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Eqns (35) to (50) are the absolute sensitivity coefficients. Let the expressions for the sensitivity of Dopt be also developed. We recall that eqn (29) gives
By taking the partial derivatives of Dopt with respect to k1, k2, k3, and Q, then:
And for the relative sensitivity analysis, the following equations hold:
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(as customers) to be served (evacuated/transferred/transported) to the final dump station by disposal trucks accumulate there as municipal solid waste. The waste are dumped both during the day and in the night hours through a discrete influx of these materials to the site. The waste disposal trucks are deployed to the various waste dump stations for service as illustrated in Figure 12.
The waste accumulation structure is dynamic in discipline. Both service in random order (SIRO) and priority service are applicable here. In both cases, the waste bins at the various dump stations waiting to be evacuated (as customers) are considered as being in a single queue and randomly served. The servers (evacuator trucks) go to any of the bin stations that is free and waiting for service. The server arrival and departure events are depicted in Figures 13a and 13b.
?
?
?
?
?
Departure to a Transfer station(s) or a dump site(s)/
landfill(s) Server/Truck 1
Server/Truck 2 Server/Truck 3 etc.
Arrival Of Server Facility
Deployment
Waste Station 1 Waste Station 2 Waste Station 3 Waste Station 4 etc.
Queue Of Customer Facility
Figure 12: Flow diagram of a typical Multiple Servers-Multiple Waste Depot System
(a): Disposal vehicle arrives dumpsite for service
Go to unit waiting in queue
Server begins service Unit waits
in queue No Yes
Server arrival event
Waste bin(s) due for service?
(b): Disposal vehicle leaves dumpsite j for final dumpsite/transfer station
Figure 13: Arrival and departure events of a service facility
Move to final dumpsite/
transfer station Go to next unit
waiting in queue
No Server fully Yes
loaded?
Server departure event
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The following assumptions are made, and where necessary they will also apply to other models developed in this research :
a. The number of waste containers (bins) in use is known but in terms of the quantities of waste to be generated it is unknown, infinite and unrestricted. That is to say, there are many batched quantities of solid waste (measured in terms of bin sizes) to be evacuated (served) in a given period (hourly, daily, weekly, etc)
b. Waste existing at various public used roadside dumpsites are in queue
c. The service/unloading time of bins is controllable
d. The waste bins are arbitrarily but strategically distributed (placed) at various locations within the research area.
e. Quantity of solid waste found at roadside dumpsites at any given time is the total waste generated by the people living/operating within the area.
f. Waste generation and disposals are stochastic, dynamic and man-machine-material interaction processes of the system
g. The system is non-fully automated
h. The number and capacities of the disposal vehicles are known.
i. Arrival rate of waste at the dump sites is stochastic, but the evacuation is executed according to management plans.
j. State (quantity) of waste before and after evacuation at a given dumpsite in a given period(s) is always recorded.
k. Waste not brought to a public dumpsite is negligible/not considered at all.
l. Size and number of waste bins (containers) and bin locations (dumpsites) may vary, but such data is recorded and treated from the point of such a variation
m. For purposes of the present study, unless where otherwise stated, loading of an evacuator at a waste bin/dump site is measured by the fraction of full loaded chain-up bins carried by the truck per its trip from the site.
n. Any indisposed quantity of waste at a given waste bin site, after a truck's visit to the site on a day is considered a new joiner to the line of bins waiting for service during the truck's next visit to the site.
o. Any day none of the disposal trucks (servers) works in a given dumpsite is assumed as server's idle time (i.e. 𝑁(μ=0)) for that dump site.
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p. For a recording that starts at evening check of day i = 1, qd,ij = 0 and qN,ij > 0; if recording starts at morning check: qN,ij= 0 and qd,ij > 0.
To make the development of the models in the study better understood, a simulation table consisting of k inputs xij for k number of dumpsites (j = 1, 2,...,k; i = 1,2,...,n, and a number of the desired responses yij, was constructed in a Microsoft Excel spreadsheet as shown in Table 4.
Table 4: Simulation table for queue model development, demonstration and validation
Time
Dumpsite 1 … Dumpsite k All Dumpsite
Inputs xi1 Responses, y1 … Inputs xi1 Responses, y1 Σyi (Responses)
i
(days) Lq qd qN q(d+N) λq μq LR Ls Wa Wq Ws η Pt(t) … Lq qd qN q(d+N) … Pt(t) q(d+N) λq μq … Pt(t)
1 …
2 …
.
