Multi-period production scheduling The Tots Toys Company is trying to schedule production of two very popular toys for the next three months: a rocking horse and a scooter. Information about both toys is given below. Toy Rocking Horse Scooter Summer Schedule June July August Begin. Invty. June 1 25 55 Required Plastic 5 4 Plastic Available 3500 5000 4800 Time Available 2100 3000 2500 Required Time 2 3 Production Cost 12 14 Monthly Demand Horse 220 350 600 Production Cost 1 1.2 Monthly Demand Scooter 450 700 520 Develop a model that would tell the company how many of each toy to produce during each month. You are to minimize total cost. Inventory cost will be levied on any items in inventory on June 30, July 31, or August 31 after demand for the month has been satisfied. Your model should make use of the relationship Beginning Inventory + Production - Demand = Ending Inventory for each month. The company wants to end the summer with 150 rocking horses and 60 scooters as beginning inventory for Sept. 1. Don't forget to define your decision variables. Let Pij = number of toy i to produce in month j Sij = surplus (inventory) of toy i at end of month j Min s.t. 12P11 + 12P12 + 12P13 + 14P21 + 14P22 + 14P23 + 1S11 + 1S12 + 1S13 + 1.2S21 + 1.2S22 + 1.2S23 P11 - S11 = 195 S11 + P12 - S12 = 350 S12 + P13 - S13 = 600 S13 > 150 P21 - S21 = 395 S21 + P22 - S22 = 700 S22 + P23 - S23 = 520 S23 > 60 5P11 + 4P21 < 3500 5P12 + 4P22 < 5000 5P13 + 4P23 < 4800 2P11 + 3P21 < 2100 2P12 + 3P22 < 3000 2P13 + 3P23 < 2500 Pij, Sij > 0 A Make-or-Buy Decision We illustrate the use of a linear programming model to determine how much of each of several component parts a company should manufacture and how much it should purchase from an outside supplier. Such a decision is referred to as a make-or-buy decision. The Janders Company markets various business and engineering products. Currently, Janders is preparing to introduce two new calculators: one for the business market called the Financial Manager and one for the engineering market called the Technician. Each calculator has three components: a base, an electronic cartridge, and a face plate or top. The same base is used for both calculators, but the cartridges and tops are different. All components can be manufactured by the company or purchased from outside suppliers. The manufacturing costs and purchase prices for the components and manufacturing times (in minutes) for the components are summarized in Table 1. Janders' forecasters indicate that 3000 Financial Manager calculators and 2000-Technician calculators will be needed. However, manufacturing capacity is limited. The company has 200 hours of regular manufacturing time and 50 hours of overtime that can be scheduled for the calculators. Overtime involves a premium at the additional cost of $9 per hour. Table 1 MANUFACTURING COSTS AND PURCHASE PRICES FOR JANDERS' CALCULATOR COMPONENTS Cost Per Unit Component Manufacture (Regular Purchase Manufacturing Time Time) (minutes) Base $0.50 $0.60 1.0 Financial cartridge 3.75 4.00 3.0 Technician cartridge 3.30 3.90 2.5 Financial top 0.60 0.65 1.0 Technician top 0.75 0.78 1.5 Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order has been for 1200 door lock units for an apartment complex developer. The sales and production departments must work together to determine delivery schedules. Each lock unit consists of three components: the knob and face plate, the actual lock itself, and a set of two keys. Although the processes used in the manufacture of the three components vary, there are three areas where the production manager is concerned about the availability of resources. These three areas, their usage by the three components, and their availability are detailed in the table. Resource Knob and Plate Lock Key (each) Available Brass Alloy Machining Finishing 12 18 15 5 20 5 1 10 1 15000 units 36000 minutes 12000 minutes A quick look at the amounts available confirms that Larkin does not have the resources to fill this contract. A subcontractor, who can make an unlimited number of each of the three components, quotes the prices below. Component Knob and Plate Lock Keys (set of 2) Subcontractor Cost Larkin Cost 10.00 9.00 1.00 6.00 4.00 .50 Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost. 6. Let PM = the number of knob and plate units to make PB = the number of knob and plate units to buy LM = the number of lock units to make LB = the number of lock units to buy KM = the number of key sets to make KB = the number of key sets to buy Min 6PM + 10PB + 4LM + 9LB + .5KM + 1KB s.t. 12PM + 5LM + 2KM < 18PM + 20LM + 20KM < 15PM + 5LM + 2KM < PM + PB > 1200 LM + LB > 1200 KM + KB > 1200 PM, PB, LM, LB, KM, KB 15000 36000 12000 > 0 Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles. Current schedules yield the following information. Component Front Seat/Frame Rear wheel (each) Available Plastic 3 4 .5 50000 Requirements Time 10 6 2 160000 Space 2 2 .1 30000 Cost to Manufacture 8 6 1 Cost to Purchase 12 9 3 The company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles so has gathered purchase information for each component. Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased in order to provide 12000 fully completed tricycles at the minimum cost. Let FM = number of fronts made SM = number of seats made WM = number of wheels made FP = number of fronts purchased SP = number of seats purchased WP = number of wheels purchased Min s.t. 8FM + 6SM + 1WM + 12FP + 9SP + 3WP 3FM + 4SM + .5WM < 50000 10FM + 6SM + 2WM < 160000 2FM + 2SM + .1WM < 30000 FM + FP > 12000 SM + SP > 12000 WM + WP > 24000 FM, SM, WM, FP, SP, WP > 0 The Electro-Poly Corporation is the world’s leading manufacturer of slip rings. A slip ring is an electrical coupling device that allows current to pass through a spinning or rotating connection—such as a gun turret on a ship, aircraft, or tank. The company recently received a $750,000 order for various quantities of three types of slip rings. Each slip ring requires a certain amount of time to wire and harness. The following table summarizes the requirements for the three models of slip rings. Model 1 Number Ordered 3,000 Hours of Wiring Required per Unit 2 Hours of Harnessing Required per Unit 1 Model 2 2,000 1.5 2 Model 3 900 3 1 Unfortunately, Electro-Poly does not have enough wiring and harnessing capacity to fill the order by its due date. The company has only 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity available to devote to this order. However, the company can subcontract any portion of this order to one of its competitors. The unit costs of producing each model in-house and buying the finished products from a competitor are summarized below. Model 1 Cost to Make Cost to Buy $50 $61 Model 2 Model 3 $83 $97 $130 $145 Electro-Poly wants to determine the number of slip rings to make and the number to buy to fill the customer order at the least possible cost. x1 = Number of model 1 slip rings to make x2 = Number of model 2 slip rings to make x3 = Number of model 3 slip rings to make y1 = Number of model 1 slip rings to buy y2 = Number of model 2 slip rings to buy y3 = Number of model 3 slip rings to buy The Objective Function: Minimize the total cost of filling the order. MIN: 50x1 + 83x2 + 130x3 + 61y1 + 97y2 + 145y3 Demand Constraints x1 + y1 = 3,000 } model 1 x2 + y2 = 2,000 } model 2 x3 + y3 = 900 } model 3 Resource Constraints 2x1 + 1.5x2 + 3x3 <= 10,000 } wiring 1x1 + 2.0x2 + 1x3 <= 5,000 } harnessing Nonnegativity Conditions x1, x2, x3, y1, y2, y3 >= 0 Assignment: 1. PM Computer Services produces personal computers from component parts it buys on the open market. The company can produce a maximum of 300 personal computers per month. PM wants to determine its production schedule for the first 6 months of the new year. The cost to produce a personal computer in January will be $1,200. However, PM knows the cost of component parts will decline each month so that the overall cost to produce a PC will be 5% less each month from the base cost. The cost of holding a computer in inventory is $15 per unit per month. Following is the demand for the company's computers each month: Month Demand Month Demand January 180 April 210 February 260 May 400 March 340 June 320 Determine a production schedule for PM that will minimize total cost. 2. In Previous Problem, suppose that the demand for personal computers increases each month, as follows: Month Demand Month Demand January 410 April 620 February 320 May 430 March 500 June 380 In addition to the regular production capacity of 300 units per month, PM Computer Services can also produce an additional 200 computers per month by using overtime. Overtime production adds 20% to the cost of a personal computer. Determine a production schedule for PM that will minimize total cost. 3. The J. Mehta Company’s production manager is planning for a series of 1-month production periods for stainless steel sinks. The demand for the next 4 months is as follows: The Mehta firm can normally produce 100 stainless steel sinks in a month. This is done during regular production hours at a cost of $100 per sink. If demand in any 1 month cannot be satisfied by regular production, the production manager has three other choices: (1) He can produce up to 50 more sinks per month in overtime but at a cost of $130 per sink; (2) he can purchase a limited number of sinks from a friendly competitor for resale (the maximum number of outside purchases over the 4-month period is 450 sinks, at a cost of $150 each); or (3) he can fill the demand from his on-hand inventory. The inventory carrying cost is $10 per sink per month. Back orders are not permitted. Inventory on hand at the beginning of month 1 is 40 sinks. Formulate the J. Mehta production problem as a linear program. 4. 5. ABC Co. produces two products with contribution to profit per unit of Tk. 10 and Tk.9 respectively. Total labor requirements per unit produced and total hours of labor available from personnel to each of four departments are given below: Dept. 1 2 3 4 Pdt. 1 0.65 0.45 1.00 0.15 Pdt. 2 0.95 0.85 0.70 0.30 Total hours available 6500 6000 7000 1400 Suppose, the company has a cross training program that enables some employees to be transferred between departments. By taking advantages of the cross training skills, a limited number of employees and labor hours may be transferred from one department to another. From Dept. Cross training transfers permitted to Dept. Maximum Hours Transferable 1 2 3 4 1 -- Yes Yes -- 400 2 -- -- Yes Yes 800 3 -- -- -- Yes 100 4 Yes Yes -- -- 200 How to assign workforce to maximize profit?
