/Ȝ .)
4.5 Assume that service is to be provided to 400 customers uniformly scattered in a (20 mi. x 20 mi.) square region, with the depot in a corner. The following numerical constants describe the situation:
Ȝ/vmax = 0.1 (truckloads/day-mile2); Į1/r = 2 ($/ vehicle-mile) ; Į2 = 21 ($/customer visit) ; kcd = 1 ($/veh-mile) ; cs = 20 ( $/vehicle stop); ch vmax = 30 ( $ / truckload-day ) ; Ȗ = 1 (truckload of items).
Solve the minimization problem defined by Eqs. (4.25 and 4.26), es-timate the upper and lower bounds to total cost, and define a distri-bution strategy for the primary and secondary tours.
4.6 Items are to be carried from a depot to many scattered destinations.
Two modes of transportation are available: mode A is fast sA = and expensive cdA >> cdB . Mode B is slower: sB < . These features aside, the rest of the features are the same for both modes. They share the cost per stop, vehicle capacity and loading/handling cost.
Storage space at the origin and the destinations is plentiful, so rent costs are neglected, but inventory cost changes drastically across destinations. This is described by a cumulative density function which gives the density, į(ci) , for all the destinations with carrying
156 One-to-Many Distribution
cost below ci. (For any ci these are assumed to be uniformly and randomly scattered about the service region, R.) If the demand is stationary and the same at all destinations, describe qualitatively a procedure for allocating the destinations to modes and the optimal service characteristics ( A and H ) of each mode. Assume that all the customers allocated to the mode are served in every headway, and that the pipeline inventory cost cannot be neglected.
[Hint: Prove that if destination nA goes on mode A and nB goes on node B, then the inventory cost of nA must be greater than the inven-tory cost of nB . Then find the minimum cost for both transportation modes as a function of a critical ˜ci ; iff ci < ˜ci , then mode B is used.]
4.7 Derive a closed form solution for the minimum of Eq. (4.29b) sub-ject to (4.29c). Assume without loss of generality that the n are ar-ranged in order of increasing Gn [ where Gn = (D'n) (ch(n)) ] , and that the units of measurement are such that į = 1000 , and cs = 1 .
Then, if the fraction of customers with Gn below x , P(x) , is:
P(x) = 0 if x 1
= a(x-1) if 1 < x < [ 1 + a-1 ]
= 1 if x [ 1 + a-1 ]
solve for the minimum f0 and H . Do this for various values of "a"
and describe the result.
4.8 An 80 story office building is served by 20 elevators. Whether full or empty, elevators travel at a speed of one floor per 0.5 seconds.
Each time they stop at a floor, their travel time is increased by ǻt seconds:
, n +
= 't 10
where n is the number of passengers exiting or entering the elevator at that particular floor.
In the morning rush hour all the traffic originates at the lobby, at a constant rate of two passengers per minute per floor.
1. Arrange the elevators in banks so that the average passenger waiting plus riding time is minimized. (A bank is a set of "m"
elevators serving the lobby and "b" contiguous floors.) Ignore pairing and assume that the arrival rate is maintained for a long time. Assume as well that an elevator can hold as many people as needed.
2. For part 1, it was reasonable to expect every elevator to stop at every floor. During the off-peak, however, (with demand rate Ȝ
<< 2 pax./min.) elevators may skip floors. Discuss how the opti-mal banking strategy is affected by Ȝ . As an aid for thinking, you may use the following approximate expression for the total ser-vice time, T(x) , for a passenger going to floor x (a floor that is included in a bank with b floors and is served by m elevators):
where S is the elevator speed in floors/sec., and R is the elevator round trip time for that bank,
with k = 5(1 + b) + [x + (b-1)/2]/S .
This expression assumes that ǻt = 10 , regardless of n. It can be included in a spreadsheet for sensitivity analysis. Try Ȝ = 0.1 , 0.3 , and 1.0 pax/min. Note the influence that elevator speed has on the homogeneity of your bank configuration.
3. As a prelude to the material in Chapter 5, discuss how the use of skylobbies could improve the level of service and reduce the amount of floor space consumed by elevator shafts.
[Hint for part 1: The number of elevators per floor should be about the same for each bank. Note that this is an engineering de-sign problem; if you cannot identify the optimum analytically, you must still propose the best design possible. Use of a com-puter spreadsheet is recommended.]
