{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

AggregatePlanning-Multi-Product-Hopp-and-Spearman

AggregatePlanning-Multi-Product-Hopp-and-Spearman -...

This preview shows pages 1–11. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Consider a’ production line with four workstations, labeled j = 1,2, 3, and 4, in tandem (all products ﬂow through all four machines in order)! Three different products, laheied i = A B, and C, are produced on the line. The hours required on each workstation for each product and the net proﬁt per unit sold (a) are given as follows: The number of hours availabletcﬁl and the upper andlower limits on demand (tin and gt“) for each product over the next four quarters are: a. Suppose we usea quarterly holding cost of\$5 and a quarterly backorder - cost of \$10 per item on all productsand allow backordering. Formulate an LP to maximize proﬁt minus holding and backorder costs subject to the constraints on workstation capacity and min/max sales. b. Using the LP solver of your choice, solve your formulation in part (a). ' Which constraints are binding in yoursolution? _ HOWY—QEW/mw {2.552.} Phobia/w: 6. Sm : mmfor {Cm L" km Pom; J:— QzA 1 P3) C s 1,2,), gH PWFJTﬁ P£VENUE~ COST LEVENwE-iiy}; 3"” : 505M 4’ SO SA/L + SOS/53 1” E50 SAJH wA’cﬂ 4— 55 SEA 4— (OSS‘MJ— 63 \$313+ (05 33H w 053T;- HOUDJNG 1‘” EP-C/id/OB/DEIMNEJ if 1 Onwhand ear” ‘VCWE \‘r\ ”C. lot ' IE,e—¢# of wok/crew far rm L‘ Mt. J . , 2 0 lava 0r Elke/mi int“- _ i I'M: ‘ ‘Wem- (y f ,, 1 E '— .. \$5?me at ’OZZIgt w-MSTM)MT_51 wmrnagm-s for ma: SMsloo ggﬂgwo Mime? ’“f ‘ . M Eat/«NBS Sc,“ 3300 \$42.31?” so)? 67.50 gcw QM Lax/ES" EOMWDS mum—«”3 SM'Z'7/0 gg‘LZFZ'O SM? 0 33,3"; 20 S Aft-f7 0 SB‘L-rj/LS SC’H 7/ O gC/J'Z. >/ O (.73 7;, (a? YE ’c: .‘Fr-oamc/Hon (0kg Jror L 1% JC- (CJEQTPA U+g WMTFOWU: 2. L1. XA 7, Jr 1 oxgﬂ+o ﬂ XLJZ.‘ (0% ng‘opz7LEMTELA M ‘7. L; XAJg + '2. 0x8 gj—J-O ‘Qxaﬂj é \cho EUODBJCEMWQ.“ 1 I+ Xpﬁ + 7, 02mm 0 Ci xqLf s 180 {@104} CENTEJM Camemhzaj'uomn E " ' " a, l'LfXPut +1-0ngk3“ OH fot g d‘lﬂé 162417—1311 V— Myﬁsd”§”:f__’ Wﬂ§¥raﬂnﬂ 60" Whit, Léﬁher— 1- \-\ XAt': +2“; XE): + O‘ﬁxcvtédzjt {’5‘}; 3:9? WM“ mw-Hum- m Hm .. H",m_:_mww3‘ _ W ,.....‘ “WMWA-«wlwhm. .m-nwn W.»- ""“ “WM verWMM.“ mJJ-an'VbHalvau,‘ C/qPOH/H‘y WMTK‘OIM‘J far wart, (jg/mfg, 2 max/m- +1-1ng inXLJc< ad“: mama, W WL W417 3 Eat/14‘? wm’ I’mmn 76m MVP/hwy Ewan me 16361 maﬂom= [Mg/”Mg 5r0‘rh%g I‘mﬂ {1 wt). gem/W pﬂ’bd FOP A only); QSAJLS yF/1+1AJ'\ I 3ij; \ﬂgJ/h-W‘i S tSXC/t-lﬂl—‘t-A'IQJt. C) Li.) We Aha/Md alSo defmgi . . + I, LL/t‘ 51L}? IL/C k ‘,A,P2,c, ill/21“" lf-l— Peﬁbcl Multiple Product Workforce Planning Model (Follows Hopp and Spearman) We use the following deﬁnitions: c 1: = machine capacity in work center j for time period t al.1- = machine requirements per unit of production of product i ( of the m products) in work center j xi, = units produced of product i- in time period t The machine capacity restrictions are: 2(1ng Sci.” for alljand t :21 _ by. = worker hours required per unit of product i'in work center 1' u, = conversion from workers to hours in time period t wj, = workers assigned to work center 1' during time t o 1: = overtime workers in work center 1’ during time t 5’ The worker capacity_re\$ttiction are: I} it ‘Zboc. Su,(wj,+oﬁ), for alljandt i=1 or} = overtime fraction limit for workers assigned to work center j Over time restrictions: osd<mw,mmmmm‘ I‘- 3;. 