OPRE 6302 on 20 March 2008 - The University of Texas at Dallas
OPRE 6302 on 20 March 2008 - The University of Texas at Dallas
OPRE 6302 on 20 March 2008 - The University of Texas at Dallas
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Q3. RichChick is an east-Richards<strong>on</strong> company th<strong>at</strong> produces two types <strong>of</strong> Chicken cutlets for sale to supermarkets.<br />
Cutlets are called Silverlet and Brownlet. Each cutlet c<strong>on</strong>sists <strong>of</strong> white me<strong>at</strong> and dark me<strong>at</strong>. Silverlet sells for<br />
$12/kg and must c<strong>on</strong>sist <strong>of</strong> <strong>at</strong> least 80% white me<strong>at</strong>. Brownlet sells for $10/kg and must c<strong>on</strong>sist <strong>of</strong> <strong>at</strong> least 40%<br />
white me<strong>at</strong>. At most 300 kg <strong>of</strong> Silverlet and 500 kg <strong>of</strong> Brownlet can be sold in the next m<strong>on</strong>th. Three types <strong>of</strong><br />
chicken are used to manufacture the cutlets: Farm-grown chicken, Free Range chicken and Organic-fed chicken.<br />
Each chicken’s cost, and its dark and white me<strong>at</strong> yield are below:<br />
Chicken type Farm-grown Free-Range Organic-fed<br />
Cost in $ 6 7 8<br />
White me<strong>at</strong> yield in kg 1 2 2<br />
Dark me<strong>at</strong> yield in kg 2 1 1<br />
[4pts] a) To formul<strong>at</strong>e a LP to maximize RichChick’s pr<strong>of</strong>it in the next m<strong>on</strong>th, define decisi<strong>on</strong> variables.<br />
ANSWER: F, R, O: Number <strong>of</strong> Farm-grown, free-Range, Organic-fed chicken purchased. DS, DB: Kilograms <strong>of</strong><br />
Dark me<strong>at</strong> used in Silverlet, Brownlet. WS, WB: Kilograms <strong>of</strong> White me<strong>at</strong> used in Silverlet, Brownlet.<br />
[4pts] b) Pr<strong>of</strong>it is revenue minus costs, express RichChick’s pr<strong>of</strong>it in terms <strong>of</strong> decisi<strong>on</strong> variables.<br />
ANSWER: Max 12(WS + DS) + 10(WB + DB) − 6F − 7R − 8O<br />
[4pts] c) Write c<strong>on</strong>straints so th<strong>at</strong> RichChick does not <strong>at</strong>tempt to sell more cutlets than next m<strong>on</strong>th’s demand.<br />
ANSWER: WS + DS ≤ 300 and WB + DB ≤ 500<br />
[4pts] d) Write c<strong>on</strong>straints so th<strong>at</strong> RichChick does not <strong>at</strong>tempt to use more white or dark me<strong>at</strong> than it buys in<br />
the form <strong>of</strong> Farm-grown chicken, Free Range chicken and Organic-fed chicken.<br />
ANSWER: WS + WB ≤ F + 2R + 2O and DS + DB ≤ 2F + R + O<br />
[4pts] e) Finish your formul<strong>at</strong>i<strong>on</strong> by adding any c<strong>on</strong>straints you find necessary.<br />
ANSWER: Silverlet must have <strong>at</strong> le<strong>at</strong> 80% white me<strong>at</strong>:<br />
Brownlet must have <strong>at</strong> least 40% white me<strong>at</strong>:<br />
WS<br />
WS + DS<br />
WB<br />
WB + DB<br />
N<strong>on</strong>neg<strong>at</strong>ivity c<strong>on</strong>straints: F, R, O, WS, DS, WB, DB ≥ 0<br />
≥ 0.8.<br />
≥ 0.6.