Mathematics for Management Science Notes 04 prepared by Professor Jenny Baglivo

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1 Mathematics for Management Science Notes 04 prepared by Professor Jenny Baglivo Jenny A. Baglivo All rights reserved. Application type 1: blending problems Blending problems arise when a manager must decide how to blend two or more resources to produce one or more products. Blending problems occur in the petroleum industry (e.g. crude oil to gasoline), chemical industry (e.g. chemical components to fertilizers), and the food industry (e.g. ingredients to soups). Each product (indexed using i=1, 2, etc.) is produced by blending resources (indexed using j=1, 2, etc.). We need to make decisions about how much of resource j is in product i for all possible i and j. Decision variables have double subscripts and can be visualized using a contingency table. The following table shows the table when there are i=2 products and j=3 components. Resource 1 Resource 2 Resource 3 sum Product 1 x 11 x 12 x 13 x 1+ Product 2 x 21 x 22 x 23 x 2+ sum x +1 x +2 x +3 x ++ In the table, (1) x i+ is the total production of product i. (2) x +j is the total amount of resource j used in making all products. (3) x ++ is the total production (taking all products into account). The sum variables are often called auxiliary or defined variables. They can be used in defining the model and in the spreadsheet implementation. page 1 of 24

2 Exercise 1: The Grand Strand Oil Company produces regular and premium blends for independent service stations in the southeastern United States. The Grand Strand refinery manufactures the gasolines by blending three petroleum components. The gasolines are sold at different prices, and the petroleum components have different costs. The first table gives costs for the three components and supplies in the next production cycle. The second table gives sales prices for the two blends and blending specifications. Component Cost/gallon Available. 1 $0.50 5,000 gallons 2 $ ,000 gallons 3 $ ,000 gallons Product Revenue/gallon Specification. regular $1.00 at most 30% component 1 at least 40% component 2 at most 20% component 3 premium $1.08 at least 25% component 1 at most 40% component 2 at least 30% component 3 Current commitments to distributors require Grand Strand to produce at least 10,000 gallons of regular blend in the next production cycle. The firm wants to determine how to mix or blend the three components into the two gasoline products in order to maximize profits. Summarize the problem Define the decision variables precisely page 2 of 24

3 Completely specify the LP model page 3 of 24

4 Clearly state the optimal solution. x 11 equals zero in the optimal solution, but the reduced cost is zero. Why? If Grand-Strand could obtain 1000 gallons of one of the components at current costs, which should it buy? Why? If the commitment for regular gas was only 9000 gallons, how would the profit change? Be specific. page 4 of 24

5 Exercise 1 solution and sensitivity report A B C D E Grand Strand Oil Company Components used in blending Cost per gallon Available gallons min% in regular blend 0.4 max% in regular blend min% in premium blend max% in premium blend 0.4 Types of blended gasoline R e g u l a r P r e m i u m Sales price per gallon Gallons produced min M O D E L Decision Variables Comp 1 Comp 2 Comp 3 Total: Regular blend Premium blend Total: Total Revenue: Total Cost: Maximize Profit: 9300 Subject to L H S R H S Component 1 available 5000 <= 5000 Component 2 available <= Component 3 available <= Regular needed >= 10,000 Component 1 in regular <= 0 Component 3 in regular 0 <= 0 Component 2 in regular 4000 >= 0 Component 1 in premium 1250 >= 0 Component 3 in premium 3500 >= 0 Component 2 in premium <= 0 Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$20 Regular blend Comp E+30 $C$20 Regular blend Comp $D$20 Regular blend Comp $B$21 Premium blend Comp E+30 0 $C$21 Premium blend Comp $D$21 Premium blend Comp Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $B$29 Component 1 available LHS $B$30 Component 2 available LHS $B$31 Component 3 available LHS $B$32 Regular needed LHS $B$33 Component 1 in regular LHS E $B$34 Component 3 in regular LHS $B$35 Component 2 in regular LHS E+30 $B$36 Component 1 in premium LHS E+30 $B$37 Component 3 in premium LHS E+30 $B$38 Component 2 in premium LHS E page 5 of 24

