midterm_practice_solution

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Midterm Exam Practice Solution - 1 - OpMgt 301: Principles of Operations Management Yong-Pin Zhou 1. Problem 10 in Chapter 19 of the textbook. Decision Variables: x 1 = number of containers of orange juice x 2 = number of containers of grapefruit juice x 3 = number of containers of pineapple juice x 4 = number of containers of All-in-One Parameters: Orange Juice Grapefruit Juice Pineapple Juice All-in-One Revenue per qt. $1.00 $.90 $.80 $1.10 Cost per qt. $.50 $.40 $.35 $.417 Profit per qt. $.50 $.50 $.45 $.683 Objective Function and Constraints: Maximize .50 x 1 + .50 x 2 + .45 x 3 + .683 x 4 s.t. Orange juice: 1 x 1 + .33 x 4 ≤ 1600 (qt.) Grapefr. juice: 1x 2 + .33 x 4 ≤ 1200 (qt.) Pineapple juice: 1x 3 + .33 x 4 ≤ 800 (qt.) Grapefr.: -.30 x 1 +.70 x 2 -.30 x 3 +.70 x 4 ≤ 0 Ratio: -5 x 1 +7 x 3 ≤ 0 Non-negativity: x 1 , x 2 , x 3 , x 4 ≥ 0 Note that originally the “Grapefr” constraint is written as “ x 2 +x 4 ≤ 0.3(x 1 +x 2+ x 3 +x 4 )” , which can be simplified to “ -.30 x 1 +.70 x 2 -.30 x 3 +.70 x 4 ≤ 0”. Similarly, the “Ratio” constraint is originally written as “ x 1 /x 3 ≥ 7/5”. But this is not linear as the left-hand-side is a division of two decision variables. You can simplify it to “x 1 ≥ 1.4x 3 ,” “5x 1 ≥ 7x 3 ,” or “-5x 1 +7x 3 ≤ 0.” 2. Problem 14 in Chapter 19 of the textbook. Decision Variables: x 1 = boxes of regular mix x 2 = " " deluxe " x 3 = " " cashews x 4 = " " raisins x 5 = " " caramels x 6 = " " chocolates Objective Function and Constraints: Maximize .80 x 1 + .90 x 2 + .70 x 3 + .60 x 4 + .50 x 5 + .75 x 6 s.t.

Midterm Exam Practice Solution - 2 - OpMgt 301: Principles of Operations Management Yong-Pin Zhou cashews: .25 x 1 +.50 x 2 +x 3 ≤ 120 (lb./day) raisins: .25 x 1 +x 4 ≤ 200 (lb./day) caramels: .25 x 1 +x 5 ≤ 100 (lb./day) chocolate: .25 x 1 +.50 x 2 +x 6 ≤ 160 (lb./day) Boxes regular: x 1 ≥ 20 deluxe: x 2 ≥ 20 cashews: x 3 ≥ 20 raisins: x 4 ≥ 20 caramels: x 5 ≥ 20 chocolates: x 6 ≥ 20 Non-negativity: x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ≥ 0 (Note that the non-negativity constraints are redundant due to the “boxes” constraints). 3. A farmer is raising pigs for market, and he wishes to determine the quantities of the available types of feed that should be given to each pig to meet certain nutritional requirements at a minimum cost. The number of units of each type of basic nutritional ingredients contained within a kilogram of each feed type is given in the following table along with the daily nutritional requirements and feed costs. Nutritional Ingredient Corn (Kg) Tankage (Kg) Alfalfa (Kg) Minimum Daily Requirement Carbohydrates 90 20 40 200 Protein 30 80 60 180 Vitamins 10 20 60 150 Cost (cents) 42 36 30 In addition, for weight control reasons, the farmer would like each pig to be fed no more than a total of six kilograms of feed. Finally, no more than 40% of the total diet (by weight) should consist of Alfalfa. Formulate this problem as a linear program. Decision Variables: c: kilograms of corn t: kilograms of tankage a: kilograms of alfalfa Objective Function and Constraints:

Midterm Exam Practice Solution - 3 - OpMgt 301: Principles of Operations Management Yong-Pin Zhou 0 , 0 , 0 : 0 4 . 0 4 . 0 6 . 0 or ) ( 4 . 0 : 6 : 150 60 20 10 : vitamins 180 60 80 30 : 200 40 20 90 : 30 36 42                         a t c ity nonnegativ t c a a t c a alfalfa a t c control weight a t c a t c protein a t c tes carbohydra to subject a t c z Cost Minimize 4. X-Mart is a supermarket that operates everyday from 8 a.m. to midnight. Requirement for security personnel at its Seattle store is as follows: Time Period Number of Officers Needed 8 a.m. - 12 p.m. 8 12 p.m. - 4 p.m. 15 4 p.m. - 8 p.m. 12 8 p.m. - 12 a.m. 9 Union contract requires that security officers work consecutive hours. So they can start an 8-hour shift at 8 a.m., 12 p.m., 4 p.m., or a 4-hour shift at 8 p.m. Formulate a linear program to find the minimum number of people to meet the requirement. There are four possible shifts: 8 a.m. – 4 p.m., 12 p.m. – 8 p.m., 4 p.m. – 12 a.m., (8-hour shifts) and 8 p.m. – 12 a.m. (4-hour shift). There is one decision variable for each shift. Decision Variables: x 1 : number of security officers to work 8 a.m. – 4 p.m. x 2 : number of security officers to work 12 p.m. – 8 p.m. x 3 : number of security officers to work 4 p.m. – 12 a.m. y : number of security officers to work 8 p.m. – 12 a.m. Objective Function and Constraints: 0 , , , : 9 12 15 8 Headcount Total 3 2 1 3 3 2 2 1 1 3 2 1             y x x x ity nonnegativ y x x x x x x to subject y x x x Minimize

