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Give step-by-step solution with explanation and final answer: A production scheduler must develop an aggregate plan for the next two quarters of next year. The highly automated plant produces graphics terminals for the computer products market. The company estimates that 7 0 0 terminals will need to be shipped to customers in the first quarter and 3 2 0 0 in the second quarter. It takes an average of five hours of labour to produce each terminal and only 9 0 0 0 hours of straight labour are available. Overtime can be used, but the company has a policy of limiting the amount of overtime in each quarter to 1 0 per cent of the straight time labour available. Labour costs R 1 2 0 per hour at the straight - line rate and R 1 8 0 per hour at the overtime rate. If a terminal is produced in one quarter and shipped in the next quarter, a carrying cost of R 5 0 0 is incurred. The objective is to determine how many terminals should be produced on straight - line and overtime in each of first and second quarter to minimise straight - time labour, overtime labour and carrying costs. ( a ) Define the decision variables that will be needed to formulate a linear programming ( LP ) model for this problem. ( 1 0 marks ) ( b ) Formulate the Linear Programming ( LP ) model that represents this problem. ( 3 0 marks ) ( c ) Solve the LP model using SOLVER. ( 2 0 marks ) ( d ) Interpret your SOLVER solution within the context of the original business problem. ( 1 0 marks ) Use ONLY the initial printout of the optimal solution ( the ANSWER and SENSITIVITY reports ) to answer the following questions. ( This means that you may not change the relevant parameters in the model and do re - runs. ) Explain how you arrived at your answers. e ) Determine the optimal solution and total costs if the amount of straight labour time available in quarter 2 is ( i ) 8 7 5 0 hours. ( 3 marks ) ( ii ) 8 5 0 0 hours. ( 2 marks ) f ) What would the total costs be if the amount of straight labour costs in Quarter 1 is R 1 3 0 ? ( 3 marks ) g ) Under what circumstances will it be possible to use all the available straight - labour time for the second quarter. give

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Give step-by-step solution with explanation and final answer: A production scheduler must develop an aggregate plan for the next two quarters of next year. The highly automated plant produces graphics terminals for the computer products market. The company estimates that 7 0 0 terminals will need to be shipped to customers in the first quarter and 3 2 0 0 in the second quarter. It takes an average of five hours of labour to produce each terminal and only 9 0 0 0 hours of straight labour are available. Overtime can be used, but the company has a policy of limiting the amount of overtime in each quarter to 1 0 per cent of the straight time labour available. Labour costs R 1 2 0 per hour at the straight - line rate and R 1 8 0 per hour at the overtime rate. If a terminal is produced in one quarter and shipped in the next quarter, a carrying cost of R 5 0 0 is incurred. The objective is to determine how many terminals should be produced on straight - line and overtime in each of first and second quarter to minimise straight - time labour, overtime labour and carrying costs. ( a ) Define the decision variables that will be needed to formulate a linear programming ( LP ) model for this problem. ( 1 0 marks ) ( b ) Formulate the Linear Programming ( LP ) model that represents this problem. ( 3 0 marks ) ( c ) Solve the LP model using SOLVER. ( 2 0 marks ) ( d ) Interpret your SOLVER