Vision is a large company that produces videocapturing devices for military applications such as missiles, longrange cameras, and aerial drones. Four different types of cameras (differing mainly by lens type) are made in the three plants in the system. Each plant can produce any of the four camera types, although each plant has its own individual constraints and unit costs. These constraints cover labor and ma chining restrictions, and the specific values are given in Tables 8–10. Note that even though the products are identical in the three plants, different production processes are used and thus the products use different amounts of resources in different plants. The corpo ration controls the material that goes into the lenses; the material requirements for each product are given in the last column of Tables 8–10. A total of 3,500
Production Planning and Shipping CASE MR.DANG VU TUNG SIE - HUST GROUP INTRODUCTION MODEL FORMULATION SOLUTION SENSITIVITY ANALYSIS INTRODUCTION MODEL FORMULATION SOLUTION SENSITIVITY ANALYSIS Three Plants: Asland (Plant 1) Huntington (Plant 2) Johnson City (Plant 3) For each plant: - Labor hours available - Machine hours available - Production cost - Total available material Three Customers: RAYco HONco MMco For each customer: - Product sales price - Shipping cost - Maximum product sales Inspection Capacity (Shipping from Plant and to RAYco and HONco) Using Liner Programming to create a plan for production and shipping The potential issues: - Get more material - Get more inspection capacity - Add extra machine hours - Handle RAYco’s demand increases Four Types of Products: Small Medium Large Precision Assumption: Produce less than or equal to customer’s demand INTRODUCTION MODEL FORMULATION SOLUTION SENSITIVITY ANALYSIS DECISION VARIABLES DATA Data + Liner program = Model OBJECTIVE FUNCTION CONSTRAINTS DATA DECISION VAR x_1 = small from Ashland to Rayco x_19 = large from Huntington to Rayco x_2 = small from Ashland to Honco x_20 = large from Huntington to Honco x_3 = small from Ashland to MMco x_21 = large from Huntington to MMco x_4 = medium from Ashland to Rayco x_22 = precision from Huntington to Rayco x_5 = medium from Ashland to Honco x_23 = precision from Huntington to Honco x_6 = medium from Ashland to MMco x_24 = precision from Huntington to MMco x_7 = large from Ashland to Rayco x_25 = small from Johnson City to Rayco x_8 = large from Ashland to Honco x_26 = small from Johnson City to Honco x_9 = large from Ashland to MMco x_27 = small from Johnson City to MMco x_10 = precision from Ashland to Rayco x_28 = medium from Johnson City to Rayco x_11 = precision from Ashland to Honco x_29 = medium from Johnson City to Honco x_12 = precision from Ashland to Mmco x_30 = medium from Johnson City to MMco x_13 = small from Huntington to Rayco x_31 = large from Johnson City to Rayco x_14 = small from Huntington to Honco x_32 = large from Johnson City to Honco x_15 = small from Huntington to MMco x_33 = large from Johnson City to MMco x_16 = medium from Huntington to Rayco x_34 = precision from Johnson City to Rayco x_17 = medium from Huntington to Honco x_35 = precision from Johnson City to Honco x_18 = medium from Huntington