AREC 214 Winter Term 2016 Assignment #3 Due Date: Friday, February 12 @ noon (12:00 p.m.) 1. (15 points) Use appropriate matrix methods to solve the following system of equations: 2. (25 points) An agricultural chemicals manufacturer has facilities in Winnipeg and Calgary. The business has contracts to ship chemical pesticides to Regina, Saskatoon and Edmonton. There are 90 tonnes of pesticides in storage in Calgary, and 120 tonnes in storage in Winnipeg. The contracts require 100 tonnes to be shipped to Regina, 50 tonnes to Saskatoon and 60 tonnes to Edmonton. A complicating factor in deciding upon a shipping pattern is the fact that the manager has budgeted $5000 for shipping costs in the next accounting period and cannot afford to spend more than that. If less than $5000 is spent, however, the manager risks having his/her budget cut. The per tonne costs of shipping the chemicals are shown in the following table: Destination Regina Edmonton Calgary Source Saskatoon $10/tonne $20/tonne $5/tonne Winnipeg $30/tonne $30/tonne $40/tonne For example, the cost of shipping from Calgary to Saskatoon is $20 per tonne while the cost of shipping from Winnipeg to Edmonton is $40 per tonne. Use matrix methods to determine the shipping pattern (i.e., the number of units shipped from each supply location to each demand location) that will allow the manager to get the products where they are needed, while meeting the budget for shipping costs. -1- 3. (15 points) The manager of a commercial forest is pondering how to harvest a particular tract. The forest tract of interest is composed of two separate and distinct compartments; A and B. Compartment A consists of 500 acres and compartment B consists of 200 acres. Two logging techniques may be used; tractor logging or hi-lead logging. Compartment B, however, is too steep for tractor logging. As well, the manager may choose not to log portions of each compartment. Estimates of the effects of each logging technique on timber production, stream sedimentation, and fire danger are shown in the following table. Compartment A B No Logging Hi-Lead No Logging Tractor Hi-Lead Timber (Mbft/acre) 0.57 0.57 Sediment (yd3/acre) 7.84 7.74 0.50 13.30 0.80 Fire Danger (Index/acre) 64.70 33.10 2.52 82.80 4.20 Impact 0.34 Suppose that the manager wishes to harvest these two tracts in such a way that the following requirements are met; timber output should be 200 Mbft (thousand board feet), sedimentation should be 4000 cubic yards (yd3), and fire danger should be 25000 Use matrix methods to determine an appropriate harvesting schedule; that is, the allocation of land to alternative uses, including \"no logging\" as a potential use. -2- 4. (25 points) A cannery produces six canned fruit products; mixed fruit, fruit cocktail, fruit delight, a premium fruit dessert, pure peaches, and pure pears. The main ingredients for all of these products are peaches and pears and each of the products goes through a processing line that mixes, cans and packages them into \"lots\" of products. The weekly capacity of the processing line is 750 lots in total for the six products. The cannery manager wishes to maintain production levels so that the processing line is operating at full capacity. The company has a supply of peaches of 13000 cases per week and 12000 cases of pears per week, and wants to use up the full amount available. The \"per lot\" requirements of each product for peaches and pears is specified in the table below. For example, a mixed fruit lot will consist of 20 cases of peaches and 10 cases of pears whereas a lot of pure peaches will consist of 30 cases of peaches and zero cases of pears. The net return from producing and selling one lot of mixed fruit is $100, for fruit cocktail it is $60, for fruit delight $90, for the premium fruit dessert $150, for peaches $70 and for pears $50. The company has a target net return level of $75,000 per week which needs to be exactly met. Finally, to take in account expected demand for the various products the total production of mixed fruit, fruit cocktail, fruit delight and premium fruit dessert combined, should be four times as great as the total production of pure peaches and pure pears combined. Requirement Mixed Fruit Fruit Cocktail Fruit Delight Premium Fruit Pure Peaches Pure Pears Peaches (cases/lot) 20 10 15 20 30 0 Pears (cases/lot) 10 20 15 20 0 30 Use matrix methods to determine a production plan (i.e., the number of lots of each product to be produced) that will allow the manager to meet the required target return, while using all resources (i.e., peaches, pears and processing line capacity) and meeting the demand expectations. -3- 5. (20 points) A commercial bison ranch has a capacity of 3,500 animals. The herd is divided into four age classes; calves, young adults, mature adults and old adults. From one time period to the next, calves become young adults, young adults become mature adults, and mature adults become old adults. There is a certain level of death loss from natural causes; 10 percent of young animals die before becoming young adults, and 10 percent of young adults die before becoming mature adults. The herd is replenished through reproduction. Young adults and mature adults reproduce between each time period. The rate of reproduction for these age groups is 75 percent; that is, there are 0.75 young animals produced per adult animal. A certain amount of \"culling\" takes place in the bison herd at the beginning of each time period. These represent animals taken from the herd and sold to generate revenue. Culled animals come from the adult age classes. A certain number of young adults and mature adults may be culled. The herd manager has a contract that requires 1225 animals to be culled each year. Notes: S S S S Since culling takes place at the beginning of each time period, culled animals do not contribute to the replenishment of the herd (i.e., they do not reproduce). Death loss occurs between time periods. If not culled, old adults \"disappear\A commercial bison ranch has a capacity of 3,500 animals. The herd is divided into four age classes; calves, young adults, mature adults and old adults. From one time period to the next, calves become young adults, young adults become mature adults, and mature adults become old adults. There is a certain level of death loss from natural causes; 10 percent of young animals die before becoming young adults, and 10 percent of young adults die before becoming mature adults. The herd is replenished through reproduction. Young adults and mature adults reproduce between each time period. The rate of reproduction for these age groups is 75 percent; that is, there are 0.75 young animals produced per adult animal. A certain amount of \"culling\" takes place in the bison herd at the beginning of each time period. These represent animals taken from the herd and sold to generate revenue. Culled animals come from the adult age classes. A certain number of young adults and mature adults may be culled. The herd manager has a contract that requires 1225 animals to be culled each year. Notes: - Since culling takes place at the beginning of each time period, culled animals do not contribute to the replenishment of the herd (i.e., they do not reproduce). - Death loss occurs between time periods. - If not culled, old adults \"disappear\