Midterm Exam Practice Solution - 4 - OpMgt 301: Principles of Operations Management Yong-Pin Zhou 5. The personnel department of the A & M Corporation wants to know how many workers will be needed each month for the next six-month production period. The following is a monthly demand forecast for the six-month period. Month Forecasted Demand July 1,250 August 1,100 September 940 October 900 November 1,000 December 1,150 The inventory on hand at the end of June was 500 units. The company would like to have 400 units on hand at the end of December. Each unit requires 5 employee-hours to produce; there are 20 working days each month, and each employee works 8-hour days. The workforce at the end of June was 35 workers. Hiring cost is $200 per worker and firing cost is $1,000 per employee. Monthly inventory holding cost is $5 per unit and monthly backorder cost is $10 per unit. a) Determine a chase production plan and compute its associated costs. b) Determine a level production plan and compute its associated costs. When your plan calls for a fractional number of employees, round it up. This would result in a maximum production level that exceeds your planned production level. Assume for this problem that you don't have to produce to the maximum. That is, you can produce the amount determined by your aggregate plan, even if that amount is less than the maximum production level. Chase Plan: First, here is how to deal with initial inventory and ending inventory requirement: subtract the initial inventory from the first month’s demand and add ending inventory requirement to last month’s demand to form the production quantities. As a result, productions for July and December are 750 and 1550 units respectively. Since each unit requires 5 employee-hours to produce and each employee works 160 hours per month, the monthly production rate is 32 units per employee. For July, the demand of 750 units requires 24 (750/32, rounded up). The workforce requirements for the other months are calculated similarly. Note that the assumption here is that the production for each month does not have to be exactly the same as the potential maximum production. For example, in July, 24 people are required due to rounding up. Therefore the potential maximum production is 768, but we will produce only 750. Month Demand Production Inventory Workers Beginning Ending Average Needed Hired Fired July 1250 750 500 0 250 24 0 11 August 1,100 1,100 0 0 0 35 11 0 September 940 940 0 0 0 30 0 5 October 900 900 0 0 0 29 0 1

Midterm Exam Practice Solution - 5 - OpMgt 301: Principles of Operations Management Yong-Pin Zhou November 1,000 1,000 0 0 0 32 3 0 December 1150 1550 0 400 200 49 17 0 Total 500 400 450 31 17 Inventory holding cost = 450 *$5 = $2250. Hiring Cost = 31* $200 = $6,200 Firing Cost = 17 * $1,000 = $17,000 Total Cost = $25,450 Level Plan: Total (revised) demand over the six months is 6,240. Therefore, a level plan would produce 1,040 units each month. This requires a constant workforce of 33 (1,040/32 rounded up). Note that in the following table total demand does not equal to total production because of the initial inventory and ending inventory requirement. If you adjust for that, then they both come out to be 6,240. Month Demand Production Inventory Backorder Workers Beginning Ending Average Needed Hired Fired July 1,250 1040 500 290 395 0 33 0 2 Aug. 1,100 1040 290 230 260 0 33 0 0 Sep. 940 1040 230 330 280 0 33 0 0 Oct. 900 1040 330 470 400 0 33 0 0 Nov. 1,000 1040 470 510 490 0 33 0 0 Dec. 1,150 1040 510 400 455 0 33 0 0 Total 6,340 6240 2330 2230 2280 0 0 2 In this level plan, there are no backorder or hiring costs. Inventory Holding Cost = 2280 * $5 = $11,400 Firing Cost = 2 * $1,000 = $2,000 Total Cost = $13,400 6. Mr. Meadows Cookie Company makes a variety of chocolate chip cookies in the plant in Spokane, WA. Based on orders received and forecasts of buying habits, it is estimated that the demand for the next four months is 850, 510, 1260, and 980, expressed in thousands of cookies. During a 46-day period when there were 120 workers, the company produced 1.7 million cookies. Assume that the numbers of workdays over the four months are respectively 26, 24, 20, and 16. There are currently 100 workers employed, and there is no starting inventory of cookies. a) What is the minimum constant workforce required to meet demand over the next four months? b) Assume that inventory holding cost is 10 cents per cookie per month, hiring cost is $100, and firing cost is $500. Moreover, assume that we will produce to the

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