Check answers to selected homework problems from the text and for Linear Programming notes

 

Ch. 2      2. a) Choose a person at random, have them taste the four blends and state their

preference.     b) Yes.  1/4 to each blend.  The classical method of equally  likely outcomes.     c) 0.20, 0.30, 0.35, 0.15

              4. a) 0.604     b) 0.093     c) 0.78     d) 85.8     e) $1.529

6. a) 0.31       b) 0.69

7. a) 0.4, 0.4, 0.6       b) 0.8.  Yes       c) P(Ac) = 0.6  P(Cc) = 0.4   

    d) P(AÇBc) = 0.6       e) 0.8

              9. 0.72

            11. a) Yes.  0.8     b) 0.65

            12. a) 0.667     b) 0.8     c) No

            13. a) Example-P(FÇQ) = 0.218     b) A student will most likely indicate cost

                  or convenience as the first reason with a 0.511 probability.     c) 0.473

                  d) 0.386     e) The events are dependent.

            15. a) Example-P(18to34ÇYes) = 0.375     b) Example-46% are ages 18 to 34    

     c) 0.15     d) 0.185     e) 0.120     f) 0.567     g) Probability of no health              
     insurance coverage is 0.15.  A higher probability exists for the younger group: 

approximately 18.5% of the age group.  For the 35 and over group it is approximately 12%.  Of the no insurance group, more are in

     the 18 to 34 age group:  approximately 57%.

           18. a) 0.25,  0.40,  0.10     b) 0.25     c) The program does not help.   Independent

           19. a) 0.009     b) 0.101

           21. a) 0.50     b) 0.75     c) 0.652

           23. a) 0.70     b) The revised probabilities are 0.3125, 0.50, and 0.1875.  Probabilities now favor medium-quality oil rather than high-quality oil.             
                 0.8125

           24. 0.359

 

Ch. 3     2. a) 0.05     b) 0.70     c) 0.40

             4. a) 6     b) 4.5     c) 2.121

             7. a) 445        b) –1,250         c) 9475     97.34

             8. a) 145 and 140.  Medium   b) 2725 and 12,400.  Medium

           21. a) 0.2967     b) 0.4418     c) 0.33     d) 0.5910     e) 0.8849    f) 0.2389

           23. a) - 0.80     b) 1.66     c) 0.26     d) 2.56     e) - 0.5

           24. a) 0.3830     b) 0.1056     c) 0.0062     d) 0.1587

           25. a) 0.1112     b) 0.1112     c) 40.496

           26. a) 0.7745     b) 36.32     c) Approximately 19%

           27. a) 0.0475     b) 0.0099     c) 0.7938     d) 796.75

 

Ch. 4     2. a) Optimistic-d1     Conservative-d3          Minimax regret-d3

                  c) Optimistic-d1     Conservative-d2, d3    Minimax regret-d2

              6. a) Pharmaceuticals, 3.4%     b) Financial, 4.6%

              9. a) The decision is the choice of the service type, the chance event is the level  

                  of demand, and the consequence is the amount of quarterly profit. 

                  Two decision alternatives-full price and discount service

                  Two outcomes for  the chance event-strong demand and weak demand

                  b) Optimistic-full price service       Conservative-discount service      

                  Minimax regret-discount service

                  c) Discount service            d) Full price service 

            13. a) The decision is the choice of the type of grape, the chance event is the

                  demand for the wine, and the consequence is the expected annual profit

                  contribution.

                  Three decision alternatives-plant Chardonnay, plant Riesling, and plant both

                  Four chance outcomes-week demand for Chardonnay and week

                 demand for Riesling (WW), and the rest are,WS, SW, and SS

                 c) Plant Chardonnay grapes only

                 d) Plant both grapes

           14. a) If s1 then d1, if s2 then d1 or d2, if s3 then d2     b) 192.5     c) d1.

                182.5      d) 10

           16. The optimal decision strategy will depend on the cost of the market research.

           17. a) Start the R&D project , and if it successful, build the facility, $10 million    

                 b) $25 million or more     

           19. b) Sell.     $100,000     c) $25,000     d) If favorable, produce, if unfavorable,      

                 sell     e) $1035     f) No.     $1035     g) No agency, sell the pilot          

            21. a) Purchase, $200,000      b) If high resistance, do not purchase, if low

                 resistance purchase     c) Yes, $30,400 > $10,000.  $30,400

            24. a) P(C) = 0.695     P(O) = 0.215     P(R) =0.09

                  P(S1|C) = 0.978     P(S1|O) = 0.791     P(S1|R) = 0

                  P(S2|C) = 0.022     P(S2|O) = 0.209     P(S2|R) = 1

                  c) Check the weather, take the expressway unless there is rain.  If rain, Queen

                  City Avenue.  Expected time 26.655 minutes

            25. b) EVPI = $9000     c) 0.355     d) Conduct the study if the cost is less

                  than $3650.  If it indicates a favorable condition, manufacture the component.

                  e) $3650     f) 40.6%

 

Ch. 7    11. A = 100, B = 50, and Max z = 750

            12. a) A = 3, B = 1.5       b) A = 0, B = 3       c) (0,0), (4,0), (3,1.5), and (0,3)  

            20. b) A = 3.429, B = 3.429, and Max z = 17.145        c) s1 = 2.858,  s2 = 0,   

                  s3 = 1.429,  s4 = 0   

            26. x1 = 666.667, x2 = 333.333 and Max z = 60,000             

            33. a) A = 3, B = 1 and Min z = 5          b) s1 = 14 (slack),  s2 = 0 (surplus), 

                  s3 = 10.5 (slack), s4= 0 (surplus)         c) A = 6, B = 2, and Max z = 34     

            39. a) x1 = 4000, x2 = 10,000      b) 60,000        c) Everything should be   

                   invested in the stock fund.