solution within the context of the original business problem. ( 1 0 marks ) Use ONLY the initial printout of the optimal solution ( the ANSWER and SENSITIVITY reports ) to answer the following questions. ( This means that you may not change the relevant parameters in the model and do re - runs. ) Explain how you arrived at your answers. e ) Determine the optimal solution and total costs if the amount of straight labour time available in quarter 2 is ( i ) 8 7 5 0 hours. ( 3 marks ) ( ii ) 8 5 0 0 hours. ( 2 marks ) f ) What would the total costs be if the amount of straight labour costs in Quarter 1 is R 1 3 0 ? ( 3 marks ) g ) Under what circumstances will it be possible to use all the available straight - labour time for the second quarter. give

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Asked by: Priyanshu Yadav
Created at: 2025-09-28 12:27:35

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# (a Decision Variables : - \( x_{1} \): Number of terminals produced Quarter 1 straight time - \( x_{1o} \): Number of terminals in Quarter with overtime \( x_{s} \ Number of terminals in Quarter 2 straight time \( x_{2o} \): Number of terminals produced in Quarter 2 with overtime - \( I_1 \): Inventory at the end of Quarter 1 (terminals produced in Q1 to be shipped in Q2) --- # (b) Linear Programming Model ## **Parameters** - Demand Q1: \( D_1 = 700 \) - Demand Q2: \( D_2 = 320 \) - Straight-time labor available/quarter: \( L = 900 \) hours - Overtime limit/quarter: \( .10 \times 900 = 900 \) hours - Labor hours per terminal: \( h = 5 \) - Straight-time labor cost: \( c_s = \text{R}120 \)/hour - Overtime labor cost: \( c_o = \text{R}180 \)/hour - Carrying cost: \( c_{inv} = \text{R}500 \)/terminal/quarter --- ## **Objective Function** Minimize total cost: \[ \text{Minimize:} \quad Z = 120 \times 5 (x_{1s} + x_{2s}) + 180 \times 5 (x_{1o} + x_{2o}) + 500 \times I_1 \] or simplified: \[ Z = 600(x_{1s} + x_{2s}) + 900(x_{1o} + x_{2o}) + 500I_1 \] --- ## **Constraints** ### **1. Demand satisfaction** - Q1: \( x_{1s} + x_{1o} = D_1 + I_1 \) (what's produced in Q1 = what is shipped plus what is carried to Q2) - Q2: \( x_{2s} + x_{2o} + I_1 \geq D_2 \) ### **2. Labor capacity** - Q1 straight: \( 5x_{1s} \leq 900 \) - Q1 overtime: \( 5x_{1o} \leq 900 \) - Q2 straight: \( 5x_{2s} \leq 900 \) - Q2 overtime: \( 5x_{2o} \leq 900 \) ### **3. Non-negativity** - \( x_{1s}, x_{1o}, x_{2s}, x_{2o}, I_1 \geq \) --- # (c) LP Solution using Solver ## **Step 1: Set up equations** - \( x_{1s} + x_{1o} = 700 + I_1 \) - \( x_{2s} + x_{2o} + I_1 \geq 320 \) - \( 5x_{1s} \leq 900 \implies x_{1s} \leq 180 \) - \( 5x_{1o} \leq 900 \implies x_{1o} \leq 180 \) - \( 5x_{2s} \leq 900 \implies x_{2s} \leq 180 \) - \( 5x_{2o} \leq 900 \implies x_{2o} \leq 180 \) - \( x_{1s}, x_{1o}, x_{2s}, x_{2o}, I_1 \geq \) ## **Step 2: Solve** ### **First, minimize overtime and inventory due to cost.** #### **Maximum straight time production per quarter:** 180 units #### **Quarter 1:** - If we produce all Q1 demand with straight time: - \( x_{1s} = 700 \) - \( x_{1o} = \) - \( I_1 = \) (no inventory carried) #### **Quarter 2:** - Demand: 320 - Maximum straight time: 180 - Need to produce the rest with overtime or carry inventory from Q1 If \( x_{2s} = 180 \), remaining to be produced: \[ 320 - 180 = 140 \] Maximum overtime: 180 units (\( x_{2o} \leq 180 \)), which is only 180 units. So, in Q2: - \( x_{2s} = 180 \) - \( x_{2o} = 180 \) - \( I_1 = 320 - 180 - 180 = 122 \) (must be carried from Q1) But Q1 can only produce up to 180 units in straight time and 180 in overtime, but Q1 demand is only 700. So to carry 122 units to Q2, Q1 