6 Formulas worksheet: A B C D E Grand Strand Oil Company Components used in blending Cost per gallon Available gallons min% in regular blend 0.4 max% in regular blend min% in premium blend max% in premium blend 0.4 Types of blended gasoline R e g u l a r Sales price per gallon Gallons produced min M O D E L P r e m i u m Decision Variables Comp 1 Comp 2 Comp 3 Total: Regular blend =SUM(B20:D20) Premium blend =SUM(B21:D21) Total: =SUM(B20:B21) =SUM(C20:C21) =SUM(D20:D21) =SUM(B20:D21) Total Revenue: =B14*E20+C14*E21 Total Cost: =SUMPRODUCT(B5:D5,B22:D22) Maximize Profit: =B24-B25 Subject to L H S R H S Component 1 available =B22 <= =B6 Component 2 available =C22 <= =C6 Component 3 available =D22 <= =D6 Regular needed =E20 >= =B15 Component 1 in regular =B20-B8*E20 <= 0 Component 3 in regular =D20-D8*E20 <= 0 Component 2 in regular =C20-C7*E20 >= 0 Component 1 in premium =B21-B9*E21 >= 0 Component 3 in premium =D21-D9*E21 >= 0 Component 2 in premium =C21-C10*E21 <= 0 page 6 of 24

7 Application type 2: financial planning problems Many financial planning problems involve planning for multiple periods of time, since the amount of money invested or spent at one point directly affects the amount available in subsequent time periods. In the problems considered here, the manager must determine the levels of initial funding, initial investments, and re-investments necessary to meet obligations at fixed periods of time over a fixed time horizon. (For example, a construction fund could be set aside now so that scheduled monthly payments for a new building can be made over the next two years.) The initial funds include the obligation for the first period and the amount to be invested to meet the future obligations. By rewriting, we get For subsequent time periods: Initial Funds Initial Investments = Obligation for period 1 Returns on Investments Re-Investments = Obligation for the given period In general, we write Cash In Cash Out = Obligation The manager would like to satisfy all obligations using the minimum level of initial funding. Notes: There will be flow constraints for each time period. There may be other decisions variables, and there may be additional constraints. page 7 of 24

8 Exercise 2: Hewlitt Corporation has established an early retirement program as part of its corporate restructuring. At the close of the voluntary sign-up period, sixty-eight employees had elected early retirement. As a result of these early retirements, the company has incurred the following obligations over the next eight years. Cash requirements (in thousands of dollars) are due at the beginning of each year: Year Requirement The corporate treasurer must determine how much money has to be set aside today to meet the eight-year financial obligations as they come due. The financial plan for the retirement program includes investments in government bonds as well as savings. The investments in government bonds are limited to three choices. Price/Unit Rate Maturity Bond 1 $ % 5 years Bond 2 $ % 6 years Bond 3 $ % 7 years The government bonds have a par value of $1000 (that is, they pay $1000 at maturity even with different purchase prices). The rates shown are based on the par value. For purposes of planning, the treasurer has assumed that any funds not invested in bonds at the beginning of this period will be placed in savings and earn interest at an annual rate of 4%. Hewlitt would like to minimize the total dollars that must be set aside now to meet the eight-year obligation. (Note: Units of bonds are purchased. The units do not have to be whole numbers.) Summarize the problem Define the decision variables precisely page 8 of 24

9 Completely specify the LP model Clearly state the optimal solution. page 9 of 24

10 Exercise 2 solution and sensitivity reports: A B C D E F G H I Hewlitt Corporation Y e a r Cash Requirement Bond 1 Bond 2 Bond Price (thous) Rate Maturity (yrs) Savings M u l t i p l e M O D E L Decision Variables: F B 1 B 2 B 3 Thousands of $: Units: S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 Thousands of $: Minimize Funds: Subject to INFLOW OUTFLOW L H S R H S Year = 430 Year = 210 Year = 222 Year = 231 Year = 240 Year = 195 Year = 225 Year = 255 Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$18 Thousands of $: F E+30 1 $E$18 Units: B $F$18 Units: B $G$18 Units: B $B$22 Thousands of $: S $C$22 Thousands of $: S $D$22 Thousands of $: S $E$22 Thousands of $: S $F$22 Thousands of $: S E $G$22 Thousands of $: S E $H$22 Thousands of $: S E $I$22 Thousands of $: S E Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $D$28 Year 1 LHS E $D$29 Year 2 LHS E $D$30 Year 3 LHS E $D$31 Year 4 LHS E $D$32 Year 5 LHS E $D$33 Year 6 LHS $D$34 Year 7 LHS $D$35 Year 8 LHS page 10 of 24

