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Hospital Systems, Inc (HSI)

The emergency room at HSI serves patients who arrive according to a Poisson distribution at the rate of 9 per hour.  Treatment takes an average of 6 minutes and the treatment times can be considered to follow an exponential distribution.  What is the

(a)  minimum number of doctors required so that at least 70% of the arriving patients can receive treatment immediately?

(b)  minimum number of doctors required so that the average time a patient waits for treatment is no more than 30 minutes as advertised?  No more than 15 minutes?

Broke & Down Machine Shop

Broke & Down has several hundred machines in its large manufacturing facility as it uses the PWP concept (plant-within-a-plant).  Ray Pear, Manager of Maintenance and Chief Mechanic for the Broke & Down Machine Shop, says that broken machines arrive at the repair facility according to a Poisson pattern at the average rate of 2 per 8-hour day.  Ray likes to remind his boss, Mr. I. M. Broke, that since he is such a master mechanic he is the only person needed to repair the broken machines, and he can repair a machine in just 2 hours on average.  Furthermore, the repair time follows an exponential distribution, and Ray repairs the machines in the order in which they break down.  Assuming that the machine shop operates 8 hours per day:

(a)  if a machine breaks down, what is the probability that Ray can begin work on it immediately?

(b)  on average, how long is a broken machine out of service?

(c)  on average, how many machines are awaiting repair?

(d) Mr. Churchill Down, co-owner of the machine shop is concerned that too much productive time is being lost due to machine break downs.  He agrees that Ray is a super mechanic but also thinks that Ray needs help so the shop would have less downtime waiting for repairs.  Churchill knows of Ray’s son Desmond, often called Des for short.  Des is also a master mechanic and Churchill has been considering hiring Desmond for some time if it is economically feasible.  Master mechanics are paid $20 per hour, and the machines produce revenue at $60 per hour when in service.  Should Des Pear be hired?

(e)  I. M. Broke is not too keen on the idea of hiring Des Pear as it has a flavor of nepotism.  I. M. suggests that to help Ray they could invest in more sophisticated diagnostic equipment and repair tools that would reduce Ray’s average repair time to 1 hour and 20 minutes (i.e. 4/3 hours).  If these more sophisticated tools and equipment cost Broke & Down $20 per hour to operate, and Ray is paid $20 per hour, should the investment be made?  Again, the machines produce revenue at the rate of $60 per hour when in service.

(f)  Anna Last, Director of Corporate Analytics for Broke & Down suggests that the use of the more sophisticated equipment (part e) not only shortens the average repair time but also the associated variability and changes the distribution of the repair time so it is no longer exponential.  From discussions with the manufacturer of the equipment, it is estimated that the standard deviation of the repair time is 1 hour.  Does this change what I.M. should do?  Show/explain.

City Cab, Inc.

City Cab, Inc., uses two dispatchers to handle requests for service and to dispatch the cabs.  The telephone calls that are made to City Cab use a common telephone number.  When both dispatchers are busy, the caller hears a busy signal; no waiting is allowed.  Callers who receive a busy signal can call back later or call another cab company for service.  Assume that the arrival of calls follows a Poisson distribution, with a mean of 40 calls per hour, and that the call handling time follows an exponential probability distribution with a mean service time of 2 minutes.  Based on this information, answer the following questions.

What percentage of the time are both dispatchers idle?

What percentage of the time are both dispatchers busy?

What is the probability that a caller will receive a busy signal if 2, 3, or 4 dispatchers are used?

If management wants no more than 12% of the callers to receive a busy signal, how many dispatchers should be used?

Suppose the service time distribution is not exponential, it follows some other distribution such as a normal, but the mean service time remains at 2 minutes.  Does this make any difference in the model used and in the results?  Explain.