must produce: \[ x_{1s} + x_{1o} = 700 + 122 = 192 \] But \( x_{1s} \leq 180, x_{1o} \leq 180 \), and \( x_{1s} + x_{1o} = 192 \). Let's set \( x_{1s} = 180, x_{1o} = 120 \) (total 192). Both within limits. Therefore: - Q1: \( x_{1s} = 180, x_{1o} = 120, I_1 = 122 \) - Q2: \( x_{2s} = 180, x_{2o} = 180 \) ## **Step 3: Objective Function Value** \[ Z = 600(x_{1s} + x_{2s}) + 900(x_{1o} + x_{2o}) + 500I_1 \] \[ = 600(180 + 180) + 900(120 + 180) + 500 \times 122 \] \[ = 600 \times 360 + 900 \times 300 + 610,000 \] \[ = 2,160,000 + 270,000 + 610,000 \] \[ = 3,040,000 \] --- # (d) Interpretation **Optimal production plan:** - **Quarter 1**: Produce 180 terminals with straight time, 120 with overtime. Ship 700, carry 122 units in inventory. - **Quarter 2**: Produce 180 with straight time, 180 with overtime. Meet remaining demand of 320 using Q1 inventory. - **Total cost**: R3,040,000, which includes straight time, overtime, and inventory holding costs. --- # (e) Sensitivity Analysis ## (i) If Q2 straight time labor = 875 hours (\( x_{2s} \leq 175 \)): - Now \( x_{2s} \leq 175 \) - Q2 max production: \( x_{2s} + x_{2o} \leq 175 + 180 = 193 \) - Inventory needed: \( 320 - 193 = 127 \) - Q1 must produce: \( 700 + 127 = 197 \) - \( x_{1s} = 180, x_{1o} = 170 \) (within limits) Objective: \[ Z = 600(180 + 175) + 900(170 + 180) + 500 \times 127 \] \[ = 600(355) + 900(350) + 635,000 \] \[ = 2,130,000 + 315,000 + 635,000 \] \[ = 3,080,000 \] --- ## (ii) If Q2 straight time labor = 850 hours (\( x_{2s} \leq 170 \)): - \( x_{2s} = 170 \), \( x_{2o}=180 \) - Q2 production: 188; inventory needed: 320 - 188 = 132 - Q1: \( x_{1s}=180, x_{1o}=120 \) (to total 192, enough for 700+122) - Not enough: need 700+132=202; \( x_{1s} + x_{1o} = 202 \) - Only possible if \( x_{1s}=180, x_{1o}=180 \) - Yes, total 198, but need 202, so not enough. So, problem is infeasible. But let's check with \( x_{1s}=180, x_{1o}=180, I_1=132 \), total production in Q1: 198, but 700 must be shipped, so can only carry 128. Thus, the problem becomes infeasible unless overtime or inventory constraints are relaxed. --- # (f) If Quarter 1 straight labor cost is R130/hr: - \( c_s = 130 \) - Objective: \( Z = 650(x_{1s} + x_{2s}) + 900(x_{1o} + x_{2o}) + 500I_1 \) - Using original optimal values: \[ Z = 650(180 + 180) + 900(120 + 180) + 500 \times 122 \] \[ = 650 \times 360 + 900 \times 300 + 610,000 \] \[ = 2,340,000 + 270,000 + 610,000 \] \[ = 3,220,000 \] --- # (g) When Will All Q2 Straight-Time Be Used? All straight-time labor for Q2 will be used if: - Demand in Q2 exceeds straight-time production capacity (\( D_2 > x_{2s}^{max} \)) - There is insufficient inventory carried from Q1 to cover the shortfall - Overtime and inventory options are more expensive and thus not preferred **In summary:** All available straight-labor time in Q2 will be used whenever Q2 demand minus Q1 inventory carried forward is at least the maximum Q2 straight-time production (\( D_2 - I_1 \geq x_{2s}^{max} \)). This is the case in the scenarios above, so all Q2 straight-time labor is used in the optimal solution. --- ## **Summary Table** | Quarter | Straight Time Prod | Overtime Prod | Inventory | Total Cost (R) | |---------|-------------------|---------------|-----------|----------------| | 1 | 180 | 120 | 122 | | | 2 | 180 | 180 | | 3,040,000 | --- **Note:** For (c) and (d), you should run SOLVER in Excel using the LP model above to confirm (the calculated solution matches the expected optimal plan given the constraints and costs). For (e)-(f), use the sensitivity report to determine feasibility and shadow prices.

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