. …
n …
Σ =
Mean
In estimating the quantities of waste generated and evacuated in a given dumpsite j, it is assumed that check starts in the morning of day i = 1, with Lw,(i-1)j > 0, qN,(i-1)j > 0, μq,ij = 0, qd,ij = 0, qN,ij = 0 and μq,(i-1)j = (r D)(i-1)j . Thus, in accordance with eqn. (17)
𝑞𝑀,𝑖𝑗 = (𝑄 − 𝑟𝐷)(𝑖−1)𝑗 + 𝑞𝑁,(𝑖−1)𝑗 = 𝑞𝑀,𝑜𝑗 (51a) Or rewriting eqn. (47a) in terms of queue parameters
𝐿𝑞,𝑜𝑗 = 𝑞𝑀,𝑖𝑗 = 𝐿𝑤 ,(𝑖−1)𝑗 + 𝑞𝑁,(𝑖−1)𝑗 (51b)
At the evening check of the same 1st day we also assume that qM,ij > 0, Lw,ij > qM,0j, qN,ij
= 0 and μq,ij > 0. By this we obtain that
𝑞𝑑,𝑖𝑗 = 𝐿𝑤 ,𝑖𝑗 + μ𝑞,𝑖𝑗 − 𝑞𝑀,𝑜𝑗 (51c) At the morning check of the next day, i+1, with μq,ij > 0, Lw,ij > qd,ij > 0, qN,ij > 0,
𝐿𝑞 ,(𝑖+1)𝑗 = 𝐿𝑤 ,𝑖𝑗 + 𝑞𝑁,𝑖𝑗 (52a) At the evening check of day i+1 still, with no evacuation made and qN,ij > 0. Thus
𝑞𝑑, 𝑖+1 𝑗 = (𝐿𝑤 + μ𝑞) 𝑖+1 𝑗 − 𝑞𝑀, 𝑖+1 𝑗 (52b)
From the foregoing, therefore, we conclude that for a given day i, the total quantity of waste (number of bin loads) deposited at dump site j,
𝑞 𝑑+𝑁 𝑗 = (𝑞𝑑 + 𝑞𝑁)𝑖𝑗 (52c) A waste container is considered full or overflowing if and only if:
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𝑉𝑏 = 𝑚 𝑞 𝑑+𝑁 𝑖𝑗
𝑖=1 ≥ 1 (where 1 ≤ 𝑚 ≤ 𝑛) (53a)
And number of fully and/or over loaded bins 𝑁𝐵𝑓 = 𝑞(𝑑 +𝑁 )
𝑛𝑖=1
𝑉𝑏 ≥ 1 (53b) Let for a given dumpsite j,
D = NBf x Vb (53c)
Time from t = 1 to t = m taken for ( 𝑚𝑖=1𝑞 𝑑+𝑁 𝑖𝑗 = 𝐷) = tGL (say) (53d) Number of WGLTs in a period t = 1 to t = n is obtained from the relation:
𝑁𝑡𝐺𝐿 = 𝑞(𝑑 +𝑁 )
𝑛𝑖=1
𝐷 ≥ 1 (53e)
For k number of dumpsites, daily quantity of waste arriving (deposited) in a period i to n days is obtained thus, 𝜆𝑞 = (𝐿𝑞 + 𝑞𝑑)𝑖𝑗 (54a)
While the combined mean quantity of waste arriving (deposited) in a period i to n days is obtained thus, 𝜆 = 𝑛𝑘1 𝑛𝑖=1 𝑘𝑗 =1𝜆𝑞,𝑖𝑗 (54b)
Combined mean number of waste bins served over a given period
𝜇 = 𝑛𝑘1 𝑛𝑖=1 𝑘𝑗 =1𝜇𝑞,𝑖𝑗 (55) With μq.ij > 0, number of loaded bins in the queue on the nth day
𝐿𝑞 ,𝑛 = 𝐿𝑞,𝑜+ ( 𝑛−1𝑖=1 𝑞 d+N − 𝑛−1𝑖=1 𝜇𝑞)𝑖𝑗 (56a) Number of loaded bins in the system on the nth day
𝐿𝑠,𝑛 = 𝐿𝑞 ,𝑜 + ( 𝑛−1𝑖=1 𝑞 d+N − 𝑛−1𝑖=1 𝜇 + 𝑞𝑑)𝑖𝑗 (56b) For k number of dumpsites, the average number of waste bins in the system
𝐿𝑠 = 𝑛𝑘1 𝑘𝑗 =1(𝐿𝑞 ,𝑜+ 𝑛−1𝑖=1 𝑞(d+N)− 𝑛−1𝑖=1 𝜇 + 𝑞𝑑)𝑖𝑗 (56c) And the number of loaded bins waiting for service in the system on the nth day 𝐿𝑤 ,𝑛 = 𝐿𝑞 ,𝑜 + ( 𝑛−1𝑖=1 𝑞 d+N − 𝑛𝑖=1𝜇𝑞 + 𝑞𝑑)𝑖𝑗 (57)
From the general queuing theory, we obtain
Wq = Lq λ-1 (58a) Ws = Ls λ-1 (58b) 𝑊𝑎 = 1
𝑛 𝑛𝑖=1𝑁(μ=0) (58c)
A waste management system's daily evacuation efficiency (portion of waste evacuated) is defined by the relation,
𝜂𝑖𝑗 = μ𝑞 ,𝑖𝑗
λ𝑞 ,𝑖,𝑗 x 100% (59a)
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While the overall system's efficiency is defined by the relation 𝜂 = μ𝑞 ,𝑖𝑗
𝑘𝑗 =1 𝑛𝑖=1