to MMco x_36 = precision from Johnson City to MMco OBJECTIVE FUNCTION Objective: Maximize Total Profit Max Z = Profit Per Unit (Revenue − Cost) * xj for j = (1,2,3, ,36), Where Revenue = Sales price/Unit, and Cost = Shipping cost/Unit + Production cost/Unit Therefore our objective function is z = 2x_1 + 0.4x_2 + 0.9x_3 + x_4 + 0.4x_5 − 0.1x_6 +3x_7 + 2.4x_8 + 3.9x_9 + 2x_10 − 1.6x_11 − 0.1x_12 + 2.8x_13 + 1.5x_14 + 2x_15 − 0.2x_16 − 0.5x_17 − x_18 + 0.8x_19 + 0.5x_20 + 2x_21 + 3.8x_22 + 0.5x_23 + 2x_24 + 1.6x_25 + 0.5x_26 + 0.7x_27 + 1.6x_28 + 1.5x_29 + 0.7x_30 + 1.6x_31 + 1.5x_32 + 2.7x_33 + 4.6x_34 + 1.5x_35 + 2.7x_36 CONSTRAINTS CONSTRAINTS + Resources Constraints: + Sales and shipping constraints Maximum Small Product Sales to RAYco : 17(x_1 + x_13 + x_25) ≤ 200 Maximum Medium Product Sales to RAYco : 18(x_4 + x_16 + x_28) ≤ 300 Maximum Large Product Sales to RAYco : 22(x_7 + x_19 + x_31) ≤ 500 Maximum Precision Product Sales to RAYco : 29(x_10 + x_22 + x_34)≤ 200 Maximum Small Product Sales to HONco : 16(x_2 + x_14 + x_26) ≤ 400 Maximum Medium Product Sales to HONco : 18(x_5 + x_17 + x_29) ≤ 300 Maximum Large Product Sales to HONco : 22(x_8 + x_20 + x_32) ≤ 200 Maximum Precision Product Sales to HONco : 26(x_11 + x_23 + x_35) ≤ 400 Maximum Small Product Sales to MMco : 16(x_3 + x_15 + x_27) ≤ 200 Maximum Medium Product Sales to MMco : 17(x_6 + x_18 + x_30) ≤ 400 Labor Hours for Ashland Plant : 3x_1 +3x_2 +3x_3 +3x_4 +3x_5 +3x_6 +4x_7 +4x_8 +4x_9 + 4x_10 + 4x_11 + 4x_12 ≤ 6000 Machine Hours for Ashland Plant : 8x_1 + 8x_2 + 8x_3 + 8.5x_4 + 8.5x_5 + 8.5x_6 + 9x_7 + 9x_8 +9x_9 +9x_10 +9x_11 +9x_12 ≤ 10000 Labor Hours for Huntington Plant : 3.5x_13 + 3.5x_14 + 3.5x_15 + 3.5x_16 + 3.5x_17 + 3.5x_18 + 4.5x_19 + 4.5x_20 + 4.5x_21 + 4.5x_22 + 4.5x_23 + 4.5x_24 ≤ 5000 Machine Hours for Huntington Plant : 7_x13 + 7x_14 + 7x15 + 7x1_6 + 7x_17 + 7x_18 + 8x_19 + 8x_20 + 8x_21 + 9x_22 + 9x_23 + 9x_24 ≤ 12500 Labor Hours for Johnson City Plant : 3x_25 + 3x_26 + 3x_27 + 3.5x_28 + 3.5x_29 + 3.5x_30 + 4x_31 + 4x_32 + 4x_33 + 4.5x_34 + 4.5x_35 + 4.5x_36 ≤ 3000 Machine Hours for Johnson City Plant : 7.5x_25 + 7.5x_26 + 7.5x_27 + 7.5x_28 + 7.5x_29 + 7.5x_30 + 8.5x_31 + 8.5x_32 + 8.5x_33 + 8.5x_34 + 8.5x_35 + 8.5x_36 ≤ 6000 Total Materials Used by Each Plant : 1x_1 + 1x_2 + 1x_3 + 1.1x_4 + 1.1x_5 + 1.1x_6 + 1.2x_7 + 1.2x_8 + 1.2x_9 + 1.3x_10 + 1.3x_11 + 1.3x_12 + 1.1x_13 + 1.1x_14 + 1.1x_15 + 1x_16 + 1x_17 + 1x_18 + 1.1x_19 + 1.1x_20 + 1.1x_21 + 1.4x_22 + Maximum Precision Product Sales to MMco : 27(x_12 + x_24 + x_36) ≤ 300 1.423x_23 + 1.4x_24 + 1.1x_25 + 1.1x_26 + 1.1x_27 + 1.1x_28 + 1.1x_29 + 1.1x_30 Inspection Capacity: x_1 +x_2 +x_4 +x_5 +x_7 +x_8 +x_10 +x_11 +x_13 +x_14 + 1.3x_31 + 1.3x_32 + 1.3x_33 + 1.3x_34 + 1.3x_35 + 1.3X_36 ≤ 3500 +x_16 + x_17 +x_19 +x_20 +x_22 +x_23 ≤1500 Maximum Large Product Sales to MMco : 23(x_9 + x_21 + x_33) ≤ 300 INTRODUCTION MODEL FORMULATION SOLUTION SENSITIVITY ANALYSIS OPTIMAIL SOLUTION X1 = 11.76 (rounded up 12) X4 = 16.67 (rounded up 17) X7 = 22.73 (rounded up 23) X10 = 6.897 (rounded up 7) X15 = 12.5 X21 = 13.04 (rounded up 13) X24 =11.11 (rounded up 11) Others equal OPTIMAL VALUE The maximum profit is Z = 195,48 Number of