2 units of product i sold in time period t, where there. are upper and lower-limits on sales for each product i and each time period t '- df5s_sd? it a u ' .Q - ' "k ' I“ = inventory of product i in time period t The inventory balance equations are: with In 2 0. since no backorders are allowed, 1;, =1“ +15“ —s,.,, forall i and: 14:: ﬂ be the standard loss rate of employees, then the number of workers in work center j and time t is a function of the losses, the hiring; H ﬂ. and ﬁring, Fin during period t Wit = ﬁiij-1+Hit "Fir There are worker regular and over time costs, Cf and Cf, and hiring and ring costs, C :H and Cf, which can vary by work center j. If we get a return (profit) of n for product i and it costs hi to hold one unit of this product for one period, then the objective is to maximize the total return given by: IT IT max £20m, ‘hilv)“22(Cinr +CfFj, thwj, +Cf’oj,) i=1 [:1 }=1 l=l subject to the restriction given above and all variables being non-negative. Formulate and solve the following problem from Hopp and Spearman’s textbook: Factory Physics. ,. N UMEBICAL §OLUTION . f] MAX 505A!+503A2+50\$A3+508A4+65SBl+65SBZ+6SSBS+GSSB4+7GSCI+ I 708C2+7OSC3+7OSC4-SIPA1-5IPA2—5IPA3-SIPA4-5IPBl-SIPB2—SIPBJ- I 51PB4-5IPC]~51PCZ-SIPC3-SIPC4-IOIMAI-101MA2-lOIMAZ-lOlMA4-IOIMB1-r IOIMBZ-101MB3--10H\£B4-10MCl-10MC2-101MC3-101MC4 ! SUBJECTTO I 2) SAl<= 100 3) SA2<= 50 4) sm<= so 5) SA4<= 75 6) s31<= 100 7) s132<= 100 8) 5133 <= 100 9) lsa4<= 100 ' I 10) sc1<= 300 11) scz <= 250 12) scs<=l 250 13) 8C4 411:! 400 I 14) 5A1» 0 I. 15) SA2>= 0 ' 16) SA3>= 0 17) SA4>= 0 13) SB1>= 20 19) s132>= 20 20) SB3>= 20 21) SB4>= 25 22) SCl>= 0 23) sc2>= 0 24) sc3>= 0 25) SC4>= 50 26) 2.4XA1+2XBI+0.9XCI<= 640 27) 242042452 x132+0.9 xcz <= 640 28) 2.4XA3+2XB3+0.9XC3 <= 1230 29) 2.4XA4+2XB4+0.9XC4<= 1280 1' 30) l.lXAl+2.2XBl+0.9XC1<_= 640 I. 31) 1.1m+2.2x132+0.9xc2<= 640 f 32) 1.1XA3+2.2XB3+0.9XC3 <= 640 g 33) 1.1XA4+2.2x134+0.9XC4<= 640 1 34) 0.3m1+1.2x131+xc1 <= 1920 g 35) 0.0XA2+1.2x132+xcz<= 1920 36) 0.3XA3+1.2XBs+xc3<= 1920 I 37) 0.8XA4+1.2X134+XC4<= 1920 g 33) 3x41+2.1x131+2.5x01<= 1280 ,6 39) 3XA2+2.1XBZ+2.5XC2 <= 1230 I 40) 3XA3+2.1XB3+2.5XC3<= 1230 f 41) 3XA4+2.IXB4+2.5XC4<= 25m“ 42) SA1-XA1+1A1= 0 4,) 43) ss1-x131+1131= 0 I 44) SCl-XC1+IC1= 0 I 45) SA2_-XA2—1A1+1A2= 0 I 46) SBZ-XBZ—IBI-l-IBZ= 0 47) scz-xcz-Ic1+102= 0 43) SA3-XA3-IA2+IA3= 0 49) SB3-XB3-IE2+IBS= 0 50) SC3~XC3-IC2+IC3= o 51) SA4-XA4-1A3+1A4= o ‘ 52) ss4-1m4-133+1134= 0 I 53) SC4-XC4-IC3+IC4= 0 1 I 54)-IPAI+IMA1+IA1= 0 I .- .. I 55)-IPBI+IMB1+IBI= 0 I - 1 56)-IPC1+[MCI+IC1= 0 I 57)-IPA2+IMA2+IA2= 0 53)-113132+11v032+1132= 0 I 59)-I.PC2+IMCz+IC2= 0 j 60)-[PA3+IMA3+IA3= 0 1 61)-IP133+1MB3+1133= 0 | .3.