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Question 1 A foundry smelts steel, aluminum, and cast-iron scraps to produce two types of metal ingots, I and II, with specific limits on the aluminum, graphite and silicon contents. Aluminum and silicon briquettes may be used in the smelting process to meet the desired specifications. Table 1 and Table 2 set the specifications of the problem: Input item Contents (%) Aluminum Graphite Silicon Cost/ton ($) Available tons/day Steel scrap 10 5 4 100 1000 Aluminum scrap 95 1 2 150 500 Cast Iron scrap 0 15 8 15 2500 Aluminum briquette Silicon briquette 100 0 0 900 Any amount 0 0 100 380 Any amount Toble 1. Raw material characteristic Ingredient Aluminum Ingot 1 Ingot 11 Minimum Maximum Minimum Maximum 8.1% 10.8% 6.2% 8.9% 1.5% 3.0% 4.1% GO Graphite Silicon Demand (tons/day) 2.5% 00 2.8% 4.1% 130 250 Table 2 Product Characteristic Determine the optimal input mix the foundry should smelt. Homework Furmolations 26 2018 D. Xis: is ingredient / ton (1,2,3,4,5), j ingot (1,2) obiedive tundion. Mina loo (Xu +X12) + 150(X21 +220) +15(131+352) + 900(X4:+Xuz)+ 380CX51 +Ysz). ingof 1 st ol Xit + 0.95X21 + Xun 0.081 X130 Xu to 95721 + YU 0.108x130 0.05X1 +0.0X21 +0.15X31 20.0151130 0.05%uto. X2 + .. 15731 203.1130 0.04 Xij to.o2X21 +0.08X31+X5120.0251130 0.1X12 +0.9522+%42 20.262x250 ..1912 + 0.45X22 +142 S 0.089X250 0.05X12+ 0.01422 +0.15X2200411280 0.04 X127 0.02X22 +0.08X32 + X$2 = 0.0282250 0.04X2 + 0.22722 +0.08X2 + 152 Sool 250 X1 + X21 +231 +Xx) + X5) - 130 X 12 + X27 +232 +242 + X52= 250 X1 + x n slooo X31+X32 52500 X21 +422s soo Ingot 2 2 X 11, X21 ... 151 , X12, X22, ... Xsz zo. C Question 4 .- The Canadian Parks Commission controls two tracts of land. Tract 1 consists of 300 acres and tract 2, 100 acres. Each acre of tract 1 can be used for spruce trees or hunting, or both. Each acre of tract 2 can be used for spruce trees or camping, or both. The capital (in hundreds of dollars) and labor (in worker-days) required to maintain one acre of each tract, and the profit in thousands of dollars) per acre for each possible use of land are given in Table 4. Tract Capital (in hundreds Labor (in Profit (in thousands of of dollars) worker-days) dollars) 1 Spruce 3 0.1 0.2 1 Hunting 3 0.2 0.4 1 Both 4 0.2 0.5 2 Spruce 1 0.05 0.06 2 Camping 30 5 0.09 2 Both 10 1.01 1.1 Table 4 The capital and labor required to maintain one acre of each tract, and the profit per acre for each possible use of land Capital of $150,000 and 200 man-days of labor are available. How should the land be allocated to various uses to maximize profit received from the two tracts? D.V: Let X. , A2, x3 acres allocated for tract 1 spruce, trait 4 hunting. let du, Xs uab trad a spruce, trout a hunting, objective function: Max2= lood (0.221 + a 4 X2 to 5x7 to.obx4 0,09X5 +1,1x6) st: X1 + x2 + x> $300 Xut X5 +X6 80 0.02X , to.o482 to, o3X3to.o9Xn _40 o.olxi to.olx2 to.o 483 +0.04 x 4 22o. xi +82 + x3 +Xu=10o. 22 2 loo XI , X2 X3 X4 20

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