            40. x1 = 30, x2 = 25, and Min z = 55

            52. x1 = 384, x2 = 80

            53. x1 = 50 hours, x2 = 30 hours  

 

Linear Programming:  Sensitivity Analysis and Interpretation of Solution Using Excel Solver

              1. a) U = 800     H = 1200     $8400     b) 1 and 2.  All funds have been invested         

                  and the maximum allowable risk has been reached     c) 0.09,  1.33,   0.          

                  For example, increasing the available funds by $1, will increase the annual          

                  return by $0.09     d) No, optimal solution has only 800 shares of U.S. Oil 

                 Stock

             2. a) More than $7     b) More than $3.5     c) None

             3. a) S = 4000   M = 10,000   62,000     b) Surplus.  Investment in money market funds exceeds the minimum requirement by $7000.     c) For S--  3.75
                 to No Upper Limit, for M--No Lower Limit to 6.4     d) $60,000     e) 5%     f) -0.057 risk units.  A one-dollar increase in the available funds will
                 reduce the total risk by 0.057, a one-dollar decrease in the available funds will increase the total risk by 0.057 units.   

                  g) 5.7%       h) The optimal solution does not change.  67,000

             4. a) x1 = 7.297   x2 = 0.000   x3 = 1.892     139.730     b) 2 and 3     c) 0.000,    

                 3.405,   4.432.  No improvement can be made by increasing the RHS’s

                 d) 3.   From 20 to 19

            5. a) A = 1000   C = 200   H = 0     b) 2 and 4     c) Constraints 1-3 slack,                         

                 for example, 4000 minutes of unused cutting and dying time.  Constraint 4

                 surplus.  Only the minimum number of All-Pro model is produced.

                 d) Yes.  0.20 < 0.33     e) -2.   An increase of one unit in the requirement will

                 cause a $2 decrease in profit contribution     f) Does not change.  $4200

            6. a) x1 = 25   x2 = 0   x3 = 25     $1250     b) MI--31.25 hours   MII--37.5                                 

                 hours     c) 12.5     d) Yes.   x1 = 24   x2 = 8   x3 = 16     $1440

 

Ch. 9   1. a)  x1 = 4   x2 = 14   x3 = 10       Budget    $8000   $4,200   $6,000     

                Total audience = 1,052,000       b) 5,130

            3. x1 = $630,000   x2 = $170,000   x3 = $460,000   x4 = $140,000   x5 = $600,000

                 Annual return = $188,800

            4. a) x1 = 500  x2 = 300   x3 = 200       b)  $0.55  
                c) Surplus for aroma, s1 = 0, therefore aroma rating = 75.  Surplus for taste,

                s2 = 4400, therefore taste rating = 80 + 4400/1000 = 84.4       d) $0.60

            5. x1 = 4   x2 = 4   x3 = 2

            8. x1 = 3   x2 = 3   x3 = 7   x4 = 0   x5 = 4   x6 = 2

          11. x11 = 600   x12 = 400   x13 = 0   x21 = 0   x22 = 0   x31 = 800 

                Total cost = $20,400

          13. Contribution to profit = $3233.75

          15. x11 = gallons of crude 1 used to produce regular = 266,667   x12 = gallons of     

     crude 1 used to produce high-octane = 333,333   x21 = gallons of crude 2 used  

     to produce regular = 533,333   x22 = gallons of crude 2 used to produce high-

     octane = 166,667

          17. a)  FM = 5000   FP = 0   SM = 2692   SP = 7308   StM = 0   StP = 5000      

                b) Total cost = $368,076.91       c) C--350 hours, M--259.62 hours, and

                S--375 hours          

                d) Nothing        e) Yes.   For a purchase price below $47.423, the solution

                 improves

          18. a) x1 = 5   x2 = 2   x3 = 3       b) x1 = 7.333   x2 = 0   x3 = 0

 

Ch.10  3. b) Total cost = $24,800

           5. c) Total cost = $11,900       d) With Southern Gas for the additional 100 units.

7. b) Total billing = $57,925       c) New Total Billing = $57,525

8. Total profit = $282,000

           9. Total cost = $37,810

         10. Total cost = $3990

         12. b) Total time = 64 days

         13. b) Considering Burton has saved 2 days.       c) Ellis

         16. Total profit = $61,000

         19. Total time = 590 hrs

 

Ch.12  4. a) A-D-G     b) No, the critical path activities require 15 months

                 to complete

            8. b) B, C, E, F, H       d) Yes.  Project completion time is 49 weeks.

          12. a) A, D, H, I       b) 25.666 days       c) 0.2578  

          15. c) A, B, G, H, I.   15 weeks     d) T = 17.393 weeks

          24. c) A, B, C, F   31 weeks      f) $112,500

 

 Ch.13  4. a) 1095.445     b) 240     c) 22.822 days     d) Holding cost = $273.861,  

                 Ordering cost = $273.861

            8. Approximately 12,   5 classes per year,    $225,200

          10. 1414.214,   28.284 days,   7.071 days

          11. a) 1788.854     b) 1333.333     c) 894.427     d) 826.898     EOQ--Q* = 800

                1) Q* for  Production Lot Size is always greater than the EOQ Q* for the same

                D, Co, and Ch values.   2) As P increases, Q* decreases, but remains greater

                than the EOQ Q*.

          13. a) 1078.119     b) 6.678     c) 37.435     d) 10.781     e) 767.621  

                 f) 2003.49     g) 432

          15. a) 1148.913     b) 104.447     c) 1044.466     d) 23.936     e) $522.224

          16. 240,   r = dm - S,   r = 240 - 104.447 = 135.553.  The reorder point will be   

                smaller if backorders are allowed.

          20. 100 units.