“ 62) - IPC3 + IMC3 + 1C3 = 0 63)-IPA4+IMA4+IA4= 0 64)-IPB4+IMB4+IB4= 0 65) - IPC4 + IMC4 + 1C4 = 0 (b) T'he solution is given below. Constraints (3)—(l3) are tight sales constraints. Two machine constraints are tight: constraint (26) is the machine capacity constraint on machine I in quarter 1; constraint (33) is the machine capacity constraint on machine 2 in quarter 4. OBJECTIVE FUNCTION VALUE 1.) 122240.50 VARIABLE VALUE REDUCED COST SA] 70.833340 .000000 SA2 50.000000 .000000 5A3 50.000000 .000000 SA4 75.000000 .000000 SB 1 100.000000 .000000 SB2 100.000000 .000000 SB3 100.000000 .000000 SB4 100.000000 .000000 SC 1 300.000000 .000000 SC2 250.000000 .000000 3C3 250.000000 .000000 8C4 400.000000 .000000 IPA] .000000 .000000 1PA2 .000000 .000000 IPA3 .000000 .000000 IPA4 .000000 .000000 IPBl .000000 .000000 IP32 .000000 .000000 IPB3 10.227270 A .000000 IP34 .000000 .000000 IPCl .000000 .000000 IP02 .000000 .000000 IPC3 .000000 .000000 IPC4 .000000 000000 [MAI .000000 15.000000 1MA2 .000000 15.000000 1MA3 .000000 15.000000 1MA4 .000000 15.000000 1MB 1 .000000 15.000000 IMB2 .000000 15.000000 lMB3 .000000 15.000000 IMB4 .000000 15.000000 lMCl .000000 15.000000 IMC2 .000000 15.000000 1MC3 .000000 15.000000 IMC4 .000000 15.000000 XAl 70.833340 .000000 X3 1 100000000 .000000 86 XCl XA2 X32 XC2 XA3- , X33 XC3 XA4 XB4 XC4 1A1 13 l {C 1 [AZ 1132 1C2 1A3 1B3 ' 1C3 [A4 1B4 1C4 ROW'SLACKORSURHAS DUALHMCES 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) . 17) 18) 19) 20) 21) ,22) 23) 24) 25) 20) 27) ' 23) 29) 30) 300.000000 50.000000 100000000 250000000 50.000000 110227300 ' 250.000000 75.000000 89.772730 400.000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 10.227270 .000000 .000000 .000000 .000000 29.166670 .000000’ .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 70333340 50000000 50000000 75000000 80000000 30000000 7 30000000 75.000000 300.000000 250000000 250 .000000 350.000000 .000000 95.000000 714.545 500 560.454500 72.083340 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 55.000000 46.666660 23.750000 5.000000 5.000000 5.000000 2.500000 .000000 2.954545 7.500000 10.000000 7.045455 .000000 50.000000 50.000000 47.500000 23.333340 65.000000 65.000000 60.000000 51.250000 ' 70.000000 70.000000 67.954540 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 20.833330 .000000 .000000 .000000 .000000 87 31) 32) 33) 34) 35) 3’6) 31) 33) 39) 40) 41) 42) 453 44) 45) 46) 47) 43) ~49) 50) 51) 52) 53) 54) 55) 56) 57) 53) 59) 60) 61) 62) 63) 64) 65) 140.000000 1 17.500000 .000000 1443.3 3 3000 1510000000 1497327000 13 52.273000 107. 500000 295000000 273 .522700 1 146.477000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 ' .000000 .000000 .000000 .000000 .000000 .000000 .000000 .000000 ' .000000 .000000 .000000 2.272727 .000000 .000000 .000000 .000000 . .000000 .000000 .000000 .000000 50.000000 41.666660 18.750000 .000000 .000000 .000000 .000000 .000000 .000000 2.500000 5.000000 2.045455 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 ...
View Full Document

{[ snackBarMessage ]}

Page1 / 11

AggregatePlanning-Multi-Product-Hopp-and-Spearman -...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online