Nooner Appliance

1.   Nooner Appliance Producers (NAP), a small appliance manufacturing company that specializes in clocks, must decide what types and quantities of output to manufacture for each week’s sale.  Currently Nooner makes only two kinds of clocks, regular clocks and alarm clocks, from which the product mix is selected.  Next week’s product mix can only be produced with the labor, facilities, and parts currently on hand.  These supplies are as follows:

Number of labor hours                  1,600

Number of processing hours         1,800

Number of alarm assemblies            350

      The resources are related to the two alternative manufactured outputs, regular clocks and alarm clocks, in the following way:  each regular clock produced requires 2 hours of labor and 6 hours of processing, while each alarm clock produced requires 4 hours of labor and 2 hours of processing.  The profit per unit for regular clocks is $3.00 while the company makes $8 per unit for alarm clocks.  Additionally, at least 300 clocks in total must be produced.  How many of each type of clock should Nooner produce to maximize profit?  The LP structure and solution are shown below. 

LINEAR PROGRAMMING PROBLEM:  Nooner Appliance Producers (NAP)

MAX 3X1+8X2

     S.T.

       1)  2X1+4X2<1600

       2)  6X1+2X2<1800

       3)  1X2<350

       4)  1X1+1X2>300

OPTIMAL SOLUTION

Objective Function Value =        3100.000

      Variable             Value             Reduced Costs  

   ————–     —————      ——————

         X1                   100.000                   0.000

         X2                   350.000                   0.000

     Constraint        Slack/Surplus           Dual Prices   

   ————–     —————      ——————

         1                      0.000                   1.500

         2                    500.000                   0.000

         3                      0.000                   2.000

         4                    150.000                   0.000

OBJECTIVE COEFFICIENT RANGES

   Variable       Lower Limit       Current Value     Upper Limit

 ————   —————    —————  —————

      X1                  0.000              3.000            4.000

      X2                  6.000              8.000   No Upper Limit

RIGHT HAND SIDE RANGES

  Constraint      Lower Limit       Current Value     Upper Limit

 ————   —————    —————  —————

       1               1400.000           1600.000         1766.667

       2               1300.000           1800.000   No Upper Limit

       3                300.000            350.000          400.000

       4         No Lower Limit            300.000          450.000

Given the Nooner  scenario:

(a)     Solve using the graphical solution procedure and identify all extreme points of the feasible region. 

(b)     How much of each clock should be produced and what is the associated profit??

Using the output from the Management Scientist, answer the remaining questions.

(c)     What are the values and interpretations of all slack and surplus variables?

(d)    Determine (compute manually) and interpret the range of optimality for the objective function coefficients.

(e)     Interpret each of the shadow prices.

(f)     If the profit on an alarm clock decreased to $6 per unit, what parts of the optimal solution would change and how, and what parts would not change?  Explain your rationale.

(g)     Suppose Nooner can get an additional 100 alarm assemblies at a premium price of $1 more than the current price.  Should Nooner take advantage of this offer?   Explain your rationale.

Monroe County Sheriff

The Monroe County Sheriff’s Department schedules police officers for 8-hour shifts.  The beginning times for the shifts are 8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m.  An officer beginning a shift at one of these times works for the next eight hours.  During normal weekday operations, the number of officers needed varies depending on the time of the day.  The department staffing guidelines require the following minimum number of officers on duty:

            Time of Day                           Minimum Officers on Duty

            8:00 a.m. – noon                                          5

            Noon – 4:00 p.m.                                         6

            4:00 p.m. – 8:00 p.m.                                 10 

            8:00 p.m. – midnight                                   7

            Midnight – 4:00 a.m.                                   4

            4:00 a.m. – 8:00 a.m.                                   6

         How many police officers should be scheduled to begin the 8-hour shifts at each of the six times in order to minimize the total number of officers required?  Structure this linear programming problem (decision variables, objective function, and constraints).  You do not need to solve this problem, just structure it.

