Tutorial 3 - DOC by Oe16Avx


									                                           TUTORIAL 3

Transportation Model

1.         Tri-Country Utilities, Inc, supplies natural gas to customers in a three-county area.
           The company purchases natural gas from two companies: Southern Gas and
           Northwest Gas. Demand forecasts for the coming winter season are Hamilton County,
           400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to
           provide the following quantities have been written: Southern Gas, 500 units; and
           Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon
           the location of the suppliers. The distribution costs per unit (in £’000) are as follows:

                     From                Hamilton               Butler              Clermont

                 Southern Gas               10                    20                    15
                 Northwest Gas              12                    15                    18

          a.     Prepare an appropriate initial tableau
          b.     Obtain an initial feasible solution using the least cost method
          c.      Proceed to obtain the optimal solution using the u, v method
          d.     Recent residential and industrial growth in Butler County has the potential for
                 increasing demand by as much as 100 units. Which supplier should Tri-Country
                 contract with to supply the additional capacity?

(ASW : Ch 7, Qn 5 – min - sensitivity)

Tutorial 3.doc
2.         Arnoff Enterprises manufactures the central processing unit (CPU) for a line of
           personal computers. The CPUs are manufactured in Seattle, Columbus and New
           York, and shipped to warehouses in Pittsburgh, Mobile, Denver, Los Angeles and
           Washington, D.C, for further distribution. The following transportation tableau shows
           the number of CPUs available at each plant and the number of CPUs required by each
           warehouse. The shipping costs (£ per unit) are also shown in each cell.

            Plant         Pittsburgh Mobile  Denver    Los                     Washingto       CPUs
                                                       Angeles                 n               Available
            Seattle              10       20        5        9                       10
            Columbus                  2          10           8           30          6
            New                       1          20           7           10          4
            York                                                                               8000
            Required      3000            5000         4000        6000        3000            21,000

           a.       Prepare an appropriate initial tableau
           b.       Obtain an initial feasible solution using the least cost method
           c.       Proceed using the u, v method to determine the amount that should be shipped
                    from each plant to each warehouse to minimise the total shipping cost.
          d.        The Pittsburgh warehouse has just increased its order by 1000 units, and Arnoff
                    has authorised the Columbus plant to increase its production by 1000 units.
                    Will this lead to an increase or decrease in total shipping costs? Solve for the
                    new optimal solution.

(ASW : Ch 7, Qn 6 - min - sensitivity)

3.         Klein Chemicals, Inc, produces a special oil-base material that is currently in short
           supply. Four of Klein’s customers have already placed orders that together exceed the
           combined capacity of Klein’s two plants. Klein’s management faces the problem of
           deciding hwo many units it should supply to each customers. Since the four customers
           are in different industries, different prices can be charged because of the various
           industry pricing structures. However, slightly different production costs at the two
           plants and varying transportation costs between the plants and customers make a “sell
           to the highest bidder” strategy unacceptable. After considering price, production costs,
           and transportation costs, Klein has established the following profit (£) per unit for each
           plant-customer alternative.

            Plant                D1                   D2               D3                 D4

            Clifton Springs 32                        34              32                  40
            Danville        34                        30              28                  38

Tutorial 3.doc
           The plant capacities and customer orders are as follows:
                   Plant Capacity             Distributor Orders
                       (units)                      (units)
               Clifton Springs 5000                D1 2000
                                                   D2 5000
               Danville          3000              D3 3000
                                                   D4 2000

           a. Show a network model and linear programming formulation.
           b. How many unit should each plant produce for each customer to maximise profits?
           c. Which customer demands will not be met?

(ASW: Ch7, Qn 8 – max – excess demand)

4.         Sound Electronics, Inc, produces a batter-operated tape recorder at plants located in
           Martinsville, North Carolina; Plymouth, New York; and Franklin, Missouri. The unit
           transportation cost for shipments from the three plants to distribution centres in
           Checago, Dallas and New York are as follows:

                      From              Chicago               Dallas                New York
                   Martinsville          1.45                  1.60                   1.40
                    Plymouth             1.10                  2.25                   0.60
                    Franklin             1.20                  1.20                   1.80

           After considering transportation costs, management has decided that under no
           circumstances will it use the Plymouth-Dallas route. The plant capacities and
           distributor orders for the next month are as follows:

                    Plant         Capacity                            Distributor      Orders
                                   (units)                                             (units)
                 Martinsville       400                                Chicago          400
                  Plymouth          600                                 Dallas          400
                  Franklin          300                               New York          400

           Because of different wage scales at the three plants, the unit production cost varies
           from plant to plant. Assuming the costs are £29.50 per unit at Martinsville, £31.20 per
           unit at Plymouth, and £30.35 per unit at Franklin, find the production and distribution
           plan that minimises total production and transportation costs.

(ASW : Ch 7, Qn 9 – min – unacceptable route – use TMS only)

Tutorial 3.doc
5.         Forbelt Corporation has a one-year contract to supply motors for all refrigerators
           produced by the Ice Age Corporation. Ice Age manufactures the refrigerators at four
           locations around the country : Boston, Dallas, Los Angeles and St Paul. Plans call for
           the following number (in thousands) of refrigerators to be produced at each location.

            Boston             50
            Dallas             70
            Los Angeles        60
            St Paul            80

           Forbelt has three plants that are capable of producing the motors. The plants and
           production capacities (in thousands) are

            Denver             100
            Atlanta            100
            Chicago            150

           Because of varying production and transportation costs, the profit that Forbelt earns on
           each lot of 1000 units depends on which plant produced the lot and which destination
           it was shipped to. The following table gives the accounting department estimates of
           the profit (£) per unit (shipments will be made in lots of 1000 units).

                                                             Shipped To
                 Produced At         Boston            Dallas       Los Angeles        St Paul
                   Denver               7               11               8               13
                   Atlanta             20               17              12               10
                   Chicago              8               18              13               16

           With profit maximisation as a criterion, Forbelt wants to determine how many motors
           should be produced at each plant and how many motors should be shipped from each
           plant to each destination.

           a.        Prepare an appropriate initial tableau
           b.        Obtain an initial feasible solution using the least cost method
           c.        Proceed to find the optimal solution using the u,v method.

(ASW : Ch 7, Qn 11 - max)

Tutorial 3.doc

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