Question

a) All steel manufactured by Steelco must meet the following requirements: 3.2-3.5% carbon ; 1.8-2.5 % silicon; and 0.9-1.2 % nickel; and tensile strength of at least 45,000 per square inch (psi). Steelco manufactures steel by combining two alloys.. The cost and properties of each alloy are given in the following Table. Alloy 2 $200 2.5 Alloy 1 $190 Cost per ton Percent silicon Percent nickel Percent carbon Tensile strength (psi) 2 1.5 1 42,000 50,000 Assume that the tensile strength of a mixture of the two alloys can be determined by averaging that of the alloys that are mixed together. For example, a one ton mixture consisting of 40% alloy 1 and 60 % alloy 2 has a tensile strength of 0.4(42,000) + 0.6(50,000). Formulate the problem of minimizing the cost of producing a ton of steel as an LP problem. (Please do not solve the actual problem) b) Please complete the following sentences i) Simplex algorithm concludes that a problem has unbounded solution by . .. ii) Simplex algorithm can detect if a given problem does not have a feasible solution by c) Consider the following network diagram of a supply system, where the numbers associated with the arrows represent unit shipping costs (from factories to warehouses). Our goal is to minimize the total shipping cost. Using Vogel's approximation method, find a Basic Feasible Solution to the problem and find the corresponding shipping cost. DMAND Phemets WAREHOUSE Cwwerry 15 p0 :6

Answer 1

Ans:- Le ton os Alloy J ai ^3 ton of olloy a be mnized togethes to finol poduet of Prepaче thе the steet. objective function denoting Let Z the cost OS the Steel .. Z 1901 + aoona FOmulating constrainis:- Pescentage of silicon: - a7 ta5a> 1-8 and Pescentage of Carbon- 3 2+42 > 3. a and 32143 3.5 pnickel Percentage of (: - 52 > o.9 and

Tensile Streng th - www. Mass PETCRntace Alloy n ton o See- t 201% Skel Alloy an Mass Pecentoge ton 10 0 Totai msile siveng th 50000 5000 L5000 0002, 5o00 ADgramning Papblem- * Linean constraints a objectid to Minimia e 7 :190x; Soo and 32 12 2 3. 3x 3 and and Roo6 * 500007 > 4500 o

The Cive pmblem can be epsesented in tobulot ftrnat follus house Foctories FI F3 O.8 3 Demand Lt5 Oemand total s ince, the balanced iS Padblem total copacity the tyons portation by Souttion basic feasible VAM Now , we find column PenotHes SupPly FI F2 O 2 (o.8 Denand 20 0.2 Penolity Difterence ot Smolles t A next Smallest cost In - RAyouas column espective F

he Manimum penality which cores ponds td e for column in which second Smallest cost is ^ . which Cost con be solved by manimum the Smallest 90 units to C&) cetl's allocate Qullo cation , So we find Penality for the xemaining tabler Now, we again \ Supply Penals W3 26 F2 F3 10 Demand O.3 reno liy that corres ponds to Second Manimum Penality O.5 Tou хo0 that and easl cost i. E to, 303 o nit'sto that cell min we allocote these is no since Fi lling he allocations for last Tow , to Ca,) cell lo allo cak 5 unite to Option we have onit's to Ca, 3) cell .

5uprly 15 F2 13 o Demond Shipping cost of the flow Csc Netuook COst motri 1 20 15 Demord

appoximahon method to hnd the Now, we apply voget basic possible Sof' 9 Find the difference bn he two least cost eils which Step and fox both the ous of Columns. have not be en allocokd Slep a select -Ihe largest of the te choose the ore thexe s a columa case and and in M assigned considerina be tohere the maximum quality. can the minimuUs COS as wel etep 3 :- ASsign the biggest quality with in yne demand and He column wheve the Iddns COst is mimimum eliaminate column hat has bee Satished Jhe yow Slep A requirement then stop. all the demand and eupPl Slep 5- fact/ave CP 20 .8 (15 2 3

Demand ditf 8- uayeho se CAP tfp B0 15 -8 E. -9. - Demard 39 dift ditt Cop 1 5 8 16 5 50 Demand O

ap Dermand basts teaside sol which is in tHe above obtain we the assigned as the matia shipping cost Covesponding xtS t -8 X 15 4 - 8 X 5 t Ix5 x 5 4.5 15 t 1a.0 440 * 5 cOst ts 40 5 unit So, total Shipping