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# Note for MATH 1313 with Professor Ahmed-Zaid at UH linear programming section 3.3

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Date Created: 02/06/15
83 SQ KNG 101K A Section 33 L o k r Linear Programming Consider the following gure Which is associated With a system linear inequalities SW lm Ung 9m h Sawhw u algal S l XJYgtO ii The set S is called a feasible set Eachpoint in S is a candidate for the solution of the id problem and is called a feasible solution vk r Q r NH The points inS that optimizes maximizes or minimizes the objective function is called the optimal solution Theorem 1 in book Linear Programming If a linear programming problem has a solution then it must occur at a vertex or corner point of the feasible set S associated With the problem Furthermore if the objective function P is optimized at two adjacent vertices of S then it is optimized at every point on the line segment joining these vertices in Which case there are in nitely many solutions to the problem The Method of Corners 1 Graph the feasible set graph the system of constraints 2 Find the coordinat of all comer nts ve ces of the feasible set ts 4 Find the vertex that renders the objective function a maximum minimum Section 33 7 Linear Programming 1 PW 4395736 as o 69 39 039 393 2M RU 27 5 C 3 5 LE 04 guor t 39 mod O A 13 T05 up on 4 Y ogxw m Q Ma n g 7 f l lt 1 W lo Tax XDV1 quot HUUV 5 M21 39H r193 kwcpei W MMQ z39gw 3 s iv M 4k L Hum S K 39l SK3 L L O I I I 39 6 to 3 34 er rmoakllw thga kaSf Cwm N Ra lagfl LO 6 moment a szu W g CUHO LUH100 7 CUM LVNvim Hf Example 2 RECALL THE FOLLOWING PROBLEM FROM 32 Apatient in a hospital is required to have at least 84 units ofdrug D1 and at least 120 units ofdrug D 2 each day assume that an overdosage of either drug is harmless Two substances M and N contain each of these drugs however in addition both contain an undesirable drug D3 Each gram ofsubstance M contains 10 units of drug D1 8 units of drug D Z and 3 units ofdrug D 3 Each gram ofsubstance N contains 2 units ofdrug D1 4 units of drug D2 and 1 unit ofdrug D 3 How many grams ofsubstances M andN should be mixed to meet the rninjmum daily requirements and at the same time minimize the intake 9 8 WMEJHIY 33 Sxy 1 x 1 Yx Le quot q lbw30 0 L62 Till 03 oKojfZQ39qP xg Bl oZVH N 7 a ROWAN gt 70 Na 00H6F 9 quot 0 2 20 be J39 D WO wW QHMAQ 1h mum am3 gm 3 Ugh 1 m u Mowth 112 w LLDLZQgtY alll x gt Lf bhdl 3k 0 1 gK x Q39L 10 39Lx 1306 SM 3 3 Q2 7 P Q L 9U ED 7 kc gtr Lnquot l C am 39gt 9 Example 3 RECALL THE FOLLOWING PROBLEM FROM 32 The of cers ofa high school senior class are planning to rent buses and vans for a class trip Each bus can transport 40 students requires 3 chaperones and costs 1200 to rent Each van can transpOIt 8 students requires 1 chaperone and cost 100 to rent The of cers must plan to accommodate at least 400 students Since only 36 parents have volunteeredto serve as chaperones the of cers must plan to use at most 36 chaperones How many vehicles of M w 11 419 1 v V L 36 AY 1W 5 YC O V 12 so Flvt 7 73 lulr up I Q gtQ u 5 A is qalt 3gtrfw 1g ponl vmk 0 5 2mm 36 gt1 o 530 W m 9quotquot r s 0 3x Jet 3 Lxc j 9 2 a MNA W311 i Egan2 MM anU 7Aer 2 mg 1 j gamut Lo quot 0mm 399 AKQB Lklzl ska1 5 so 7539 go o wtkrg MIIHMI ZQQ3lQOO7Kg ng kKRgQ tllfuo mug 49900 gm 1001c3k YqQDl 00 9qu MMKMM NV g q kg C9A0 K 7 94 NJ VA in ux39wi ti Q QX H57 stekww lixJAW nm c 220 213 Exam3mg lOXH Z S E S a 20 2 3 7 17C 320 Q1 MEX Hal A g 9 37 9 x 0 gK egy 560 0 K L O LLo H9 TQ3klt VKD 3 W bb g O 710 1 is Mcme 921 Q SUQQ QQHM A 52 U V W WAM prnQrf ULOOamp Mum Altqzlb QAQ L QQWS CQOTAQ lt V ampmcya K 5Q 39qK JT 2 17 v J39quot g6 00 c g Q CanU1 Maidmm V lZK f BY L7lt t 00 C VIM msk C Q 4 Mfg M I m 116 ML gym MR2 5 He GBEKYE 17 So 06M twang

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