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secs., mb
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158 One-to-Many Distribution
Glossary of Symbols
Į0: Handling and fixed pipeline inventory cost ($/item), Į1: Fixed cost per vehicle dispatch ($/dispatch), Į2: Transportation cost added by a customer detour ($),
Į3: Pipeline inventory cost added by a customer detour ($/item), Į4: Stationary holding cost for one item during the time between
de-mands ($/item5), A, A(x),A(t,x): Area of A,
a: Number of aisles with a request,
An: Area of delivery district used for type-n customers, A: Subregion of R ; delivery district,
ß: Characteristic constant used in Sec 4.4.2, ß1 , ß2 , ß2(n): Constants used in Sec 4.7.1,
cd: Cost per vehicle-"mile",
cf: Terminal handling cost constant, ch: Holding cost per item-day,
ch(n): Holding cost per item-day for type-n items, ci: Inventory cost per item-day,
cr: Rent cost per item-day,
cs: Fixed transportation cost of a vehicle stop,
c's: Added transportation cost of carrying an extra item, cr: Rent cost per item-day,
ct: Vehicle operating cost,
C, Cmax: Maximum number of stops made by a vehicle, Cmax(x): Maximum number of stops in the neighborhood of x, Cp: Maximum number of stops in the pth subregion, į, į(x), į(t,x): Spatial customer density (customers/area),
įn: Spatial density of type-n customers (customers/area), d: Tour length,
¯
D ': Time averaged demand rate for a customer, D', D'(t): Demand rate per customer (at time t),
Dmp: Demand in subregion "p" during time interval "m" , Dn(t), D(t): Cumulative demand of customer "n" by time "t", İ: A mathematical ratio in the full vehicle condition proof, E( ): Expectation of a random variable,
ĭ( ): Standard normal cumulative distribution function, f(x): Probability density (for customer location) at x, f0: Fraction of items collected as overflow,
Ȗ, Ȗ(t,x): [Variance/Mean] of items demanded in a time interval (index of dispersion),
g0: Probability of overflow,
H, H(t), H(t,x): Headway,
H0: Time constant (in Sec. 4.6.1 only), H5: 5th headway,
Hn: Headway for the nth customer class, Hs: Time between production setups,
k: Dimensionless factor for the VRP local distance; (more vehicle tours than stops per tour),
k': Dimensionless factor for the VRP local distance; (fewer vehicle tours than stops per tour),
Ȝ , Ȝ(t,x): Demand density rate (items/time-area),
Ȝn: Demand density rate for type-n customers (items/time-area), 5 = (1,...,L): Indexes for dispatching times and headways,
Lp: Number of dispatches for the pth subregion, m = (1, ...,M): Indexes for time intervals,
n = (1,...,N): Indexes for customers, destinations, and customer classes, ns, ns(t,x): Number of stops per tour , at (t,x),
ns*: Optimal number of stops per tour, ns ,i: Number of stops per tour, for tours near xi,
ns5 (and ns5(x)): Number of stops per tour (near x) for the 5th dispatch, Np: Number of destinations in the pth subregion,
p = (1,...,P): Indexes for the subregions of R, Pp : pth subregion in a partition of R,
r: Average distance from the points in a delivery region to the depot, r(x) , (or ri): Distance from the depot to point x , (or xi),
rp: Average distance from the pth subregion to the depot, R : Service region,
ŇRŇ: Surface area of R, s : Vehicle speed,
IJ(x): Time to cover a unit area around x. (Sec. 4.5.1 only), IJm: The mth time interval in a partition of the study period, t : Time,
t5: Time of the 5th dispatch,
tm: Average time spent in a vehicle (per item), tmax: End of the study period,
ts: Time duration of a vehicle stop,
T(x): Express time to point x. (Sec. 4.5.1 only),
v (and v5): Delivery lot size to a customer (for the 5th shipment), vmax: Vehicle capacity (items),
vo , vo(x): Maximum allowable accumulation at a destination (items), x = (x1,x2): Spatial coordinates of a point,
xo: Inner point of A, z: Cost per item, z*: Optimal cost per item,
zh and z5: Upper and lower bounds to z*. (Sec. 4.4.2),
zmp: Cost per item in subregion "p" for time interval "m", zm: Motion cost per item,
160 One-to-Many Distribution
zp: Pipeline inventory cost per item, zs: Stationary inventory cost per item,
[ ]+: Closest integer from above to the argument in brackets, [ ]-: Integer part of the argument in brackets.
Transshipments
Readings for Chapter 5
Designing a one-to-many logistics system with transshipments is a com-plex task, as one must decide how many terminals will be operated, their location, the routes and schedules of the various vehicle types operated, and the allocation of customers to specific terminals and routes. Daganzo and Newell (1986) shows how the design problem can be reduced to a simpler terminal sizing and location problem, as explained in Sec. 5.2.
This reference is also at the core of the design discussion in Secs. 5.3 and 5.5. The discretization approach presented in Sec. 5.6 is taken from Ouy-ang and Daganzo (2004).