unit sales is 94.71 Result: Would not meet small, medium, large, precision demand for HONco and medium for MMco Suggestion - Production & Shipping Assuming that we round up the products to integer values, we get the following results: 12 small products from Ashland to RAYco 17 medium products from Ashland to RAYco 23 large products from Huntington to RAYco precision products from Ashland to RAYco 12.5 small products from Huntington to MMco 13 large products from Huntington to MMco 11 precision products from Huntington to MMco - Cost & Revenue We have calculated the total cost (=production costs + shipping costs) and the revenue that Vision company will receive according to each plant For the Ashland plant, the total cost is $1095 and the total revenue is $1219 For the Huntington plant, the total cost is $722.45 and the total revenue is $796 For the Johnson City plant, the total cost is $0 and the total revenue is $0 INTRODUCTION MODEL FORMUALTION SOLUTION SENSITIVITY ANALYSIS Material "If you could get more material, how much would you like? What would you be willing to pay for it?” No, it is not necessary to get more material because the shadow price of the toal material constraint is This is also because we have a lack of 3500 − 11.764706 = 3488.235294 units for the total material constraint Inspection Capacity "If you could get more inspection capacity, how much would you like? How would you use it? What would you be willing to pay for it?” No, it is not necessary to get more inspection capacity because the shadow price of the inspection constraint is This is also because we have a lack of 1500 − 58.055197 = 1441.944803 units for the inspection capacity constraint Machine Hours "At what plant(s) would you like to add extra machine hours? How much would you be willing to pay per hour? How many extra hours would you like?” No, it is not necessary to add extra machine hours at any plants because the shadow price of the machine hours for each plant constraint is This is also because we have a lack of 10000 − 360.73207 = 9639.26793 units for the machine hours for plant constraint and 12500 − 291.84783 = 12208.15217 units for the machine hours for plant RAYco's Demand +50% "Marketing is trying to get RAYco to consider a 50% increase in its demand Can we handle this with the current system or we need more resources? How much more money can we make if we take on the additional demand?” Increase our profit to $257.48, which is an increase of $62 by selling 124.21 the number of units (including: small, medium, large and precision products) THANK FOR LISTENING ... 3.5x_13 + 3.5x_ 14 + 3.5x_15 + 3.5x_16 + 3.5x_17 + 3.5x_18 + 4. 5x_19 + 4. 5x_20 + 4. 5x_21 + 4. 5x_22 + 4. 5x_23 + 4. 5x_ 24 ≤ 5000 Machine Hours for Huntington Plant : 7_x13 + 7x_ 14 + 7x15 + 7x1_6... 8x_21 + 9x_22 + 9x_23 + 9x_ 24 ≤ 12500 Labor Hours for Johnson City Plant : 3x_25 + 3x_26 + 3x_27 + 3.5x_28 + 3.5x_29 + 3.5x_30 + 4x_31 + 4x_32 + 4x_33 + 4. 5x_ 34 + 4. 5x_35 + 4. 5x_36 ≤ 3000 Machine... Plant : 3x_1 +3x_2 +3x_3 +3x _4 +3x_5 +3x_6 +4x_7 +4x_8 +4x_9 + 4x_10 + 4x_11 + 4x_12 ≤ 6000 Machine Hours for Ashland Plant : 8x_1 + 8x_2 + 8x_3 + 8.5x _4 + 8.5x_5 + 8.5x_6 + 9x_7 + 9x_8 +9x_9 +9x_10