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{"sheet":"Problem4","index":0,"n_rows":50,"n_cols":3,"rows_kept":50,"cut":false,"hidden":false,"merged":[[13,1,15,2],[20,1,22,2],[31,1,36,2],[49,1,51,2]]}
["Parameter","Value","Formula"]
["MTTF",{"$f":"8760*1.5","v":13140}]
["alpha",3]
["CPM",8000]
["CCM",15000]
["CU",10000]
["CT",75000]
["MDT",16]
["C_F",{"$f":"C_CM+C_U*MDT+C_T","v":250000},{"$f":"_xlfn.FORMULATEXT(B9)","v":"=C_CM+C_U*MDT+C_T"}]
[]
[]
["4a)","Cost. Eq"]
[]
[null,"We can solve the task by (i) deriving the cost equation C(τ) wrt to the interval, τ, setting the derivative equal to 0, and then solving in terms of τ, (ii) using a numerical minimization routine, e.g., the Solver in Excel, or (iii) creating a plot for C(τ), and finding the minimum."]
[]
[]
[]
["4b)"]
["tau",{"$f":"MTTF/_xlfn.GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+MDT*C_U+C_T)))^(1/alpha)","v":3707.8930437106205},{"$f":"_xlfn.FORMULATEXT(B19)","v":"=MTTF/GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+MDT*C_U+C_T)))^(1/alpha)"}]
[]
[null,"The downtime, D, is exponentially distributed with parameter 1/MDT. Then the probability that D is > 20 is given as 1-Pr(D <=20) = 1 - FD(20) "]
[]
[]
["4c)"]
["tau_B",20,"Buffer size"]
["Pr(D>20)",{"$f":"EXP(-tau_B/MDT)","v":0.28650479686019009},{"$f":"_xlfn.FORMULATEXT(B26)","v":"=EXP(-tau_B/MDT)"}]
["MDT|Empty",{"$f":"MDT","v":16},"Since downtime is exponentially distributed, we do not have \"memory\" after the tank is emtpy, i.e. we still have MDT=16"]
[]
["4d)"]
["C_0",1000,"Baseline cost"]
["C_tau_B",250,"Cost per hour capacity installed"]
[null,"The cost equation with buffer changes. We calculate p(buffer size)=Pr(Empty buffer) for buffer size 20 and 30. Then, the cost equation is modified by multiplying the cost element (C_U*MDT+C_T) with p, i.e., p(buffer size)*(C_U*MDT+C_T). Then we optimize to find the optimal t for this buffer size 20 and 30 respectively. For each of these t values, we calculate the effective failure rate, and then the cost equation. For the cost equation, we add the cost of the buffer depending on the size of the buffer."]
[]
[]
[]
[]
[]
["Buffer=20:"]
["tau_20",{"$f":"MTTF/_xlfn.GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+EXP(-20/MDT)*(MDT*C_U+C_T))))^(1/alpha)","v":5369.3730217667116},{"$f":"_xlfn.FORMULATEXT(B39)","v":"=MTTF/GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+EXP(-20/MDT)*(MDT*C_U+C_T))))^(1/alpha)"}]
["lambda_E(tau_20)",{"$f":"(_xlfn.GAMMA(1+1/alpha)/MTTF)^alpha*tau_20^(alpha-1)","v":4.315119074736911E-6},{"$f":"_xlfn.FORMULATEXT(B40)","v":"=(GAMMA(1+1/alpha)/MTTF)^alpha*tau_20^(alpha-1)"}]
["C(tau_20,tau_B=20)",{"$f":"C_PM/tau_20+lambda_E_tau_20*(C_CM+EXP(-20/MDT)*(C_U*MDT+C_T))+C_0/8760+C_tau_B*20/8760","v":3.197748874113556},{"$f":"_xlfn.FORMULATEXT(B41)","v":"=C_PM/tau_20+lambda_E_tau_20*(C_CM+EXP(-20/MDT)*(C_U*MDT+C_T))+C_0/8760+C_tau_B*20/8760"}]
["(per year)",{"$f":"B41*8760","v":28012.280137234749}]
[]
["Buffer=30:"]
["tau_30",{"$f":"MTTF/_xlfn.GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+EXP(-30/MDT)*(MDT*C_U+C_T))))^(1/alpha)","v":6297.1127367016488},{"$f":"_xlfn.FORMULATEXT(B45)","v":"=MTTF/GAMMA(1+1/alpha)*(C_PM/((alpha-1)*(C_CM+EXP(-30/MDT)*(MDT*C_U+C_T))))^(1/alpha)"}]
["lambda_E(tau_20)",{"$f":"(_xlfn.GAMMA(1+1/alpha)/MTTF)^alpha*tau_30^(alpha-1)","v":1.2445757225927231E-5},{"$f":"_xlfn.FORMULATEXT(B46)","v":"=(GAMMA(1+1/alpha)/MTTF)^alpha*tau_30^(alpha-1)"}]
["C(tau_20,tau_B=20)",{"$f":"C_PM/tau_30+lambda_E_tau_30*(C_CM+EXP(-30/MDT)*(C_U*MDT+C_T))+C_0/8760+C_tau_B*30/8760","v":2.875954884023102},{"$f":"_xlfn.FORMULATEXT(B47)","v":"=C_PM/tau_30+lambda_E_tau_30*(C_CM+EXP(-30/MDT)*(C_U*MDT+C_T))+C_0/8760+C_tau_B*30/8760"}]
["(per year)",{"$f":"B47*8760","v":25193.364784042373}]
[]
[null,"The cost is lower for buffer size 30 comparred to buffer size 20"]