Problem 1 Food Manufacturing. The final product sells for $150 per ton.

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1 Problem 1 Food Manufacturing A food is manufactured by refining raw oils and blending them together. The raw oils come in two categories, and can be bought for the following prices per ton: vegetable oils VEG 1 VEG 2 non-vegetable oils OIL 1 OIL 2 OIL 3 $ 110 $ 120 $ 130 $ 110 $ 115 The final product sells for $150 per ton. Vegetable oils and non-vegetable oils require different production lines for refining. In any month it is not possible to refine more than 200 tons of vegetable oils or more than 250 tons of non-vegetable oils. There is no loss of weight in the refining process and the cost of refining may be ignored. There is a technological restriction of hardness on the final product. In the units in which hardness is measured this must lie between 3 1

2 and 6. It is assumed that hardness blends linearly and that the hardness of the refined oils are: VEG VEG OIL OIL OIL What buying and manufacturing policy should the company pursue (each month) in order to maximise profit? 2

3 Problem 2 Investment Decision The superannuation fund, PYJAMAS, has $750 million to invest in three different investment opportunities named A, B and C. The fund expects to invest this money for a period of four years at the end of which a large sum of money is required for an already committed project. Projects, A and C need funds invested in the current year but project B only starts in the next year. The cash flows of each of these investments for a $1 investment are given in the table below. Option Current Year End of Year 1 End of Year 2 End of Year 3 End of Year 4 A ($1) $0.10 $0.50 $1 $0.25 B - ($1) $0.10 $0.50 $0.75 C ($1) $0.25 $0.25 $0.25 $0.75 The charter of the PYJAMAS fund stipulates that no more than 500 million be committed to any one of these investments. Investment C on the other hand needs only $200 million at most. 3

4 Any uninvested cash may be placed in a cash management account which earns an interest of 10% per annum. PYJAMAS must determine an investment strategy which makes sure that the maximum possible amount is available at the end of the four year period for its committed project. 4

5 Problem 3 Project Management The construction company HomeOrDoom specialises in building houses. They are known to be very efficient with their work and have a good reputation in the industry for their capacity to build houses in a very short time. A simplified list of activities for building a standard house is given in the Table 1 with the precedence requirements and expected duration for each activity. Activity Description Duration (days) Immediate Predecessors A Foundation & related 12 - work B Walls & Windows 9 A C Utility Connections 11 A D Roof construction 8 B, C E Tiling & Fittings 7 C F Finishing & Painting 6 D, E Table 1 : Project Activities, Durations and Predecessors An activity on arc (AoA) network diagram representing the above activities is given in Figure 1. The proper scheduling of the project 5

6 activities ensured that all HomeOrDoom houses are completed in 37 days. Recently, BuildOrDie, a competitor in the industry, has started offering houses to be built in 33 days and HomeOrDoom is losing market share. The management of HomeOrDoom have decided to compete by building a standard house in 29 days. They plan to do this by hiring additional resources at increased costs. The cost estimates for decreasing the duration of activities and the maximum number of days by which the duration of each activity can be decreased by are given in Table 2. 6

7 Activity Description Cost per day of reducing duration of activity ($) A Foundation & related work B Walls & Windows C Utility Connections D Roof construction E Tiling & Fittings F Finishing & Painting Table 2: Project Activity Crashing Costs and Limits Maximum possible reduction in the duration of the activity (days) The management of HomeOrDoom wishes to achieve their target completion time with the minimum possible additional cost. 7

8 Problem 4 Telco Network Design A telecommunication company wishes to build a network consisting of five nodes. Nodes are labelled 1 through 5 and are connected to each other as shown in the figure below. For each node pair of the network, the symmetric traffic flow requirement is known (see, the table below). These traffic flow requirements are satisfied by using the direct path between the two nodes concerned or by two-hop paths. The total traffic flow along a link cannot exceed the capacity provided in that link. The costs of providing one unit of transmission capacity for each link of the network are also given in the table below. # Node Pair Traffic Flow Capacity Cost # Node Pair Traffic Flow Capacity Cost

9 The company needs to design the network with the minimum possible capacity cost to satisfy the traffic flow requirements in case of any single link failure We shall only construct part of this large linear program. 9

10 Problem 5 Scheduling with job selection We are given n jobs with processing times p i and values v i, for i =1,, n. We have available m parallel identical machines. We have at most T units of time available on each machine. We also assume that the jobs cannot be split and that a machine can only do one job at a time Formulate an integer program whose solution would determine which of the n jobs are to be selected for processing and also allocate the jobs to the machines so that all processing is completed within T time units and the sum of the values of the jobs completed is maximized. Now change the objective to be the maximization of the minimum value of the jobs allocated to any machine 10

11 Problem 6 Clustering We are given n items to be placed in c mutually exclusive and exhaustive clusters. The degree of similarity between items i and j is specified by a non-negative distance d ij. The greater the similarity between items the smaller is the distance between them. We wish to cluster the items so as to minimize the sum of the distances between items in the same clusters. Formulate the above problem as a linear integer program. 11

Project Planning. Jesper Larsen. Department of Management Engineering Technical University of Denmark

Project Planning. Jesper Larsen. Department of Management Engineering Technical University of Denmark Project Planning jesla@man.dtu.dk Department of Management Engineering Technical University of Denmark 1 Project Management Project Management is a set of techniques that helps management manage large-scale