# Calculate the expected project length, variance of the critical path, standard deviation of the critical path

What is the probability the activities will not be completed within the due time of nine and a half years?

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Question 1:

A company producing hand-held games is considering introducing a new game called ‘Winging’ into the competitive market.  The company identifies the fixed cost to be \$34,000 and the variable cost per game produced to be \$6.  The selling price per unit is set at \$14.  What is the break-even point?

Question 2:

A company that specialises in a particular brand of vinyl wallpapers wishes to outsell other stores next year in the total number of rolls of this brand.  The estimated demand function is as follows:

Number of rolls of wallpaper sold = 20 x dollars spent on advertising,  + 6.8 x dollars spent on in-store displays, + 12 x dollars invested in on-hand wallpaper inventory, – 65000 x percentage markup taken above wholesale cost of a roll.

The store budgets a total of \$17,000 for advertising, in-store displays, and on-hand inventory of the wallpaper for next year.  It decides it must spend at least \$3,000 on advertising; in addition, at least 5% of the amount invested in on-hand inventory should be devoted to displays.  Markups on the wallpaper seen at other local stores range from 20% to 45%.  The company decides that its markup had best be in this range as well.

a) Manually show the decision variables, objective function and constraints. Method is shown on lecture slides.

Question 3:

The advertising director for a hardware company with a chain of four retail stores is considering two media possibilities:  One plan is for a series of half-page advertisements in the Sunday newspaper, and the other is for advertising time on television.  The stores are expanding their do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city and 60% in the suburban areas.  The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in suburban homes.  The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad.  The cost of a half-page advertisement is \$925; a television spot costs \$2,000.  The advertising director would like to select the least costly advertising strategy that would meet desired exposure levels.

a)Show the decision variables

b)Show the objective function

c) Show and name the constraints in standard form (suitable for using Excel)*

*Constraints must be shown in ‘standard form’ as required by the question.  You also need to do this in the exam.  Note X’s are on the LHS of the equation and numbers are on the RHS.

d)  Imagine you have been hired by the advertising director to solve the problem by using Excel.  Do so and write a conclusion of your findings for the director.

Question 4:

A farm in the southwest of Western Australia comprises 300 hectares and has available water of 250,000 litres per season, and costs \$100 per planted hectare per season to operate.  Wheat, corn and barley can be grown on the property.  The following information is pertinent:

 Crop Selling Price (\$/tonne) Water Requirement (litres/hectare) Yield (tonnes/hectare) Wheat 25 1,000 20 Corn 20 1,500 15 Barley 15 1,800 40

The farm forecasts a maximum demand of wheat of 5,000 tonnes; for corn, 6,000 tonnes, and for barley, 4,000 tonnes.  Also, the farm has a contract to supply at least 2,500 tonnes of wheat to a particular customer.  The objective is to determine the number of hectares of each crop that should be planted on the farm in order to maximise profit?  You are not required to solve this problem.  Simply show:

a) Objective Function

b) The constraints on water and on land.

Question 5:

A company manufactures office desks at two locations:  O’Connor and Murton. The firm distributes the desks through regional warehouses located in Auburn, Stirling and Australind.  Estimates of the monthly supplies available at each factory and the monthly office desk demands at each of the three warehouses are shown in the following table:-

 Factory Supply Capacity (# desks per month) O’Connor 130 Murton 120

The desks are shipped to three warehouses.

 Warehouse Number of Desks Required per month Auburn 60 Stirling 120 Australind 30

To ship office desks from a factory to a warehouse involves transportation costs (dollar costs per desk shipped).

 To From Auburn Stirling Australind O’Connor 2 1 3 Murton 3 1 4

All costs are in dollars

a)Solve this problem using Excel.  Write down the numbers desks shipped from each factory to each warehouse and the total cost of shipping involved (i.e. form a worded conclusion).

b)  Is this a balanced or unbalanced transportation problem?  Show the signs for the demand and supply constraints

Question 6:

a)Determine the bicycle pathway that will require the minimum amount of construction to connect all the attractive areas of a park.  Also show the total distance (arcs are measured in metres).

b)  A large department store chain has stores in seven country areas.  What route do you recommend for the driver of the delivery truck at the warehouse (represented by node 1) to take to each of the seven suburbs (represented by nodes 2 to 8)?  The network of roads is shown in the diagram below (the diagram is not drawn to scale but the number on each road gives the road distances in kilometres between the junctions).  Use a relevant process or algorithm to find the shortest route and state the distance.

c) The following network shows a telecommunications network connecting various trunk terminals to relay telephone calls. Equipment automatically routes a particular call over the first clear path found between the trunks connected to the respective telephones.  The physical configuration of the system for determining how many calls can be made between any two connected trucks must be set in advance.  How well a particular telephone system works can be quantified in terms of the maximum number of calls it can accommodate.  Each call routed between two trunks can be treated as a flow.   Determine the maximum flow that can be achieved for the network from the source at A to the sink at L by completing Table 1 below.

Question 7:

The following time estimates (in months) for ten activities required for the planning of ahuge hospital complex are shown in the table below:

a)In the headings of the last two columns show the formulas for time and variance.  Show the times and variances for activities A, B, C and D.  The times and variances have been shown for all other activities. Obey the rules of rounding (if the last digit is 5 or more, round the previous digit up.  If the last digit is 4 or less, keep the previous digit as is.  This applies to all your problems).

b)Draw a CPM/Pert diagram of the activities using the following convention.  You may use PAINT to copy your network onto your word document or attach a separate sheet to your assignment.  Show ‘start’ and ‘finish’ nodes:

c) Calculate the expected project length, variance of the critical path, standard deviation of the critical path and the activities along the critical path.

d) What is the probability the activities will not be completed within the due time of nine and a half years? Is 9.5 years an unrealistic estimate?

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