Question

1 HAL produces two types of computers: PCs and VAXes. The computers are produced in two locations: New York and Los Angeles. New York can produce up to 800 computers and Los Angeles up to 1,000 computers. HAL can sell up to 900 PCs and 900 VAXes. The profit associated with each production site and computer sale is as follows: New York-PC, $600; VAX, $800; Los Angeles-PC, $1,000; VAX, $1,300. The skilled labor required to build each computer at each location is as follows: New York- PC, 2 hours; VAX, 2 hours; Los Angeles-PC, 3 hours; VAX, 4 hours. A total of 4,000 hours of labor are available. Labor is purchased at a cost of $20 per hour. Let XNP = PCs produced in New York XLP = PCs produced in Los Angeles XNV = VAXes produced in New York XLV = VAXes produced in Los Angeles Use the LINDO printout in Figure 17 to answer the follow- ing questions: If 3,000 hours of skilled labor were available, what would be HAL's profit? b Suppose an outside contractor offers to increase the capacity of New York to 850 computers at a cost of $5,000. Should HAL hire the contractor? C By how much would the profit for a VAX produced in Los Angeles have to increase before HAL would want to produce VAXes in Los Angeles? d What is the most HAL should pay for an extra hour of labor? a

Answer 1

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Solin Let xnp is the number of PCs produced in New York, XLp is the number of PCs produced in Los Angeles, XNV is the numbeos of Vaxes produced in New York, XLV is the number of Vaxes produced in Los Angles And I is the labor hours. Also consider the following LINDO output. Max 660 XNP + 1000 XLP + 800 XNV +1300 XLV-20L SUBJECT TO 2) 2XNP +3XLP + 2XNV +4XLV - L LEO 3) XNP + XNY L- 800 4) XLP+ YLV 2= 1000 5) XNP + XLP La 900 6) XNP+XLV La 900 7) LZ- 4000 END LP OPTIMUM FOUND AT STEP OBTECTIVE FUNCTION VALUE. 1 1360000 VARIABLE -VALUE REDUCED COST XNP 0.00 XLP 200.000000 800.00 -0.00 XNV 800.00 0.00 XLV 0.00 33.333332 4000.00 0.00

SLACK OR SURPLUS oo DUAL PRICES 333.333344 133.333328 ŠonEwe - 200 200 100 100 313.333344 NO ITERATIONS RANGES IN WHICH The BASIS IS UNCHANGED OBY COFF VARIABLE XNP XLP XNV XLV Current boo 1000 800 1300 -20 LOEFFICIENT RANGES ALLOWABLE INCREASE ALLOWABLE DECRET 200 199 25.000031 IWANITY (60) 133.333328 33333374 313.333344 Right hand side Ranges | ALLOWABLE ALLOWABLE INCREASE DECREASE Row Current RH 300 २५०० 150 Soo a mohon 1000 900 400 (oo 100 2400 300 4000 » Let us consider only 3000 hours of skilled were available, since the "Allowable decrease" in sixth constraint is 2400 and we decreasing right side of this constraint by only 1000,

Thus the current basis remains optimal. The optimal values of decision Variable remain unchanged That is XNP = XND = 800 , XNU = 800, XLV=0 & L=4000 Now, Ab = -1000 & from the LINDO output we can see that the shadow price for this constraint is $313.3333441 Hence the new optimal solution is given by max Z = 1360000 - 1000 X 313:333344 6 The allowable increase to = 10u6, 667. able increase for the second constraint is too. - Because the shadow price of this constraint es 133.333328, the cost for extra so computers is - 50 x 133.333328 = 6; 666.6664 And Also the contractor pays 5,000. 1. Thus company should hire the contractor © Since from the LINDO output we can see that the "AllowABLE INCREASE for the cost of the XLV that is number of VAXES poroduced in Los Angeles is 33. 333332. This profit for a VAX produced in Los Angeles have to increase by approximately $33.33 before company would want to produced Vanes in Los Angeles.

Since from the LINDO output we can see that the Shadow price for the labor constraint that is sixth Constraint is 313.3333440 This means that it company pays 20 for an extra labor this will earn a profit of 313.333344 Hence for an extra hour of the labor company lan Pays- 313333344 +20 = 333 3333 44.