Berrymore P/L has signed a contract to deliver 15 tonnes of fresh strawberries to Woolworths supermarkets in Brisbane. The strawberries will be sourced from the Berrymore P/L farm in Beerwah where 10 tonnes of premium-grade strawberries and 12 tonnes of second-grade strawberries are available. The contract specifies that the strawberries must be supplied in the form of 350g premium-grade pattern packs, 250g premium-grade tumble packs, and 500g second-grade tumble packs (a pattern pack is a punnet where the strawberries have been carefully packed in a pattern that pleases the eye, usually with all the calyxs pointing in the same direction; a tumble pack is a punnet that has simply been volume-filled). The contract specifies that at least 10% of the strawberries (i.e., at least 1.5 tonnes) must be in 350g packs, and no more than 40% of the strawberries can be second grade. The plastic containers (or 'clamshells') used for the 350g packs cost $0.10 each. The clamshells used for the 250g and 500g packs cost $0.08 and $0.15 each respectively. Berrymore P/L packers can pack 350g pattern packs at an average rate of 160 per hour. They can pack 250g and 500g tumble packs at average rates of 300 and 160 packs per hour respectively. All packers are paid at a rate of $12 per hour. There is no limit to the number packers who can be hired or the number of 350g and 500g clamshells that can be purchased. However, the number of 250g clamshells that can be purchased is limited to 10,000. (a) (6 marks) Let x, x2 and x3 denote the numbers of 350g premium-grade pattern packs, 250g premium-grade tumble packs, and 500g second-grade tumble packs supplied. Formulate a linear program to determine how many packs of each type should be sup- plied in order to fulfil the contract at minimum cost. (b) (5 marks) Ignore the fact that the solution values should be integers and use the Linear Programming module in Excel QM to solve the LP in part (a). Report the number of packs of each type that should be supplied. Also report the total cost of clamshells and the total cost of packing labour. (c) (4 marks) Determine the cost savings, if any, that would result if the number of 250g clamshells that could be purchased was increased to 11,000. (d) (4 marks) Suppose the price of 500g clamshells was increased to $0.20 and all other prices were left unchanged. How, if at all, would the optimal solution change? (e) (6 marks) Formulate and solve the dual to the LP in part (a).