( 𝐿𝑞 ,𝑜𝑗+ 𝑛𝑖=1 𝑘𝑗 =1 𝑞(𝑑 +𝑁 ))𝑖𝑗 (59b) And the waste disposal system utilization
𝜌 = λ
𝑠μ (60) Each idle time of server in the system is defined by the conditions that:
μ = 0
q(d+N) > 0 NBf > 1 (61)
tos = tμ = 0
and Nμ=0 = 1 (say)
Thus, total number of idle times of disposal truck(s) serving k number of dumpsites in a given period,
𝑁𝑠,𝑜 = 𝑛𝑖=1 𝑘𝑗 =1𝑁 μ=0 ,𝑖𝑗 (62) Therefore, probability that a server is idle
(Po)s = (Ns)o x n-1 (63) Probability that waste container(s) in the system is full and/or overflowing,
𝑃𝐵𝑓 = (𝑁𝐵𝑓
𝑛 )𝑖𝑗
𝑘𝑗 =1
𝑛𝑖=1 {where 𝑁𝐵𝑓: 𝑁𝐵𝑓 = (𝑁𝐵𝑓 + 𝑁𝐵𝑜𝑓)} (64) It is assumed that disposal trucks (servers) visit dumpsites either in the morning or evening hours of a day and that during the visits, the waste bins should be totally emptied such that
Nμ >0 > Nb
q(d+N) = 0 (65)
and Nλ=0 = 1 (say)
By this assumption, the probability of no waste in dumpsite j at morning check of a given period,
𝑃𝑜,𝑑 ,𝑗 = 𝑁λ=0
𝑛 𝑑,𝑖𝑗 ≤ 0.5 (66a) Probability of no waste in dumpsite j at evening check of a given period,
𝑃𝑜,𝑁 ,𝑗 = 𝑁λ=0
𝑛 𝑁,𝑖𝑗 ≤ 0.5 (66b) Therefore, probability of no waste in dumpsite j is
𝑃𝑜,,𝑗 = 𝑁λ=0
2𝑛 𝑑,𝑖𝑗 + 𝑁λ=0
2𝑛 𝑁,𝑖𝑗 ≤ 1 (66c) For the entire system, the probability of no waste in k number of dumpsites is
111 𝑃𝑜,𝑘 = 1
𝑘 𝑁λ=0
2𝑛 𝑑 + 𝑁λ=0
2𝑛 𝑁
𝑖𝑗
𝑘𝑗 =1 ≤ 1 (66d)
Probability that new stock of waste arrives at dumpsite j in a given period, 𝑃𝜆 = 𝑞 𝑑 +𝑁
𝑛𝑖=1
(𝐿𝑞 ,𝑜𝑗 + 𝑛𝑖=1𝑞 𝑑 +𝑁 ) (67)
The transient probability can be based specifically on either the daily arriving stock q(d+N) or on daily total stock λq. That is to say,
𝑃(𝑡)𝑞(𝑑+𝑁 ) = 𝑞(𝑑 +𝑁 )
𝑛𝑖=1𝑞 𝑑 +𝑁 (68a) Or 𝑃(𝑡)𝜆𝑞 = 𝜆𝑞
𝑛𝑖=1𝜆𝑞 (68b)
It can be seen clearly from the above equations that the performance measures are functions of two basic queuing parameters - waste arrival rate (the average rate of waste container fill) and the container service rate (the average rate of waste evacuation). Values computed for these parameters give how well the management service mechanism handles volumes of waste generated in the given system.
3.3.6.1 Flow chart for systematic application of the waste queuing models
Flow chart for application of the waste queuing models developed in this study in Excel spreadsheet is depicted in Figure 14. Values for two basic queuing parameters - λ and μ - for i = 1 to n and j = 1 to k are required as input data which Excel uses to generate values for other parameters based on coded models of interest.
Start
Create columns k number of dumpsites and code in for each dumpsite, the model equations for: Lq, qd, (Nq=0)d, qN, (Nq=0)d, q(d+N), λq, μq, Nμ=0, Lw, Ls, Wq, Wa, Ws, η, Pq(t)
Input values for λq,ij, μq,ij for i = 1 to n,
j = 1 to k
Compute mean values λ for λq,ij = , μ for μq,ij, for i = 1 to n,
j = 1 to k
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