4.1. The three plants that process and pack potatoes are coordinated from a central office that facilitates delivery of sack of potatoes to the mentioned towns/cities. The cost of transporting a sack of potatoes per unit (i.e. 7kg potatoes sack) from each plant to each town/city is shown below. Moreover, the numbers of units of potatoes available from each plant as well as the number of requested units of potatoes at each city/town are provided in the table below. Bloemfontein Bethlehem Qwaqwa Plant 1 R8 R9 R11 Ladysmith R16 Supply 50 Plant 2 R12 R7 R5 R8 80 Plant 3 R14 R10 R6 R7 120 Demand 90 70 40 50 250/250 Formulate the linear programming model (i.e. objective function) and the corresponding constraints including the lower bounds of the decision variables. (5) How many shipments must be made from each plant to each different town/city so that the cost can be minimized (5) 4.2. Suppose that as a supervisor, you need to optimally assign drivers to different locations to deliver purchased stock to factories located in different provinces. The hours required for each job on each machine are given by: Destination 1 Destination 2 Destination 3 Destination 4 Driver 1 10 12 9 14 Driver 2 14 13 12 16 Driver 3 16 15 12 18 Driver 4 13 12 11 16 11 1 1 4.2. Suppose that as a supervisor, you need to optimally assign drivers to different locations to deliver purchased stock to factories located in different provinces. The hours required for each job on each machine are given by: Destination 1 Destination 2 Destination 3 Destination 4 Driver 1 10 Driver 2 14 Driver 3 Driver 4 16 13 12 13 15 12 9 12 12 11 14 16 18 16 Assuming that as a supervisor you want to allocate drivers such that the total time is minimized, what is the optimal solution? (Write the linear programming model in full before using some software to solve this question) (10) 4.3. Appliances Inc. manufactures industrial fridges and its manufacturing plants are based in Mozambique and Madagascar. Suppose that these industrial fridges cannot be shipped directly from plant to factories that need them. Hence, they need to be shipped first to regional distributors located in Durban (ND), Port Elizabeth (PE) and Cape Town (CT). Thereafter, they can be shipped to Johannesburg (JHB), Polokwane (PL), Pretoria (PTA) and Kimberley (KY). The transportation cost are given by: ND PE CT JHB PL PTA KY Supply Mozambique R500 R800 R1000 Madagascar R600 R850 R1100 350 400 Durban Port Elizabeth Cape Town Demand R400 R550 R750 R890 R1000 R1200 250 90 R450 R800 R800 R600 R1050 R650 250 80 Using, for instance MS Word, draw the "Network Representation' of this transportation problem, include the costs, demands, supplies and nodes on the figure. (5) Formulate the linear programming model (i.e. objective function) and the corresponding constraints including the lower bounds of the decision variables and the nodes. (5)