11 Exercise 2 formulas sheet: Hewlitt A B C Corporation Cash Y e a r Requirement M O D E L Thousands of $: 0 Decision Variables: F S 1 S 2 Thousands of $: 0 0 Minimize Funds: =B18 Subject to I N F L O W OUTFLOW Year 1 =B18 =SUMPRODUCT(E6:G6,E18:G18)+B22 Year 2 =SUMPRODUCT(E7:G7,E18:G18)+E11*B22 =C22 Year 3 =SUMPRODUCT(E7:G7,E18:G18)+E11*C22 =D22 Year 4 =SUMPRODUCT(E7:G7,E18:G18)+E11*D22 =E22 Year 5 =SUMPRODUCT(E7:G7,E18:G18)+E11*E22 =F22 Year 6 =(1+E7)*E18+F7*F18+G7*G18+E11*F22 =G22 Year 7 =(1+F7)*F18+G7*G18+E11*G22 =H22 Year 8 =(1+G7)*G18+E11*H22 =I22... continued on next page page 11 of 24

12 (continuation) D E F G H I Bond 1 Bond 2 Bond 3 Price (thous) Rate Maturity (yrs) Savings M u l t i p l e 1.04 B 1 B 2 B 3 Units: S 3 S 4 S 5 S 6 S 7 S L H S R H S =B28-C28 = =B5 =B29-C29 = =B6 =B30-C30 = =B7 =B31-C31 = =B8 =B32-C32 = =B9 =B33-C33 = =B10 =B34-C34 = =B11 =B35-C35 = =B12 page 12 of 24

13 Application type 3: make-or-buy problems In make-or-buy problems, the manager must determine how much of several components the company should manufacture (or make) and how much it should purchase from an outside supplier (or buy). There may be other decisions to make as well. The components are indexed using i=1, 2, etc. Convenient variables are M i equals the quantity of component i to make (i = 1, 2, etc.) B i equals the quantity of component i to buy (i = 1, 2, etc.) Exercise 3: 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 costs are summarized below, as well as the manufacturing time (in minutes) for the components. Cost per unit. To Make To Buy Manufacturing Time Component (regular time) (in minutes). Base $0.50 $ minutes Financial cartridge minutes Technician cartridge minutes Financial top minutes Technician top minutes 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 calculators. Overtime involves a premium at the additional cost of $9.00 per hour. The problem for Janders is to determine how many units of each component to manufacture and how many units of each component to purchase in order to minimize total cost. page 13 of 24

14 Summarize the problem Define the decision variables precisely Completely specify the LP model page 14 of 24

15 Clearly state the optimal solution. Exercise 3 solution sheet: A B C D E Janders Company Financial Technician Demand: Overtime Premium Regular Overtime ($/hr): Available (hrs): Cost ($) Cost($) Time (mins) To Make: To Buy: To Make: Base Fin. Cart Tech. Cart Fin. Top Tech.Top M O D E L Decision Variables: Make Buy 1 Base: F.Cart: T.Cart: F.Top: T.Top: Overtime hours: 0 Minimize Cost Subject to: L H S R H S #Bases 5000 = 5000 #Fin. Carts = 3000 #Tech. Carts = 2000 #Fin. Tops 3000 = 3000 #Tech. Tops 2000 = 2000 Overtime 0 <= 50 Mfg. time <= page 15 of 24

16 Exercise 3 sensitivity report: Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$20 1 Base: Make E+30 $C$20 1 Base: Buy E $B$21 2 F.Cart: Make $C$21 2 F.Cart: Buy $B$22 3 T.Cart: Make E+30 $C$22 3 T.Cart: Buy E $B$23 4 F.Top: Make E $C$23 4 F.Top: Buy E+30 $B$24 5 T.Top: Make E $C$24 5 T.Top: Buy E+30 $B$27 hours: Overtime E+30 4 Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $B$32 #Bases LHS $B$33 #Fin. Carts. LHS E $B$34 #Tech. Carts. LHS $B$35 #Fin. Tops LHS E $B$36 #Tech. Tops LHS E $B$37 Overtime LHS E $B$38 Mfg. time LHS Notice that: (1) The range of optimality of the coefficient for overtime is C11 5 (dollars per hour). (2) The shadow price for the manufacturing time constraint (dollars per minute) for minutes in the range 10,000 to 19,000. Converting to hours: dcost/dr7 = 5 (dollars per hour) for hours in the range to Thus, if additional hours of manufacturing time come with a premium on cost per hour, then Janders management should be willing to pay up to 5 additional dollars per hour. page 16 of 24