Chase Shipping

Sean Skippers is in charge of load planning for cargo ships for Chase Shipping Inc. (CSI).  Sean is preparing a load plan for a CSI freighter destined for China.  An agricultural commodities dealer would like to transport three commodity products aboard this ship.  The shipping characteristics of these commodities are shown below.

                                          Amount                 Volume per Ton          Profit per Ton

         Commodity           Available (tons)           (cubic feet)                        ($)          

                  1                         4800                            40                                70

                  2                         2500                            25                                50

                  3                         2900                            60                                80

Sean can elect to load any and/or all of the available commodities.  However, the ship has three cargo holds with the following capacity restrictions:

         Cargo Hold           Weight Capacity (tons)        Volume Capacity (cubic feet)

         Forward                         3000                                        145,000

         Center                            6000                                        180,000

         Rear                                4000                                        155,000

More than one type of commodity can be placed in the same cargo hold.  However, because of balance considerations, the weight in the forward cargo hold must be within + 10% of the weight in the rear cargo hold.  Structure as a linear programming problem (decision variables, objective function, and constraints).  You do not need to solve this problem, just structure it.

EPC

The executives of Electronic Products Company (EPC) have to decide which of two products to introduce, A or B.  Each has been in the R&D process for about 4 years and the company feels that they are now ready for introduction.  Additionally, considerable market research has been done to assses the market for each product.  Each of these new products is expected to have a five year life before renewal or phase-out.

Product A is essentially a lower risk proposition.  For this product, market research has made estimates of likely market sizes and profitability as follows:  Sales may be high, with a resulting profit of $10 million, medium with a net profit of $5 million, or low, in which case the company just breaks even.  The probabilities for these outcomes are respectively, 0.50, 0.20, and 0.30.

Product B is a little more risky in that the plant size has to be matched to the demand for the product.  There are two key decisions to be made which deal with plant size, namely, the initial size and the final size.  The possibilities for the initial and final size are either small or large as demand could be either low or high initially.  Follow-on demand could also be low or high but it is possible that demand changes so that a low initial demand could eventually become high and a high initial demand could eventually sharply drop to a low volume.  The initial plant size will be for the first two years, at which point a decision has to be made on whether to expand the plant if demand exceeds capacity.  The additional cost of a plant expansion is $1.0 million.  If a large plant is initially built, then there is no need to expand but it is difficult to downsize the plant.  If the plant is initially too small to meet demand, lost sales and unhappy customers will be the result but the plant can be expanded after two years to alleviate the situation.  However, if the plant has too much excess capacity, then profitability is greatly reduced and some losses become a possibility.  The probability of initial sales being low or high is 0.4 and 0.6 respectively.  The likelihood of follow-on sales being low or high is dependent somewhat on initial sales and is shown below along with the corresponding profitability.

        Probability Analysis for Product B

Initial Sales        Follow-on Sales (Years 3-5)

(First 2 Years)       Low        High      

               Low                    0.60        0.40

               High                   0.30        0.70

The Corporate Financial Analysis Division, working with Sales and Marketing, has come up with the following profitability estimates for Product B based on the possible scenarios.  The total profit for the 5-year planning period would be the profit made in the first two years plus that made in years 3-5. The estimated cost of the plant expansion ($1.0 million) is not included in these figures.

                                                Profit Table ($ in Millions)

                                                         Years 1-2                                    Years 3-5

                                          Low Demand     High Demand    Low Demand     High Demand

 Small Plant               1.0                    1.5                      1.5                    2.0

  Large Plant             -1.0                     5.0                      -2.0                  10.0                      

(a)     Which product should EPC introduce?  (show work/analysis)

(b)     Create a risk profile for EPC. 

(c)     How risky is EPC’s best solution?

(d)    Suppose the CEO of EPC has reservations about your analysis of his key decision on new product introduction.  He wants to know if there is a way to account for his view of risk and how he would make some of the decisions other than just EMV.  Explain what you would have to do to satisfy his fears.