17 Exercise 3 formulas sheet: A B C D E Janders Company Financial Technician Demand: Overtime Premium Regular Overtime ($/hr): Available (hrs): Cost ($) Cost($) Time (mins) To Make: To Buy: To Make: Base Fin. Cart Tech. Cart Fin. Top Tech.Top M O D E L Decision Variables: Make Buy 1 Base: F.Cart: T.Cart: F.Top: T.Top: 0 0 Minimize Overtime hours: 0 Cost =SUMPRODUCT(B10:C14,B20:C24)+E6*B27 Subject to: L H S R H S #Bases =B20+C20 = =B3+C3 #Fin. Carts. =B21+C21 = =B3 #Tech. Carts. =B22+C22 = =C3 #Fin. Tops =B23+C23 = =B3 #Tech. Tops =B24+C24 = =C3 Overtime =B27 <= =C6 Mfg. time =SUMPRODUCT(D10:D14,B20:B24)-60*B27 <= =60*B6 page 17 of 24

18 Application type 4: production scheduling problems In production scheduling problems, the manager s job is to determine an efficient multi-period production and inventory schedule that allows the company to meet product demand and other requirements. There may be additional decisions to make as well. If there is more than one product, then products are indexed using i=1, 2, etc. Each period is indexed using j=1, 2, etc. Let X ij equal the amount of product i produced during period j S ij equal the amount of product i stored at the end of period j for all possible i and j. Since the current period s demand can be met from the current period s production or from inventory carried over from previous periods, there will be constraints of the form Beginning Inventory + Current Production Ending Inventory = Current Demand for each product and period in the problem. Exercise 4 Bollinger Electronics Company produces two different electronic components for a major airplane engine manufacturer. The airplane engine manufacturer notifies the Bollinger sales office each quarter of its monthly requirements for components for each of the next three months. The monthly requirements for the components may vary considerably, depending on the type of engine the airplane engine manufacturer is producing. The order for the next three-month period is shown below. Component April May June 322A B After the order is processed, a demand statement is sent to the production control department. The production control department must then develop a three-month production plan for the components. In arriving at the desired schedule, the production manager will want to identify the total production cost and the inventory holding cost. Component 322A costs $20.00 per unit to produce and component 802B costs $10.00 per unit to produce. Monthly basis inventory holding costs are 1.5% of the production cost. Machine, labor, and storage requirements for each component are given below. Machine Labor Storage Component (hrs/unit) (hrs/unit) (sqr.ft./unit) 322A B page 18 of 24

19 Capacities for the next three months are as follows: Capacities. Machine Labor Storage Month (hours) (hours) (square feet) April ,000 May ,000 June ,000 Further, assume there are 500 units of component 322A and 200 units of component 802B inventoried at the beginning of the three-month period and that Bollinger specifies a minimum inventory level of 400 units of component 322A and 200 units of component 802B at the end of the three-month period. Bollinger would like to determine a three month production schedule that minimizes total cost. Summarize the problem Define the decision variables precisely page 19 of 24

20 Completely specify the LP model page 20 of 24

21 Clearly state the optimal solution. If the ending inventory requirement was 200 of each type, how much would be saved? If an additional 20 hours of machine time in June could be obtained at a premium of $5.00 per hour, would it make sense for Bollinger to obtain it? If it does make sense, find the new cost; if not, explain why. page 21 of 24

22 Exercise 4 solution sheet: A B C D E F G Bollinger Electronics M O D E L April May June Initial Final Demand: Demand: Demand: Inventory: Inventory: 322A: B: Production Cost: Storage Cost: April May June April May June 322A: B: Per Unit: Machine Labor Storage Time (hrs) ( h r s ) (sq.ft.) 322A: B: Per month: Labor Storage April May June ( h r s ) (sq.ft.) Machine Hrs: Production Decision Variables: Storage Decision Variables: Xi1 Xi2 Xi3 Si1 Si2 Si3 322A B Minimize Cost Subject to L H S R H S Demand 1/Apr 1000 = 1000 Demand 2/Apr 1000 = 1000 Demand 1/May 3000 = 3000 Demand 2/May 500 = 500 Demand 1/Jun 5000 = 5000 Demand 2/Jun 3000 = 3000 Ending Inventory >= 400 Ending Inventory >= 200 Machine time Apr 350 <= 600 Machine time May 500 <= 500 Machine time Jun 400 <= 400 Labor hours Apr <= 300 Labor hours May 250 <= 300 Labor hours June <= 300 Storage sq.ft. Apr 7500 <= Storage sq.ft. May <= Storage sq.ft. Jun 1400 <= page 22 of 24

23 Exercise 4 sensitivity report: Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$27 322A Xi E $C$27 322A Xi E+30 $D$27 322A Xi E+30 $E$27 322A Si E $F$27 322A Si E $G$27 322A Si E $B$28 802B Xi E+30 $C$28 802B Xi E $D$28 802B Xi E $E$28 802B Si E+30 $F$28 802B Si E+30 $G$28 802B Si E Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $B$34 Demand 1/Apr LHS $B$35 Demand 2/Apr LHS $B$36 Demand 1/May LHS $B$37 Demand 2/May LHS $B$38 Demand 1/Jun LHS $B$39 Demand 2/Jun LHS $B$40 Ending Inventory 1 LHS $B$41 Ending Inventory 2 LHS $B$42 Machine time Apr LHS E $B$43 Machine time May LHS $B$44 Machine time Jun LHS $B$45 Labor hours Apr LHS E $B$46 Labor hours May LHS E $B$47 Labor hours June LHS E $B$48 Storage sq.ft. Apr LHS E $B$49 Storage sq.ft. May LHS $B$50 Storage sq.ft. Jun LHS E Notice that: The April demand shadow prices are exactly equal to the costs of producing an item of each type. Ane, the May demand shadow prices are exactly equal to the costs of producing an item in April and storing that item for one month. page 23 of 24

24 Exercise 4 formulas sheet: Bollinger M O D E L A B C D E F G Electronics April May June Initial Final Demand: Demand: Demand: Inventory: Inventory: 322A: B: Production Cost: Storage Cost: April May June April May June 322A: B: Per Unit: Machine Labor Storage Time (hrs) ( h r s ) (sq.ft.) 322A: B: Per month: Labor Storage April May June ( h r s ) (sq.ft.) Machine Hrs: Minimize Production Decision Variables: Storage Decision Variables: Xi1 Xi2 Xi3 Si1 Si2 Si3 322A B Cost =SUMPRODUCT(B11:G12,B27:G28) Subject to L H S R H S Demand 1/Apr =F5+B27-E27 = =B5 Demand 2/Apr =F6+B28-E28 = =B6 Demand 1/May =E27+C27-F27 = =C5 Demand 2/May =E28+C28-F28 = =C6 Demand 1/Jun =F27+D27-G27 = =D5 Demand 2/Jun =F28+D28-G28 = =D6 Ending Inventory 1 =G27 >= =G5 Ending Inventory 2 =G28 >= =G6 Machine time Apr =SUMPRODUCT(B18:B19,B27:B28) <= =B22 Machine time May =SUMPRODUCT(B18:B19,C27:C28) <= =C22 Machine time Jun =SUMPRODUCT(B18:B19,D27:D28) <= =D22 Labor hours Apr =SUMPRODUCT(C18:C19,B27:B28) <= =F22 Labor hours May =SUMPRODUCT(C18:C19,C27:C28) <= =F22 Labor hours June =SUMPRODUCT(C18:C19,D27:D28) <= =F22 Storage sq.ft. Apr =SUMPRODUCT(D18:D19,E27:E28) <= =G22 Storage sq.ft. May =SUMPRODUCT(D18:D19,F27:F28) <= =G22 Storage sq.ft. Jun =SUMPRODUCT(D18:D19,G27:G28) <= =G22 page 24 of 24